aa575f1
================================================================================
INTERIOR REGULARITY OF PLANAR MUMFORD--SHAH MINIMIZERS
a complete proof, written to be read straight through
================================================================================
0. HOW TO READ THIS
A soap film on a wire frame minimizes area. Allow the film to tear, and charge the
tear by its length; the planar Mumford--Shah problem then asks where tears may run,
with area replaced by the Dirichlet energy of a function and tear by the set across
which the function jumps. The conjecture, proved here, is that a minimizing tear is
never ragged: away from the boundary it consists of smooth arcs that continue, end
freely, or meet in threes at 120 degrees.
The proof is long only because the tear set is completely arbitrary. Section 3
collects five facts imported from the literature, and everything afterwards is
proved from those five facts and nothing else. Sections 4 to 9 prove the one new
statement, that a minimizing tear cannot contain an isolated compact piece;
sections 10 to 12 classify the blow-ups, transfer the classification back, and read
off the integrability of the gradient. Each section opens with a plain statement of
what it is for and why the previous sections are not enough.
Throughout, "crack" and "tear" both mean the closed set K on which the function may
jump, and "length" means one-dimensional Hausdorff measure H^1.
================================================================================
1. THE PROBLEM
================================================================================
Let Omega be a bounded Lipschitz domain in the plane R^2. Let g be a bounded
measurable real function on Omega; think of g as the observed image and u as the
reconstruction we are looking for.
1.1 DEFINITION (admissible pair).
A pair (u,K) is admissible if
(a) K is a relatively closed subset of Omega, and H^1(K) < infinity;
(b) u belongs to L^2(Omega);
(c) the restriction of u to Omega minus K belongs to W^{1,2}(Omega minus K),
that is, u has a square integrable weak gradient off K.
No smoothness of K is assumed. Values of u on K itself are irrelevant, because
K has zero area and is only ever measured by its length.
1.2 DEFINITION (energy).
For an open set V contained in Omega put
F_g(u,K;V) = INTEGRAL over V minus K of |grad u|^2 dx
+ H^1(K intersect V)
+ INTEGRAL over V of |u-g|^2 dx.
The three terms are, in order, the cost of bending, the cost of tearing, and the
cost of disagreeing with the data.
1.3 DEFINITION (absolute minimizer).
An admissible pair (u,K) is an absolute minimizer if for every open V whose
closure is a compact subset of Omega, written V contained in the interior of
Omega and denoted V << Omega, and for every admissible pair (w,L) satisfying
(L symmetric difference K) union support(w-u) is compactly contained in V,
one has F_g(u,K;V) <= F_g(w,L;V).
This is a comparison principle, not a differential equation. It says: no local
surgery, however wild, lowers the energy. Both the function and the crack may be
changed, but only inside V. Note "compactly contained": near the boundary of V
the competitor must agree exactly with (u,K). That is what makes the condition
local.
1.4 DEFINITION (reduced).
An admissible pair is reduced if
K = support in Omega of (H^1 restricted to K).
Equivalently, every disc centred at a point of K and compactly contained in
Omega contains a positive length of K.
Reducedness only removes junk. If K contains a closed piece of zero length, that
piece contributes nothing to the energy and may be deleted without changing any
comparison; reducedness forbids this bookkeeping waste. Every minimizer can be
made reduced, and the geometric conclusion is a statement about K itself, so we
lose nothing.
================================================================================
2. THE MAIN THEOREM
================================================================================
2.1 THEOREM.
Let Omega be a bounded Lipschitz domain in R^2 and g in L^infinity(Omega). Let
(u,K) be a reduced absolute minimizer of the energy of 1.2 in Omega. Then every
point x of K has an interior neighbourhood in which K has one of the following
three forms, the arcs being of class C^{1,alpha} for a single alpha in (0,1)
that does not depend on x:
(i) a single embedded arc, with x in its interior;
(ii) a single embedded arc, with x as one of its endpoints;
(iii) three embedded arcs, meeting only at x, with pairwise angles 120 degrees.
Moreover for every open U compactly contained in Omega,
the number of connected components of K that meet U is finite,
where the components counted are the components of the whole closed set K, not
of the truncation K intersect U.
Three remarks, because each one is a place where a careless reading goes wrong.
First, the compact-set qualification in the component count is not a defect.
Consider a single straight line crossing Omega: it is one arc, but it can meet an
open set U in infinitely many components if U is badly shaped, and near the
boundary of Omega nothing prevents infinitely many pieces from accumulating. The
theorem is strictly an interior statement.
Second, the exponent alpha is uniform. It comes from the epsilon-regularity
theorem we import, not from any argument in this paper.
Third, the theorem does not assert that minimizers exist. Existence is classical
and is not our subject; we are given a minimizer and describe its crack.
The rest of the document proves 2.1.
================================================================================
3. THE FIVE IMPORTED FACTS
================================================================================
Everything the literature supplies is collected here, once, in the exact form we
shall use. After this section there are no further appeals to outside results.
The price of that discipline is that the statements must be precise, so this is
the densest section of the document; on a first reading it is safe to skim the
statements and read only the one-paragraph justification at the end.
3.1 DEFINITION (homogeneous energy and absolute minimality in the plane).
For a closed rectifiable set H in R^2 of locally finite length, and a function v
which is W^{1,2} off H in every bounded open set, put
E_0(v,H;V) = INTEGRAL over V minus H of |grad v|^2 dx + H^1(H intersect V).
A pair (v,H) is an absolute minimizer of E_0 if, for every open V and every
admissible competitor (w,L) with
(L symmetric difference H) union support(w-v) compactly contained in V,
we have E_0(v,H;V) <= E_0(w,L;V).
Two features of this definition matter. The set H is part of the variable: a
competitor may move the crack, not merely change the function. And the support of
a change is allowed to meet H or cross it; nothing forces the competitor to agree
with v on the crack, and no vanishing trace condition is ever imposed. Both
features are essential, and both are the reason the imported facts below are
stated for this particular comparison class rather than for a generic one.
3.2 DEFINITION (generalized minimizer, and the relative constants).
Let (u_j,K_j) be a sequence of rescaled minimizing pairs as in (T1) below, and let
H be their local Hausdorff limit: for every R > 0, the symmetric Hausdorff
distance between K_j intersect B_R and H intersect B_R tends to zero. Let D_k,
k ranging over a countable set, be the connected components of R^2 minus H, and
choose a reference point z_k in each D_k. For all large j the point z_k lies off
K_j, so u_j(z_k) is defined. Normalize on each component by subtracting the
reference value and record the relative constants:
v_{j,k}(y) = u_j(y) - u_j(z_k) for y in D_k,
p_{kl} = limit as j tends to infinity of ( u_j(z_k) - u_j(z_l) ), in
[-infinity, infinity].
Along a further subsequence the normalized functions converge on each component
to a harmonic function. The resulting triple (v,H,{p_kl}) is a generalized
minimizer; it is global when the limiting domain is all of R^2.
The array {p_kl} is the only subtle part. When the complement of H is
disconnected there is no meaningful way to compare the values of the limit on two
different components, only their difference, and that difference may be infinite.
Carrying the constants along is bookkeeping, but necessary bookkeeping: a
classification of the limiting object which ignored them would be false.
3.3 PROPOSITION (the five imported facts).
Let Omega be bounded and Lipschitz, let g be bounded, and let (u,K) be a reduced
absolute minimizer of the energy of 1.2. Then the following hold.
(T1) BLOW-UPS EXIST.
Assume in addition that u is locally bounded in Omega and that K is rectifiable.
Fix x in K and radii r_j decreasing to 0. Put
K_j = (K - x)/r_j, u_j(y) = u(x + r_j y) / sqrt(r_j).
Then, after passing to a subsequence, the pairs (u_j,K_j) converge locally in the
bilateral Hausdorff sense to a global generalized minimizer (v,H,{p}) in the
sense of 3.2. The set H is closed, rectifiable, of locally finite length, and
equals the support of H^1 restricted to H. It contains the origin. The function v
belongs to W^{1,2} off H on every bounded open set, including square integrability
up to H.
(T2) A CONNECTED COMPLEMENT TURNS THE LIMIT INTO A MINIMIZER IN THE PLANE.
If a global generalized minimizer has connected complement D = R^2 minus H, then
(v,H) is an absolute minimizer of the homogeneous energy E_0 of 3.1 under every
compactly supported change of the pair, changes whose support meets H included.
(T3) DENSITY, ENERGY GROWTH, AND THE REGULAR ARC.
Every global generalized minimizer has reduced, closed, rectifiable H of locally
finite length, its function v is W^{1,2} off H on every bounded open set, and
there are constants 0 < c < C, depending only on the dimension, with
c r <= H^1(H intersect B_r(y)) <= C r for all y in H and r > 0,
INTEGRAL over B_R minus H of |grad v|^2 dx <= 2 pi R for all R > 0.
Moreover H^1-almost every point of H has a neighbourhood U with H intersect U a
single C^1 arc.
(T4) CLASSIFICATION OF GLOBAL LIMITS.
Let (v,H,{p}) be a global generalized minimizer. Then:
(a) H has at most one unbounded connected component;
(b) if H disconnects the plane, then H is a line, or three half-lines meeting
at 120 degrees at their common endpoint;
(c) if all but at most one connected component of H lie in one compact set,
then H is empty, a line, three half-lines meeting at 120 degrees, or a
single half-line.
(T5) EPSILON REGULARITY.
There are constants alpha in (0,1) and epsilon > 0, determined by the bounds on
the datum g, with the following property. Suppose a reduced absolute minimizer is
defined in a neighbourhood of the closed disc of radius 2s about a point x, with
s <= 1. If its crack inside that disc has bilateral Hausdorff distance less than
epsilon s from one of the models
a diameter, a radius, three radii meeting at 120 degrees,
then its whole crack inside the disc of radius s about x is C^{1,alpha}
diffeomorphic to that model. In addition, any three arcs of the crack which meet
inside the smaller disc do so at 120 degrees. The endpoint or junction produced by
the conclusion need not be the point x.
3.4 PROVENANCE.
The items above are, in order: Assumption 2.2.1 with Theorem 2.2.3, Theorem 2.1.3
and Definition 2.2.4; Definition 2.2.2(ii) with Theorem 2.2.3(ii); Lemma 2.1.2
with identity (1.4.1) for the energy bound, Theorem 2.1.3 for the density bounds
and Theorem 1.5.2(i),(v) for the almost-everywhere arc statement; Corollary
4.4.3, Theorem 4.4.1 and Theorem 1.4.6 for (T4); and Theorem 1.3.3 for (T5). All
numbers refer to De Lellis and Focardi, "The Regularity Theory for the
Mumford--Shah Functional on the Plane", EMS Monographs in Mathematics, 2025,
author version dated 4 January 2025.
Two of these applications deserve a word, because their hypotheses are easy to
misread.
For (T1): fix a bounded open W compactly contained in Omega and containing x, and
apply the compactness theorem there to the constant sequence of original minimizers,
with scale parameter 1, base point x and radii decreasing to 0. Local boundedness
supplies the uniform bound required of the unscaled functions, and the rescaled
domains exhaust the plane. The conclusion v in W^{1,2} up to H is part of the
theorem, not a consequence of componentwise convergence; we rely on it whenever we
multiply by a cutoff whose support meets H. That the limit stays reduced follows
from the lower density bound, applied at a limit point in a small ball and then in a
slightly larger ball whose boundary carries no limiting length. Finally 0 lies in H
because 0 lies in every K_j.
For (T2): a topological competitor must preserve the separation of exterior points
lying in different components of the complement of the crack. If the complement is
connected there are no such pairs at all, so every compact change is admissible and
the comparison theorem specializes to absolute minimality for E_0. In particular a
change may cross H, which is what section 9 needs.
3.5 WHY EACH ITEM IS NEEDED. (T1) is the only source of a limit object at a
point, (T2) removes the fidelity term from that limit, (T3) supplies the density
and growth bounds plus the regular arc, (T4) classifies everything but one
configuration, and (T5) converts blow-up information into local information.
================================================================================
4. FIRST STEP: THE MINIMIZER IS BOUNDED, ITS CRACK IS A JUMP SET
================================================================================
Theorem 2.1 is proved by blowing up at interior points, and (T1) requires two
things we were not given: that u is locally bounded and that K is rectifiable.
Neither is in the definition of admissibility. This section supplies both, and in
doing so identifies K exactly with the closure of the set across which u genuinely
jumps. Both come from competition: were u too large somewhere, clipping it would
lower the energy; were K to carry length across which u does not jump, a Dirichlet
replacement would lower the energy.
Throughout, (u,K) is a fixed reduced absolute minimizer of the energy F_g of 1.2
in the bounded Lipschitz domain Omega, with g bounded.
4.1 LEMMA (Euler--Lagrange, and smoothness off the crack).
The function u satisfies, for every phi in C_c^infinity(Omega minus K),
INTEGRAL over Omega minus K of ( grad u . grad phi + (u-g) phi ) dx = 0,
and consequently Delta u = u - g weakly on Omega minus K. It follows that u
belongs to W^{2,p}_loc(Omega minus K) for every finite p, and therefore has a
representative of class C^{1,alpha}_loc(Omega minus K) for every alpha in (0,1).
We fix that representative once and for all and call it u.
PROOF. Take phi in C_c^infinity(Omega minus K) and an open V with the support of
phi compactly contained in V compactly contained in Omega. For real t the pair
(u+t phi, K) is admissible with unchanged crack and support of the change inside
V, so minimality gives
F_g(u,K;V) <= F_g(u+t phi,K;V).
The length terms are identical, so the difference of the two sides is the change
of
t -> INTEGRAL over V minus K of |grad(u+t phi)|^2 + |u+t phi-g|^2 dx,
a quadratic polynomial in t which has a minimum at t = 0. Its linear coefficient
vanishes, which is the displayed identity. As V was arbitrary this holds on all
of Omega.
The weak equation reads Delta u = u - g on Omega minus K. In the plane,
W^{1,2}_loc is contained in L^p_loc for every finite p, so u - g lies in L^p_loc,
and interior elliptic estimates give u in W^{2,p}_loc off K. Taking p larger than
2 and applying the Sobolev embedding gives a C^{1,alpha}_loc representative. QED
4.2 LEMMA (good circles).
Fix q in Omega and let R_q = dist(q, boundary of Omega). For every s in (0,R_q)
except a null set of s, all three of the following hold:
(a) the set K intersect the circle of radius s about q is finite;
(b) the integral over that circle minus K of |u|^2 + |grad u|^2 with respect to
H^1 is finite;
(c) u is bounded on that circle minus K.
Such radii are called good for q, and the corresponding disc is a good disc.
PROOF. (a) The map d(x) = |x - q| is 1-Lipschitz, so by the coarea (area) formula
applied to the closed set K, which is Borel of finite length,
INTEGRAL over s in (0,R_q) of #(K intersect circle of radius s) ds
= INTEGRAL over K intersect B_{R_q}(q) of |grad d| dH^1 <= H^1(K) < infinity,
because |grad d| = 1 at H^1-almost every point. An integrable nonnegative function
is finite almost everywhere, which gives (a).
For readers who prefer the elementary version: cover the compact set K intersect
the closed disc of radius R_q by finitely many sets E_i of diameter less than
delta. A delta-separated family of level points meets each E_i at most once, and
the set of levels at which E_i is met is the image d(E_i), an interval of length
at most diam(E_i). Hence the integral above is at most the sum of the diameters,
which by definition of H^1 can be made as close to H^1(K) as we please.
(b) Polar integration gives
INTEGRAL over s in (0,R_q) of INTEGRAL over the circle of |grad u|^2 dH^1 ds
= INTEGRAL over the disc of |grad u|^2 dx < infinity,
and the same with |u|^2 in place of |grad u|^2, because u lies in L^2. Again an
integrable function is finite almost everywhere.
(c) By (a) the circle minus K is a finite disjoint union of open arcs. On each
arc u is C^1 by 4.1, and by (b) the tangential derivative of u is square
integrable along the arc. A function on an interval with square integrable
derivative is Holder of order 1/2, hence bounded. Taking the largest of the
finitely many bounds gives (c). QED
4.3 PROPOSITION (local boundedness).
u lies in L^infinity_loc(Omega).
PROOF. Fix a good disc B of radius s about q, and let f be the trace of u on the
circle. By 4.2(c) f is bounded. Choose
M > max( (sup of |f| on the circle minus K), (sup of |g| on Omega) ),
and let T_M(t) = max(-M, min(M, t)) be the clipping map. Define
u~ = T_M(u) on B, u~ = u off B.
Then u~ lies in W^{1,2}(Omega minus K) intersect L^2(Omega). Indeed, T_M is a
contraction, so |grad T_M(u)| <= |grad u| on B; the two definitions agree on the
circle minus K because there |f| <= M is unchanged by clipping; and the arc
circle minus K has full measure in the circle, so Sobolev gluing applies.
The difference u~ - u is supported in the closed disc B. Choose an open V with
the closed disc B compactly contained in V compactly contained in Omega, and use
(u~,K) as a competitor in the definition of absolute minimality on V. The crack is
unchanged, so the length terms cancel, and the Dirichlet term does not increase.
For the fidelity term, on the set where u > M we have, using u > M >= |g|,
|T_M(u)-g|^2 - |u-g|^2 = (M-g)^2 - (u-g)^2 = (M-u)(M+u-2g) < 0,
and symmetrically wherever u < -M. Hence minimality forces |u| <= M almost
everywhere in B. Since good radii are a full measure set, discs of this kind exist
with arbitrarily small radius about every point of Omega, and a finite number of
them cover any compact subset. QED
4.4 LEMMA (u belongs to SBV_loc).
Let W be compactly contained in Omega. Then u has bounded variation on W, its
distributional derivative has no Cantor part there, and the jump set J_u of u
satisfies J_u contained in K.
PROOF. Choose a second open set W' with W compactly contained in W' compactly
contained in Omega. By the definition of H^1 and compactness of K intersect the
closure of W, there are, for each index j, finitely many open balls of radii a_ij
with
the maximum of the a_ij tends to 0 as j grows,
the sum of the a_ij is at most C(1 + H^1(K)),
the union G_j of those balls has closure inside W',
and K intersect the closure of W is contained in G_j, since the union is open and
covers it. In particular the boundary of G_j misses K on W. The discs have total
area tending to 0, and the perimeter of G_j is at most 2 pi times the sum of the
radii, because the boundary of G_j is contained in the union of the corresponding
circles.
Define u_j = 0 on G_j intersect W and u_j = u elsewhere in W. On W minus G_j the
function u_j coincides with u, whose distributional derivative there is the
absolutely continuous measure with density grad u. On G_j the function vanishes,
and across the boundary of G_j the function jumps by |u|, so
|D u_j|(W) <= INTEGRAL over W minus K of |grad u| dx
+ 2 pi (sup of |u| on W') times (sum of the a_ij).
The supremum is finite by 4.3. Also u_j tends to u in L^1(W), because the two
differ only on G_j intersect W, whose area tends to 0. Lower semicontinuity of
total variation under L^1 convergence gives |Du|(W) at most the limit inferior of
|D u_j|(W), which is finite. Hence u lies in BV(W).
Off K the derivative of u is absolutely continuous, so the Cantor part of Du is a
measure supported on K. But the Cantor part of a BV function gives zero mass to
every set of sigma-finite H^1 measure, and K is such a set because H^1(K) is
finite. Hence the Cantor part vanishes, that is, u lies in SBV(W). Finally u is
C^1 off K by 4.1, so u has no jump points outside K, which is the last assertion.
QED
================================================================================
4.5 LEMMA (the smooth-data Dirichlet input).
Let D be a disc in the plane, let h be bounded on D, and let w be a C^1 function
on the plane with compact support. Consider
J_D(z) = INTEGRAL over D of ( |grad z|^2 + |z-h|^2 ) dx + H^1(J_z intersect
the closed disc D),
minimized over functions z of class SBV on the plane which agree almost
everywhere with w off D. Here J_z is the jump set of z. Then:
(a) a bounded minimizer z exists;
(b) the difference (closure of J_z) minus J_z, intersected with the closed disc
D, has zero H^1 measure;
(c) consequently C := (closure of J_z) intersect the closed disc D is compact of
finite length, and z lies in W^{1,2}_loc off the closure of J_z;
(d) on each relatively open piece of the circle on which C is absent, the
interior and exterior Sobolev traces of z agree almost everywhere.
PROOF OF (a),(c),(d). This is the smooth-boundary, smooth-exterior-datum case of
the Dirichlet existence and essential-closure theorem; see Appendix B of the
monograph quoted in 3.4, specifically Theorem B.3.1(c), Corollary B.3.2(b) and
Proposition B.4.2(c). Items (c) and (d) also follow directly from the SBV
decomposition: on an open set which misses the closed jump set the singular part
of the derivative is absent while the absolutely continuous part is square
integrable, and the same observation applied to the whole-plane extension of z,
whose exterior value is the smooth function w, gives agreement of the traces across
a jump-free piece of the circle.
PROOF OF (b), WHY THE CIRCLE CONTRIBUTES NO EXTRA LENGTH. The content of (b) is
that the closed jump set of a minimizer does not creep along the boundary of the
Dirichlet disc. Suppose otherwise. The boundary density theorem for jump sets of
SBV minimizers supplies a positive constant and, for every point y of the closed
jump set,
H^1(J_z intersect B_r(y)) / r >= positive constant for small r > 0.
Put mu = H^1 restricted to J_z and nu = H^1 restricted to the circle. The part of
mu carried by the circle is nu restricted to J_z intersect the circle, while the
rest of mu is singular with respect to nu. Differentiation of measures, together
with nu(B_r(y))/(2r) tending to 1 for points on the circle, gives
mu(B_r(y)) / r tending to 0
at H^1-almost every point y of the circle which does not belong to J_z. At any
such point which also lay in the closure of J_z, this would contradict the lower
bound. Hence the circle part of the difference in (b) is null, and the interior
part is the usual interior essential-closure theorem. No matching of normal
derivatives across the circle is used. QED
4.6 PROPOSITION (the crack is exactly the closure of the jump set).
Assume additionally that (u,K) is reduced. Then
H^1(K minus J_u) = 0, K = closure in Omega of J_u = support in Omega of
(H^1 restricted to J_u),
and K is countably 1-rectifiable. Consequently the conclusions of (T1) apply at
every point of K.
PROOF. Fix a good disc D of radius s about q, and write Sigma for the finite set
K intersect the circle, which is finite by 4.2(a) and hence of zero length. The
trace f of u on the circle is bounded by 4.2(c).
Step 1, smoothing the boundary datum. Choose closed finite unions of circular arcs
E_m contained in the circle, each containing Sigma in its relative interior, with
H^1(E_m) =: l_m tending to 0. There are C^1 functions a_m on the circle which
agree with f off E_m: on each of the short complementary arcs interpolate the
endpoint values and the first tangential derivatives, and keep f on the rest of
the circle. Extending radially with a cutoff produces w_m in C^1 of the plane with
compact support, whose trace on the circle is a_m. No bound uniform in m is needed
or used.
Let z_m be the minimizer of 4.5 for exterior datum w_m.
Step 2, an upper bound for the Dirichlet energy. The function which equals u on D
and w_m off D belongs to SBV of the plane by the trace gluing formula, Theorems
3.84 and 3.87 with Corollary 3.89 of Ambrosio, Fusco and Pallara. Its jump set
inside the disc is J_u together with a subset of E_m, so
J_D(z_m) <= INTEGRAL over D of (|grad u|^2 + |u-h|^2) dx + H^1(J_u intersect D)
+ l_m, (*)
where h stands for the datum g restricted to D.
Step 3, building an admissible competitor for the original problem. Put
C_m = (closure of J_{z_m}) intersect the closed disc D,
L_m = (K minus D) union C_m union E_m,
v_m = z_m on D, v_m = u off D.
The set L_m is relatively closed and has finite length by 4.5(b),(c). The function
v_m is locally W^{1,2} away from L_m: on D minus C_m because z_m is, off D because
u is. At a point of the circle outside L_m the interior trace of z_m equals w_m
almost everywhere by 4.5(d), and that trace equals f equals the trace of u,
because the point lies outside E_m. Sobolev gluing is therefore available at every
possible seam, and v_m belongs to W^{1,2}(Omega minus L_m) intersect L^2(Omega).
Step 4, comparison, with the bookkeeping made explicit. Both L_m symmetric
difference K and the support of v_m - u lie in the closed disc D. Choose an open V
with the closed disc D compactly contained in V compactly contained in Omega, and
compare there.
Volume terms. Off D the two pairs agree. Inside D the crack has zero area, so
INTEGRAL over V minus K of (|grad u|^2 + |u-h|^2)
= INTEGRAL over V minus D of (|grad u|^2 + |u-h|^2)
+ INTEGRAL over D of (|grad u|^2 + |u-h|^2),
and the same expansion holds with z_m in place of u on D. Subtracting, the
difference of the two volume terms is exactly
INTEGRAL over D of (|grad z_m|^2 + |z_m-h|^2)
- INTEGRAL over D of (|grad u|^2 + |u-h|^2).
Length terms. Outside D the two cracks coincide. Inside D the crack of the
competitor is C_m, because the arcs E_m lie on the circle and K minus D misses
D. The set C_m is the closure of the jump set of z_m inside the closed disc, so
by 4.5(b) its part inside the open disc has the same length as the jump set:
H^1(L_m intersect D) = H^1(C_m intersect D) = H^1(J_{z_m} intersect D).
The circle itself carries no length in either pair: the original crack meets it
only in the finite set Sigma, and the arcs E_m contribute to L_m only along the
circle, which is not part of the open disc D.
Minimality therefore gives
H^1(K intersect D) <= INTEGRAL over D of (|grad z_m|^2 + |z_m-h|^2)
- INTEGRAL over D of (|grad u|^2 + |u-h|^2)
+ H^1(J_{z_m} intersect D).
The right-hand side is at most
J_D(z_m) - INTEGRAL over D of (|grad u|^2 + |u-h|^2),
because J_D(z_m) is that same integral plus the length of the jump set of z_m on
the closed disc, and dropping the part of that length carried by the circle can
only lower the right-hand side. Combining with (*) gives
H^1(K intersect D) <= H^1(J_u intersect D) + l_m.
Notice where l_m entered: it came in through (*), that is, through the choice of
smooth boundary data, and not through the length of the competitor. The arcs E_m
were needed only to produce a piece of C^1 boundary data close to the trace of u.
Letting m tend to infinity gives H^1(K intersect D) <= H^1(J_u intersect D). The
reverse inequality holds because J_u is contained in K by 4.4, so the two lengths
agree and K minus J_u has zero length inside D.
Step 5, globalising. The good discs are those whose radii lie in a full-measure
set, so countably many of them, with centres at the points of a countable dense
set and radii chosen from a full-measure set, cover Omega. A countable union of
null sets is null, so H^1(K minus J_u) = 0 in all of Omega.
Rectifiability of K follows at once: J_u is countably 1-rectifiable as the jump
set of an SBV function, and a null set may be added without destroying
rectifiability. Finally, reducedness says K is the support of the length measure
restricted to K, and since that length measure is the same as the one restricted
to J_u, the support is contained in the closure of J_u; the reverse inclusion
holds by closedness of K. This proves the displayed identity. QED
4.7 COROLLARY (the reduced form of the minimizer, used from now on).
Let (u,K) be as in Theorem 2.1. Then u lies in L^infinity_loc(Omega) intersect
SBV_loc(Omega); K is closed, rectifiable, of finite length, and equals the closure
of J_u; and for every smooth open W compactly contained in Omega the pair
restricted to W is again a reduced rectifiable absolute minimizer, now with
bounded function. In particular (T1) of section 3 applies at every point x of K,
and (T5) applies in every such W.
PROOF. The first sentence collects 4.3, 4.4 and 4.6. For the restriction: a
compactly supported competitor in W, extended by the original pair off W with an
unchanged collar, is an admissible competitor in the original problem, so absolute
minimality is inherited; reducedness is a local property and is inherited as well.
Boundedness on W is 4.3. A competitor of finite energy automatically has square
integrable gradient, and on its bounded comparison region V it satisfies
INTEGRAL over V of |w|^2 <= 2 INTEGRAL over V of |w-g|^2 + 2 |V| (sup |g|)^2,
so it belongs to the global L^2 class used in section 1; a competitor of infinite
energy cannot improve on a finite one. This matches the locally Sobolev comparison
class of the classical theory. QED
================================================================================
5. SECOND STEP: A BOUNDED PIECE CAN BE CUT FREE
================================================================================
Here begins the geometric half. By (T4) everything reduces to one situation: a
global minimizer of the homogeneous energy whose complement is connected but whose
crack has a bounded connected component. This section is the topological
preparation. A bounded component of a closed set need not be separated from the
rest, since other pieces may crowd up against it; we repair that by enlarging the
component, inside the crack, to a compact set A at positive distance from the
remainder B. That separation is what later lets us translate A without touching B.
5.1 LEMMA (isolating a compact piece).
Let H be a closed subset of the plane, of locally finite length, equal to the
support of H^1 restricted to H. Let C be a nonempty bounded connected component of
H. Then there is a nonempty compact set A contained in H with C contained in A
such that, writing B = H minus A,
B is closed, distance(A,B) > 0, 0 < H^1(A) < infinity,
where the distance to the empty set is read as +infinity. If in addition the
complement of H is connected, then the complement of B is connected as well.
PROOF. The component C is compact, being closed and bounded, so pick R with C
contained in the open disc of radius R and put X = H intersect the closed disc of
radius R. The component of X containing a point p of C is exactly C. Indeed C lies
inside X and is connected. Conversely, if Y is a connected subset of X containing
p, then Y is a connected subset of H containing p, and a connected subset of a
space which meets a component lies inside that component, so Y lies in C. Hence
the component of p in X equals C.
We need one standard fact about compact spaces: in a compact metric space, a
component is the intersection of all its clopen neighbourhoods. Here is the
argument, since everything rests on it. In a compact metric space, say that points
y and z are n-linked if there is a finite chain from y to z with consecutive
gaps smaller than 1/n. Fix p and let Q_n be the set of points n-linked to p. Each
Q_n is both open and closed: open because it is a union of balls of radius 1/(2n),
closed because its complement is the union of the other equivalence classes, each
of which is open for the same reason. The sets Q_n are decreasing. Their
intersection Q is connected: if Q split into two nonempty relatively closed
pieces, those pieces would be compact and at positive distance, and for large n
the set Q_n, which is n-linked, could not meet both. So Q is a connected set
containing p, hence contained in the component of p, while conversely every
clopen neighbourhood of the component contains Q_n for some n. Hence Q is the
component of p.
Now apply the fact inside the compact space X. For each point z of X lying on the
circle of radius R, the component C does not contain z, so there is a clopen
neighbourhood of C in X which omits z. Finitely many of the complements of these
neighbourhoods cover the compact set X intersect the circle, so the intersection
of the corresponding finitely many neighbourhoods is a clopen neighbourhood of C
in X which misses the circle. Call it A; if X happens to miss the circle
altogether, take A = X. In either case:
A is compact and contains C;
A is open in X;
A is disjoint from the circle of radius R.
Because A stays at a positive distance from the circle, and because X is H
intersected with the closed disc, being open in X implies being open in H: a
sufficiently small ball about a point of A meets H inside X, hence inside A.
Therefore B = H minus A is closed, and since A is compact and B is closed and
disjoint from it, the distance from A to B is positive.
For the lengths: H^1(A) is finite because H has locally finite length and A is
bounded. And H^1(A) is positive because A is open in H, nonempty, and H is the
support of its own length measure, so every relatively open nonempty piece of H
carries positive length.
Finally the complements. The complement D of H is dense, because H is closed with
locally finite length and therefore has empty interior. The complement of B
contains D and is contained in the closure of D, which is the whole plane. A set
sandwiched between a connected set and its closure is connected, so the complement
of B is connected. QED
5.2 REMARK. The set A may contain several connected components of H, and that is
harmless. What matters downstream is only the positive separation from the fixed
remainder B, which permits every sufficiently small translate of the whole piece.
================================================================================
6. THIRD STEP: SPLITTING THE GRADIENT WITH A WHOLE-PLANE PROJECTION
================================================================================
Here is the idea of the whole remaining argument. Suppose a minimizer v in the
plane has a compact piece A of its crack, cut free as in section 5. Sliding alone is
useless: the energy is translation invariant, so sliding changes nothing to first
order. But grad v splits into two fields, one attached to A and one blind to it.
Sliding A moves the first and leaves the second alone, so the interaction between
them becomes a function of the slide parameter. Minimality makes zero slide a
minimum, and the interaction is harmonic, hence constant; section 9 shows that
forces everything to vanish.
This section builds the split, the only place we leave punctured topologies behind
and work with a single global orthogonal projection.
6.1 LEMMA (projection of a compactly supported function).
Let A be a compact subset of the plane with zero area. Let h be a compactly
supported function in W^{1,2} of the plane minus A, and let e = grad h be its
gradient density, which is a vector field in L^2 of the plane. Let
Z = the closure in L^2(plane; R^2) of the set of gradients grad(zeta) of smooth
compactly supported functions zeta,
and let Q be the orthogonal projection of L^2 onto Z. Then:
(a) there is a unique function phi in W^{1,2}_loc(plane) with grad phi = Qe and
phi tending to zero at infinity.
Put F = h - phi on the plane minus A, and p = e - grad phi on the plane. Then
(b) p lies in L^2(plane; R^2), and F lies in W^{1,2}(B_R minus A) for every R;
(c) grad F = p off A, and in the sense of distributions on the whole plane
div p = 0, support of (curl p) is contained in A,
Delta p = grad-perp of (curl p),
where curl(a_1,a_2) = d_1 a_2 - d_2 a_1 and grad-perp of b = (-d_2 b, d_1 b);
(d) outside a sufficiently large disc, F and p are smooth, and
F(x) = O(1/|x|), |grad^k F(x)| = O(1/|x|^{1+k}) for k >= 1.
In particular p(x) = O(1/|x|^2) there.
PROOF OF (a): EXISTENCE AND UNIQUENESS OF THE POTENTIAL.
First, every element of Z is the gradient of a function in W^{1,2}_loc. Let zeta_n
be smooth compactly supported functions whose gradients converge in L^2 to some w
in Z. Subtracting from each zeta_n its mean over the unit disc does not change
grad(zeta_n), so we may assume those means vanish. Poincare's inequality on the
unit disc then bounds the L^2 norm of zeta_n on that disc by the L^2 norm of its
gradient, which is bounded. To pass to a larger disc B_k, integrate the gradient along rays from the unit disc;
this bounds the L^2 norm of zeta_n on B_k by a constant depending on k times the L^2
norm of its gradient there. Extracting diagonally, a subsequence converges weakly in W^{1,2}(B_k) and
strongly in L^2(B_k) for each k, and the limit phi has grad phi = w. Apply this
with w = Qe.
Uniqueness: two potentials with the same gradient differ by a constant, and the
condition of tending to zero at infinity pins that constant. We must check that a
potentially attainable limit at infinity exists; this is part of (d) below, which
shows phi = b + O(1/|x|), after which we replace phi by phi - b.
PROOF OF (b),(c): THE TWO DISTRIBUTIONAL EQUATIONS.
Since p = e - grad phi with both terms in L^2, (b) holds, and grad F = e - grad phi
= p off A because F = h - phi. The projection identity is
INTEGRAL over the plane of (e - grad phi) . grad(zeta) = 0
for every smooth compactly supported zeta, since e - Qe is orthogonal to Z and
grad(zeta) lies in Z. That identity is exactly the statement div p = 0 in the
sense of distributions. Next, off A there is a genuine weak second derivative of
h, and mixed partials of a W^{1,2} function commute, so curl e vanishes off A in
the distributional sense; hence curl p = curl e is a distribution supported in A.
Finally div p = 0 gives, in the sense of distributions,
Delta p = grad(div p) + grad-perp(curl p) = grad-perp(curl p),
which is the third identity of (c).
PROOF OF (d): DECAY.
The equation div p = 0 reads Delta phi = div e = Delta h. Since h vanishes outside
a large disc, Delta phi vanishes there too, so phi is harmonic outside a large
disc. Its gradient has finite energy on the whole plane. Expand a single-valued
harmonic function outside a disc of radius R in Fourier series: it is a constant,
plus a logarithmic term, plus modes proportional to r^n and r^{-n} for integers
n >= 1. Finite Dirichlet energy excludes the logarithmic term, whose gradient has
size 1/r and whose energy diverges logarithmically, and excludes every growing
mode r^n. What remains is a constant plus a series of modes r^{-n}, so
phi(x) = b + O(1/|x|), |grad^k phi(x)| = O(1/|x|^{1+k}) for k >= 1,
outside a larger concentric disc. Replacing phi by phi - b, which changes neither
grad phi nor anything else used above, gives (a) with the normalization. Finally
h = 0 outside a disc, so there F = -phi, which gives the stated decay of F, and
p = grad F gives the stated decay of p. QED
6.2 REMARK. The field p carries the jump across A, its curl living there and
nowhere else, while grad phi sees straight through A. Two warnings. First, p is not
compactly supported, only its curl is; that is why every comparison carries a
cutoff, and why the decay estimate (d) is essential rather than cosmetic. Second,
the projection must be taken over the whole plane: a disc-local projection dies the
moment A is translated.
================================================================================
7. FOURTH STEP: SLIDING THE COMPACT PIECE, AND THE ENERGY INEQUALITY
================================================================================
Now the actual competition. The compact piece A is translated by a small vector t,
the split of section 6 is carried along, and the part F attached to A is translated
with it, optionally modified by an extra compactly supported jump field P. Because
A_t has the same length as A and stays away from B, the length terms cancel
exactly; because the split fields decay at infinity, a distant cutoff returns us to
v at no cost in the limit. What survives is one scalar inequality, and everything
after this section analyses it.
Throughout this section, and in sections 8 and 9, we are in the following
situation.
7.0 THE SITUATION.
Let H be a closed rectifiable subset of the plane, of locally finite length, and
let A, B be closed subsets of H with A compact, B = H minus A, and
distance(A,B) > 0, the distance to the empty set being +infinity. Let v be W^{1,2}
off H on every bounded open set, with
INTEGRAL over B_R of |grad v|^2 dx <= C (1 + R) for every R >= 1.
Let (v,H) be an absolute minimizer of the homogeneous energy E_0 of 3.1 under every
compactly supported comparison, and suppose the complement of H is connected. Write
D = plane minus H and O = plane minus B.
Assume v = f + F off H where:
f is W^{1,2} off B on every bounded open set;
(F,p) is obtained from a compactly supported h in W^{1,2}(plane minus A) by
Lemma 6.1;
putting G = grad v - p, we have G = grad f off B and
INTEGRAL over B_R of |G|^2 dx <= C' (1 + R) for every R >= 1.
For a vector field a write a_t(x) = a(x - t) for the translate, and similarly
A_t = A + t for a set.
7.1 LEMMA (the translated comparison).
Assume 7.0. Fix any compactly supported h' in W^{1,2}(plane minus A) and let
(P,q) be its counterpart of (F,p) under Lemma 6.1. Choose delta > 0 so small that
A_t is disjoint from B whenever |t| < delta. Define
I_p(t) = INTEGRAL over the plane of G . p_t,
I_q(t) = INTEGRAL over the plane of G . q_t.
Then both integrals converge absolutely, and for every |t| < delta and every real
sigma,
0 <= 2 ( I_p(t) - I_p(0) ) + 2 sigma ( I_q(t) + <p,q> ) + sigma^2 ||q||^2, (7.2)
where <p,q> and ||q|| are the L^2 inner product and norm on the plane.
PROOF. Fix t with |t| < delta and a real sigma. We first build the competitor and
then compute.
CONSTRUCTION OF THE COMPETITOR.
For large R let eta_R be a smooth radial cutoff equal to one on the disc of radius
R, vanishing outside the disc of radius 2R, with |grad eta_R| <= c/R. Put
J = B union A_t,
V_R = f + F_t + sigma P_t on B_R minus J,
V_R = v + eta_R ( F_t - F + sigma P_t ) on (B_{2R} minus closure of B_R)
minus J,
V_R = v outside B_{2R} union J.
This is admissible in the sense needed for the comparison. Indeed:
J is closed, because B is closed, A_t is compact, and they are disjoint;
J is rectifiable with locally finite length, because A and B are subsets of the
rectifiable set H and translation preserves rectifiability and length;
on B_R minus J the function is W^{1,2}, because f is W^{1,2} off B, and F_t, P_t
are W^{1,2} off A_t;
the two formulas agree near their interface, because near the sphere of radius R
we have v = f + F off H;
the change from (v,H) is compactly supported in a disc slightly larger than
B_{2R}, so the comparison applies.
LENGTH TERMS CANCEL EXACTLY.
Because A_t is disjoint from B we have H^1(J intersect V) = H^1(B intersect V) +
H^1(A_t intersect V) for any open V, and H^1(A_t) = H^1(A). Hence for every R,
H^1(J intersect B_{2R+1}) = H^1(H intersect B_{2R+1}),
so the comparison contains no length term at all. This is the whole point of
requiring positive separation in Lemma 5.1.
THE ANNULUS CONTRIBUTES NOTHING IN THE LIMIT.
On the annulus between radii R and 2R the exterior decay of Lemma 6.1(d) gives
|| grad [ eta_R ( F_t - F + sigma P_t ) ] ||_{L^2}
<= c_{t,h,h'} (1 + |sigma|) / R.
Consequently the change of Dirichlet energy on that annulus is at most
c (1+|sigma|) R^{-1/2} + c (1+|sigma|)^2 R^{-2},
where we used ||grad v||_{L^2(B_{2R})} <= c R^{1/2} from 7.0. Both terms tend to
zero as R grows.
ABSOLUTE INTEGRABILITY OF THE INTERACTION.
Look at a dyadic annulus where |x| is comparable to s. There
||G||_{L^2} = O(s^{1/2}), ||p_t||_{L^2} + ||q_t||_{L^2} = O(s^{-1}),
so the integral of |G . p_t| and of |G . q_t| over that annulus is O(s^{-1/2}),
which is summable over the dyadic shells. On bounded sets the products are
integrable by Cauchy--Schwarz. Hence I_p(t) and I_q(t) converge absolutely.
THE INEQUALITY.
Inside the disc of radius R the two gradients under comparison are
grad V_R = grad f + p_t + sigma q_t = G + p_t + sigma q_t,
grad v = G + p,
where for the second identity we used grad v = G + p off H and the fact that H has
zero area. Expanding squared norms,
|G + p_t + sigma q_t|^2 - |G + p|^2
= ( |p_t|^2 - |p|^2 ) + 2 ( G.p_t - G.p ) + 2 sigma ( G.q_t + p_t.q_t )
+ sigma^2 |q_t|^2.
Integrate over the disc of radius R and let R tend to infinity. Translation
invariance of the L^2 norm and of the L^2 inner product gives
||p_t|| = ||p||, ||q_t|| = ||q||, <p_t,q_t> = <p,q>,
so the first and last terms integrate to 0 and to sigma^2 ||q||^2, while the
middle terms give 2(I_p(t) - I_p(0)) and 2 sigma (I_q(t) + <p,q>). In particular
no subtraction of infinite whole-plane Dirichlet energies is involved: every
quantity that appears is either finite or cancels before the limit. The comparison
says the total change is nonnegative, the annulus contribution vanishes, and the
length terms cancel, which is (7.2). QED
7.3 REMARK. With sigma = 0, (7.2) says only that I_p is minimal at the origin.
The amplitude sigma is the freedom that eventually kills the whole field.
================================================================================
8. FIFTH STEP: THE INTERACTION IS HARMONIC, THEREFORE CONSTANT
================================================================================
Inequality (7.2) concerns a function of the slide parameter t. We show it is
harmonic in t, which together with minimality at the origin forces it constant. The
split is forced by smoothness: p_t carries its curl on the moving set A_t, so
differentiating there is delicate, while G = grad f is analytic near A_t because f
is harmonic. So we cut the interaction into a piece near A_t, smooth times a moving
distribution, and a piece away from A_t where everything is honest and we may
differentiate under the integral.
8.1 LEMMA (smoothness of the interaction, and constancy).
Assume 7.0 and suppose in addition that f is harmonic on O = plane minus B. Then,
after decreasing delta if necessary, the function t -> I_p(t) is harmonic on the
disc of radius delta about the origin. Consequently, for every compactly supported
h' in W^{1,2}(plane minus A), one has
I_p(t) = I_p(0) and I_q(t) = - <p,q> for every |t| < delta. (8.2)
PROOF.
PRELIMINARY. Since f is harmonic on O, its gradient G = grad f is real analytic
on O, in particular smooth and bounded on compact subsets of O. This is the only
analyticity input we need.
CUTTING. Let zeta be smooth and compactly supported in O, equal to one near the
union of the translates A_t for |t| < delta. Such a zeta exists because that union
is compact and is contained in O; shrinking delta if necessary keeps it so. Write
I_p(t) = INTEGRAL of G . p_t = INTEGRAL of p_t . ( zeta G ) + INTEGRAL of p_t .
( (1 - zeta) G ),
and treat the two summands separately.
FIRST SUMMAND.
The function zeta G is smooth and compactly supported in the plane, so the pairing
t -> J(t) = INTEGRAL of p_t . ( zeta G )
is meaningful. Compute its Laplacian in t using the distributional identity (c) of
Lemma 6.1, namely Delta p = grad-perp of curl p. For fixed t the distribution p_t
satisfies the same identity translated. Hence
Delta_t J(t) = < grad-perp of curl p_t , zeta G >
= - < curl p_t , curl ( zeta G ) >,
where the second equality is the integration by parts identity for grad-perp. Now
curl p_t is a distribution supported in A_t, and curl(zeta G) = curl(grad f) = 0 on
a neighbourhood of A_t, because zeta equals one there and f is harmonic there. A
distribution supported in a set annihilates every smooth test function vanishing
on a neighbourhood of that set, so the pairing is zero. Therefore J is harmonic.
SECOND SUMMAND.
On the support of (1-zeta) the field p_t is harmonic and smooth, uniformly
separated from A_t on bounded sets: recall that p satisfies Delta p = 0 off A, and
the translate satisfies the same off A_t. At infinity the k-th translate derivative
of p_t is O(1/|x|^{2+k}) by part (d) of Lemma 6.1, applied to the translates.
Combined with the growth bound for G from 7.0 and the dyadic estimate of the proof
of Lemma 7.1, each differentiated integrand is dominated by an integrable quantity,
locally uniformly in t. Differentiation under the second integral is therefore
justified, and its translate Laplacian vanishes too, since Delta p_t = 0 on the
support of (1-zeta). Hence the second summand is harmonic as well.
I_p IS HARMONIC. Adding the two summands, t -> I_p(t) is harmonic on the disc of
radius delta.
I_p IS CONSTANT. Put sigma = 0 in (7.2). This gives 0 <= 2(I_p(t) - I_p(0)), so
I_p has a minimum at the origin. A harmonic function on a disc with an interior
minimum is constant, so I_p(t) = I_p(0) for all |t| < delta.
SQUEEZING SIGMA. With I_p constant, (7.2) reads
0 <= 2 sigma ( I_q(t) + <p,q> ) + sigma^2 ||q||^2 for every real sigma.
The right side is a quadratic polynomial in sigma whose constant term is zero and
which is nonnegative on the whole real line. If ||q|| > 0, its minimum is attained
at sigma = -(I_q(t)+<p,q>)/||q||^2 and equals minus the square of the coefficient
divided by four times the leading coefficient; nonnegativity therefore forces the
coefficient I_q(t) + <p,q> to vanish. If ||q|| = 0 the polynomial reduces to
2 sigma (I_q(t) + <p,q>), and nonnegativity for both signs of sigma again forces
the coefficient to vanish. This is (8.2). QED
8.3 REMARK (the shape of the conclusion).
The first identity of (8.2) says that sliding A does not change the interaction
between the fixed field G and the translated field p. The second says the same for
any other compactly supported jump field q, with an explicit constant. Since h' is
free, the second identity holds simultaneously for all admissible q. That is an
enormous amount of information: it says the flux of G through an arbitrary
perturbation of the jump is independent of where A has been translated to. Section
9 cashes it in.
================================================================================
9. SIXTH STEP: A MINIMIZER IN THE PLANE HAS NO ISOLATED COMPACT PIECE
================================================================================
This is the theorem the whole document has been building towards. It says that in
the plane, with the fidelity term gone and the complement connected, a minimizing
crack cannot contain a compact piece that is cut free from the rest and carries a
regular arc. Combined with the classical classification (T4), it will finish the
proof of Theorem 2.1 in section 11.
9.1 PROPOSITION (exclusion of a separated compact piece).
Let H be closed, rectifiable, of locally finite length, with connected complement
D = plane minus H. Let v be W^{1,2} off H on every bounded open set with
INTEGRAL over B_R of |grad v|^2 dx <= C (1 + R) for every R >= 1,
and let (v,H) be an absolute minimizer of the homogeneous energy E_0 of 3.1 under
every compactly supported comparison. Then there is no compact set A contained in
H with all of the following:
(i) H^1(A) > 0;
(ii) B = H minus A is closed and distance(A,B) > 0, the distance to the empty
set being +infinity;
(iii) A contains an open subarc of H which is an embedded C^1 arc, meaning that
some open neighbourhood of some point of H meets H in exactly that arc and
the arc is contained in A.
9.2 STRATEGY. There is a classical intuition here, Earnshaw's theorem: charges
cannot sit in stable equilibrium in the field they create, because the potential is
harmonic and a harmonic function has no interior maximum. Our compact piece A plays
the role of the charges, the part of grad v attached to A is the field they create,
and translating A displaces them. Minimality says the displacement is a stable
equilibrium, so the interaction must be constant, and a constant interaction of a
harmonic field with itself forces the field to vanish.
The caution is that the analogy proves nothing alone: it gives no sign for a
bounded-domain Dirichlet variation and says nothing about the length term. Lemmas
7.1 and 8.1 supply the signs, as (7.2) and (8.2); steps 6 to 11 then extract the
contradiction.
PROOF OF 9.1. Suppose such an A exists. Set B = H minus A and O = plane minus B.
We proceed in eleven explicit steps. Steps 1 to 5 build the split; steps 6 and 7
exploit the freedom in the jump; steps 8 to 11 extract the contradiction.
STEP 1. O IS CONNECTED, AND H HAS ZERO AREA.
The crack H is rectifiable, hence has planar area zero. Therefore D is dense in
the plane. The set O contains D and is contained in the closure of D, which is the
whole plane. A set sandwiched between a connected set and its closure is connected,
so O is connected.
STEP 2. THE GRADIENT OF v IS DIVERGENCE FREE ON THE WHOLE PLANE.
Let m = grad v on D, defined arbitrarily on H. For a smooth compactly supported psi
and any real s, the pair (v + s psi, H) is admissible and differs from (v,H) on a
compact set, with the same crack. Hence the energy does not decrease at s = 0 in
either direction, and the length term is unchanged, so the linear coefficient
vanishes:
INTEGRAL over D of m . grad psi = 0 for every smooth compactly supported psi.
That is the statement that div m = 0 in the sense of distributions on the whole
plane.
STEP 3. APPLY THE PROJECTION LEMMA.
Choose chi smooth and compactly supported in O and equal to one on a neighbourhood
of the compact set A; this is possible because A is a compact subset of the open
set O. The function h = chi v, extended by zero near B, belongs to W^{1,2} of the
plane minus A: on O we have h equal to chi times a locally W^{1,2} function, and
the support of chi is a compact subset of O, so no singularity of v is encountered.
Also h has compact support. Apply Lemma 6.1 to h, obtaining the potential phi, the
function F = h - phi, and the field p = grad h - grad phi.
STEP 4. THE SPLIT, AND THE HARMONIC PART.
Define on O
f = (1 - chi) v + phi, G = m - p.
Then, first, f extends across A: near A the factor (1 - chi) vanishes, so f agrees
there with phi, which is smooth on the whole plane. Second, off H we have
f + F = (1 - chi) v + phi + chi v - phi = v,
so v = f + F as required in 7.0. Third, off H,
G = m - p = grad v - ( grad(chi v) - grad phi )
= grad( (1 - chi) v ) + grad phi = grad f,
which is the identity G = grad f used in 7.0. Fourth, div G = div m - div p = 0 by
steps 2 and Lemma 6.1(c), so div G vanishes on the whole plane; since G = grad f on
O, it follows that f is harmonic on O. Fifth, for every R,
INTEGRAL over B_R of |G|^2 <= 2 INTEGRAL over B_R of |m|^2 + 2 ||p||^2
<= c (1 + R),
using the growth of v assumed in 9.1 and the fact that p lies in L^2 of the plane.
All the hypotheses of 7.0 are therefore met, with this A, this B, this f, this F
and this G.
STEP 5. APPLY SECTIONS 7 AND 8.
By Lemma 7.1 and Lemma 8.1, for every compactly supported h' in W^{1,2}(plane
minus A), with (P,q) its projection-lemma counterpart, we have the identities of
(8.2):
I_p(t) = I_p(0), I_q(t) = - <p,q>, for every |t| < delta.
This is the whole content extracted from minimality. Everything that follows is
algebra and one integration by parts.
STEP 6. THE PROJECTED POTENTIAL DROPS OUT OF THE INTERACTION.
Let h' be compactly supported in W^{1,2}(plane minus A), and let (P,q) be its
counterpart, so that grad h' = q + grad phi' with phi' its normalized potential.
We claim
INTEGRAL over the plane of G . grad phi'_t = 0 for every t, (9.3)
and therefore I_q(t) = INTEGRAL of G . (grad h')_t.
For the proof, let theta_R be a smooth radial cutoff equal to one on the disc of
radius R and vanishing outside the disc of radius 2R, with |grad theta_R| <= c/R.
Because div G = 0 by step 4, testing against theta_R phi'_t gives
INTEGRAL of theta_R G . grad phi'_t
= - INTEGRAL of G . grad theta_R phi'_t.
The test field theta_R phi'_t is compactly supported and locally of Sobolev class,
so the identity is justified by smooth approximation, since G is locally square
integrable. On the annulus the normalization of Lemma 6.1(d) gives phi'_t =
O(1/R), and grad theta_R is O(1/R), while by step 4 the L^2 norm of G on the
annulus is O(R^{1/2}); the annulus has area O(R^2). Hence the right-hand side is
O(R^{-1/2}) and tends to zero. The left-hand side tends to the uncut integral,
because the integrand is absolutely integrable by the dyadic estimate of the proof
of Lemma 7.1. This gives (9.3).
Note what (9.3) does and does not say. It does not claim that G belongs to global
L^2; it uses only the distributional equation div G = 0 together with the decay of
the potential. This is the point of carrying the normalization in Lemma 6.1 at all.
STEP 7. JUMP VARIATIONS, AND THE FLUX FORMULA.
By hypothesis (iii), A contains an open subarc of H which is a C^1 arc. Discarding
all but a smaller piece of it, and applying a rigid motion of the plane, we may
assume the arc is the graph
gamma(s) = ( s, kappa(s) ), s in an interval I,
with kappa of class C^1 on a neighbourhood of the closure of I. Choose I and a
number a > 0 so small that the open strip
S = { (s, kappa(s) + r) : s in I, |r| < a }
is relatively compact in O and meets H only in the continuation of the arc. Let
n(s) be one of the two continuous unit normals to the graph. Choose eta smooth and
compactly supported in (-a,a) with eta(0) = 1.
For beta smooth and compactly supported in I define, on the strip,
h'_beta(x_1,x_2) = 1_{ x_2 > kappa(x_1) } beta(x_1) eta( x_2 - kappa(x_1) ),
and extend it by zero outside the strip. Its support avoids the ends and the outer
edges of the strip. Because kappa is C^1, this function is Sobolev on each side of
the graph, and it has no other jump: hence h'_beta lies in W^{1,2} of the plane
minus A, and it is compactly supported. Its two trace values on the arc differ by
beta. In particular only C^1 regularity of the arc is used, not C^{1,alpha}, and
no regularity of the rest of H is assumed.
Let q_beta be the field attached to h'_beta by Lemma 6.1. We compute I_{q_beta}(t).
By step 6,
I_{q_beta}(t) = INTEGRAL of G . (grad h'_beta)_t = INTEGRAL of G(x+t) .
grad h'_beta(x) dx,
after the change of variables x -> x - t. On the lower side of the graph the
gradient of h'_beta vanishes, so the integral runs only over the upper side. There
it is an integral of a smooth bounded vector field against the gradient of a smooth
bounded function, over the strip; the divergence of x -> G(x+t) is div G translated
and therefore vanishes. Integration by parts inside the upper side gives
I_{q_beta}(t) = - INTEGRAL over the upper side of (div G)(x+t) beta eta dx
+ a boundary integral along the graph.
The first term vanishes identically. On the graph the factor eta equals eta(0) = 1,
and the value of the test function there is beta(x_1); the normal direction is
n(s) and the arc element is |gamma'(s)| ds. What remains is
I_{q_beta}(t) = plus or minus INTEGRAL over I of beta(s) n(s) . G(gamma(s)+t)
|gamma'(s)| ds. (9.4)
The sign depends only on which normal was chosen and can be absorbed; it plays no
role anywhere.
There are no other boundary terms, because the test function vanishes near all the
other edges of the strip. Note also that translation preserves the normal n(s),
which is why no factor of t appears on the left of (9.4).
STEP 8. THE ARC FORCES THE HARMONIC PART TO BE AFFINE.
By (8.2), the left side of (9.4) does not depend on t, for every choice of beta.
Since G = grad f on O and the strip is relatively compact in O,
s -> n(s) . grad f( gamma(s) + t ) |gamma'(s)|
is a continuous function of s on I, for each t, and the family of test functions
beta is dense in the continuous functions on I. Independence of t for every beta
therefore gives, pointwise in s and jointly in t,
n(s) . grad f( gamma(s) + t ) = n(s) . grad f( gamma(s) )
for every s in I and every |t| < delta. (9.5)
Now fix a point s_0 in the interior of I and put u_0 = n(s_0), c = u_0 . grad
f(gamma(s_0)). Relation (9.5) at s = s_0 says that the fixed directional derivative
u_0 . grad f is identically equal to c on the disc of radius delta about
gamma(s_0). Rotate coordinates so that u_0 is the first basis vector. Then
d_1 f = c on that disc, so f(x) = c x_1 + A(x_2) there for some function A of one
variable, and the harmonicity of f gives A'' = 0. Hence f is affine on that disc.
Because f is harmonic on the connected open set O, it is real analytic there. The
difference of f and the affine function just found is harmonic and vanishes on a
nonempty open subset of O, so it vanishes on all of O. Therefore f is affine on O,
and
G = grad f is a constant vector field on O. (9.6)
STEP 9. THE GROWTH BOUND KILLS THE CONSTANT.
The plane minus O is the closed rectifiable set B, of zero area, so (9.6) holds
almost everywhere in the plane. A nonzero constant vector field c has
INTEGRAL over B_R of |c|^2 = pi |c|^2 R^2,
which is not O(R). Since step 4 gives the bound O(1+R), the constant must vanish:
G = 0 almost everywhere in the plane. (9.7)
STEP 10. THE FIELD ATTACHED TO THE PIECE ALSO VANISHES.
Take h' = h, so that q = p. Then (8.2) reads I_p(t) = - <p,p>, while by (9.7) the
integrand of I_p(t) is identically zero, so I_p(t) = 0. Hence
||p||_{L^2}^2 = 0, that is, p = 0. (9.8)
Now m = G + p = 0 almost everywhere on D.
STEP 11. THE CONTRADICTION.
Because D is connected and grad v vanishes on it, v is constant on D; call the
constant c. Since H has zero area, v equals c almost everywhere in the plane.
Consider the competitor
(v', H') = (c, B).
It is admissible: B is closed by hypothesis (ii), it has finite length because
H^1(B) <= H^1(A) + H^1(B) = H^1(H) is finite, and the constant function is
trivially W^{1,2} off B. The support of v' - v is empty in the essential sense,
because the two functions agree almost everywhere, so the set of conditions for a
compactly supported comparison is satisfied with the compact set A. The Dirichlet
terms of the two pairs are both zero. But
H^1(B intersect V) = H^1(H intersect V) - H^1(A) < H^1(H intersect V)
for every open V containing A, because A meets B in nothing and H^1(A) > 0 by
hypothesis (i). So the competitor has strictly smaller energy, contradicting the
absolute minimality of (v,H). This contradiction completes the proof of 9.1. QED
9.3 REMARK. Hypothesis (i) gives the strict drop in step 11; (ii) makes the
translates disjoint from B so that the length terms cancel in 7.1; (iii) is used
only in step 7, for clean trace differences on the two sides of the crack. Nothing
else about A or B is assumed: A may have infinitely many pieces, and B may be wild.
================================================================================
10. SEVENTH STEP: THE ONLY POSSIBLE BLOW-UPS
================================================================================
Now combine the exclusion of section 9 with the imported classification (T4). What
(T4) leaves open is exactly a bounded component inside a connected complement, and
that is the single gap section 9 fills.
10.1 COROLLARY (classification of the generalized limits).
Let (v,H,{p}) be a global generalized minimizer arising as a limit of reduced
absolute minimizers of the energy of 1.2, as in (T1). Then H is one of
empty, a line, three half-lines meeting at 120 degrees, a half-line.
In particular H has at most one connected component which is unbounded, and
at most one point at which the local structure is a junction or an endpoint.
PROOF. By (T1) and (T3) the set H is closed, rectifiable, of locally finite length,
reduced in the sense that H is the support of the length measure restricted to H,
and v is W^{1,2} off H on every bounded open set, with the energy growth of (T3).
Case 1: H disconnects the plane. Then (T4)(b) applies and H is a line or a triod of
three 120-degree half-lines, which is the claim.
Case 2: the complement D of H is connected. Then by (T2) the pair (v,H) is an
absolute minimizer of E_0 under every compactly supported comparison, so the
hypotheses of Proposition 9.1 are met.
We claim H has no bounded connected component. Suppose C were one. By Lemma 5.1
there is a compact set A contained in H, containing C, with B = H minus A closed,
distance(A,B) > 0, and 0 < H^1(A) < infinity. We must show that A contains an open
subarc of H. Since A is relatively open in H and H^1(A) > 0, we may choose a point
y of A which is a density point of the length measure of A with respect to the
length measure of H; then any sufficiently small neighbourhood of y meets H inside
A. By (T3), H^1-almost every point of H has a neighbourhood U with H intersect U a
single C^1 arc. Choosing y to be such a point as well, and shrinking U until it
meets H inside A, gives a nonempty relatively open piece of a C^1 arc which is
contained in A. Thus condition (iii) of 9.1 holds, and so do (i) and (ii), so
Proposition 9.1 gives a contradiction. Hence H has no bounded component.
Combined with (T4)(a), which says H has at most one unbounded component, we
conclude that H is empty or connected. In either case at most one connected
component of H fails to lie in a compact set, so (T4)(c) applies and yields the
four alternatives. This proves the corollary; the last sentence is a restatement
of (T4) together with the list. QED
10.2 REMARK. Once 10.1 is in hand, every blow-up of every interior point of the
original minimizer has one of those four elementary shapes.
================================================================================
11. EIGHTH STEP: FROM BLOW-UPS BACK TO THE ORIGINAL MINIMIZER
================================================================================
We have classified the possible limits of rescalings. The remaining work is
quantitative: we must transfer the classification to the original crack, at a
uniform scale, using epsilon regularity (T5). This is where the two sheets of the
argument are glued, and where the endpoint subtlety of (T5) has to be handled
honestly.
PROOF OF THEOREM 2.1.
STEP 1. REDUCE.
By Corollary 4.7 we may, and do, assume u is locally bounded, K is closed
rectifiable of finite length, K equals the closure of the jump set J_u, and the
pair restricted to any smooth open set compactly contained in Omega is again a
reduced absolute minimizer. So (T1) and (T5) are available.
STEP 2. BLOW UP AT A POINT.
Fix x in K. Take any sequence of radii r_j decreasing to 0 and form the rescalings
K_j = (K - x)/r_j, u_j(y) = u(x + r_j y)/sqrt(r_j).
By (T1) a subsequence converges locally in the bilateral Hausdorff sense to a
global generalized minimizer (v,H,{p}). The limit set H contains the origin.
STEP 3. THE LIMIT SET IS ONE OF THREE MODELS NEAR THE ORIGIN.
By Corollary 10.1, H is empty, a line, a half-line, or a union of three half-lines
at 120 degrees. It is not empty, because it contains the origin. So H is a line, a
half-line, or a triod.
Choose rho > 0 with the following two properties. First, inside the disc of radius
3 rho about the origin, H coincides with the model it is: a full line through the
origin, or a ray from the origin, or three rays from the origin. Second, if the
model is a half-line or a triod whose vertex is some point w different from the
origin, choose rho so small that 3 rho is smaller than the distance from the origin
to w. In that case the part of H inside the disc of radius 3 rho is a single line
segment through the origin, so the model we see there is a diameter, and the vertex
plays no role at this scale. This is the only place where the vertex-shift
phenomenon of (T5) has to be anticipated.
STEP 4. HAUSDORFF CLOSENESS ON A TRUNCATED BALL.
Local bilateral Hausdorff convergence gives
bilateral Hausdorff distance between K_j intersect B_{2 rho}(0) and
H intersect B_{2 rho}(0) is o(rho).
The subtlety is the truncation at the sphere of radius 2 rho: convergence is local,
so it controls balls of radius slightly less than 2 rho, not the ball itself. To
fix this, move each model point near the sphere of radius 2 rho slightly inward
along its own ray, apply convergence in the larger ball of radius 3 rho, and then
let the inward displacement tend to zero. Approximating points are then inside the
ball of radius 2 rho. The reverse distance follows by projecting onto the truncated
model radially. Hence the displayed closeness holds.
STEP 5. RESCALE BACK AND APPLY EPSILON REGULARITY.
Put s_j = r_j rho. Returning to the original coordinates, the crack inside the disc
B_{2 s_j}(x) has model distance o(s_j), the enlarged disc is compactly contained in
Omega for large j, and s_j <= 1. The hypotheses of the fixed-data epsilon
regularity statement (T5) are therefore satisfied, with the datum g of the original
problem and the same bounds. Conclusion: for large j, the whole set K intersect
B_{s_j}(x) is C^{1,alpha}-diffeomorphic to the corresponding model.
STEP 6. THE SHIFTED VERTEX CAUSES NO TROUBLE.
The diffeomorphism in (T5) need not send the original point x to the vertex of the
model. So after transporting back, x is either an interior point of one of the
model arcs, or the endpoint, or the junction. Either way, restricting the chart to
a smaller neighbourhood of x gives exactly one of the three alternatives (i), (ii),
(iii) in the statement of Theorem 2.1, with the same exponent alpha. This is why
(T5) was stated with the vertex unspecified: the statement we need is a
neighbourhood statement about x, and the chart already provides it.
We have now produced, for each x in K, an interior neighbourhood with the asserted
structure, with a single alpha coming from (T5). This is the first half of Theorem
2.1.
================================================================================
STEP 7. FINITELY MANY COMPONENTS MEET A COMPACT SET.
For each point x of K we produced a disc B_{s_x}(x) compactly contained in Omega
whose entire intersection with K is C^{1,alpha} diffeomorphic to a model, and in
particular is connected: the intersection of a line, a ray, or a triod of three
rays with a disc is connected.
Let U be open with closure compactly contained in Omega. The set K intersect the
closure of U is compact, so finitely many of these discs, say N of them, cover it.
Now take any global connected component C of K which meets U. Pick a point z of C
in the closure of U; it lies in one of the covering discs, say B. The set C
intersect B contains z and is contained in K intersect B, which is connected, so
C intersect B is connected. Since B intersects K in a connected set which lies in
a single global component, C is that component. Hence every component of K meeting
U is one of the at most N components determined by the covering discs, and there
are finitely many. This proves the second half of Theorem 2.1 and completes its
proof. QED
================================================================================
12. NINTH STEP: INTEGRABILITY OF THE GRADIENT AT THE TIP
================================================================================
The geometric conclusion has an analytic consequence worth extracting: it bounds how
singular the gradient may be near an endpoint. The tip profile is r^{1/2}
sin(theta/2), whose gradient has size r^{-1/2}, so the gradient lies in L^p near the
tip exactly for p < 4, and at p = 4 the integral diverges logarithmically. The sharp
statement is therefore a weak-type bound at the critical exponent 4, and every
minimizer in Theorem 2.1 attains it.
12.1 COROLLARY (local weak L^4 bound for the gradient).
Let (u,K) satisfy the hypotheses of Theorem 2.1, and let grad u denote the
approximate gradient supplied by Corollary 4.7; it agrees almost everywhere on
Omega minus K with the gradient occurring in the energy of 1.2. Then for every open
U compactly contained in Omega there is a finite constant C_U, depending on U and
on the minimizer, such that
area of { x in U minus K : |grad u(x)| > t } <= C_U t^{-4} for every t > 0.
In particular grad u belongs to the weak Lebesgue space L^{4,infinity} locally, and
to the ordinary Lebesgue space L^p locally for every 0 < p < 4.
PROOF. Fix U and choose a smooth open W with U compactly contained in W compactly
contained in Omega. By Corollary 4.7 the restricted pair is a reduced rectifiable
absolute minimizer for the De Lellis--Focardi energy, with bounded datum g, so it
falls in their class.
The content of Theorem 2.1 is that the crack has no irregular points: every point
of K lies in an arc chart, an endpoint chart, or a junction chart. These are
precisely the regular models of the De Lellis--Focardi theory, namely the pure jump
model, the loose end model and the triple junction model: an arc can be flattened
to its tangent ray by a C^1 chart tangent to the identity at the base point, and
the three angles are 120 degrees. Hence the irregular stratum of the restricted
pair is empty.
Their theorem on the equivalence between the absence of irregular points and the
weak L^4 bound then gives the displayed estimate on U, after increasing the
constant to absorb the range 0 < t < 1, where the bound is trivial because the
left side is at most the area of U. Finally, K has finite length hence zero area,
so the same estimate holds for the set where the approximate gradient is large,
with K removed or not.
The final assertion follows by the layer-cake formula. For 0 < p < 4,
INTEGRAL over U of |grad u|^p dx
= p INTEGRAL over t > 0 of t^{p-1} (area of { |grad u| > t } intersect U) dt
<= p INTEGRAL over t > 0 of t^{p-1} min( area of U, C_U t^{-4} ) dt,
and the last integral is finite because the integrand is bounded near t = 0 and
behaves like t^{p-5} near infinity, which is integrable exactly when p < 4. QED
12.2 REMARK. The exponent cannot be improved to a strong L^4 bound: the tip
profile r^{1/2} sin(theta/2) has gradient of size r^{-1/2}, so the integral of the
fourth power diverges logarithmically.
================================================================================
13. INDEX, AND WHAT DEPENDS ON WHAT
================================================================================
13.1 THE DEPENDENCE GRAPH.
Each item below lists what it uses; nothing else is used anywhere.
section 4 (nothing but the definitions)
Lemma 5.1 (nothing but topology and the definition of H^1)
Lemma 6.1 (nothing but the Poincare inequality and Fourier series for
harmonic functions)
Lemma 7.1 Lemma 6.1
Lemma 8.1 Lemmas 6.1 and 7.1, plus harmonicity of f
Prop. 9.1 Lemmas 5.1, 6.1, 7.1, 8.1
Cor. 10.1 Prop. 9.1, Lemma 5.1, and (T1),(T2),(T3),(T4)
Thm. 2.1 Cor. 4.7, Cor. 10.1, and (T5)
Cor. 12.1 Thm. 2.1 and the De Lellis--Focardi weak L^4 equivalence
One imported input, (T1)--(T5); one new idea, the translation argument of sections
6 to 9.
13.2 WHAT IS NOT PROVED HERE, AND WHERE TO FIND IT.
(T1) through (T5) and the weak L^4 equivalence of 12.1 are taken from De Lellis and
Focardi, "The Regularity Theory for the Mumford--Shah Functional on the Plane", EMS
Monographs in Mathematics, 2025, author version dated 4 January 2025. The SBV
structure and trace theory of section 4 is from Ambrosio, Fusco and Pallara,
"Functions of Bounded Variation and Free Discontinuity Problems", Oxford
Mathematical Monographs, 2000. Existence of minimizers is not needed here: a
minimizer is assumed given.
================================================================================
END
================================================================================