Ground-state uniqueness on convex domains and sharp logarithmic concavity on balls
Henry Zweiman
September 16, 2026
Abstract
We prove uniqueness of the positive Dirichlet ground state of on every bounded convex domain, without boundary smoothness assumptions. Every positive log-concave solution satisfies and is this ground state. A positive-level regularization gives the Hessian bound, while a local strict transport remainder in the logarithmic Sobolev inequality yields a sharp energy comparison on with equality precisely for multiples of the ground state. This answers the ground-state uniqueness and identification questions of Gallo, Mosconi and Squassina, whose log-concave existence theorem is an essential input. On balls, we prove a globally convergent positive radial series for , with an explicit recurrence. It yields concavity of , the sharp exponent among powers of the negative normalized logarithm, and positive power concavity of . We also prove that the normalized square-root-log property fails under arbitrarily small smooth uniformly convex deformations near every ball in dimensions at least two. A degree-three shape derivative creates a cubic-jet obstruction at the moving maximum. The deformed solutions remain strictly log-concave and are therefore ground states. Uniqueness among all positive solutions remains open here.
where . All concavity statements about a logarithmic transform concern the open ball. Statements about extend to the closed ball by continuity.
Gallo, Mosconi and Squassina [GMS] establish existence of a solution with locally negative-definite logarithmic Hessian on every bounded convex domain. Their Theorem 6.5 gives radial symmetry and radial decrease of every positive solution on a ball. In Sections 6.2--6.3 they ask for stronger concavity in balls: positive power concavity of , and concavity of the negative square root of its negative normalized logarithm. Their Theorem 6.6 proves both properties in dimension one. The second question remains explicitly stated on page 44 of the final article.
The positive-series representation below answers both ball questions in every dimension. Theorem 1.5 additionally resolves the ground-state questions of [GMS, Remark 1.6] on every bounded convex domain. Write , , and set
Theorem 1.1 (Positive radial series). Every solution of (1.1) satisfies . There are strictly positive coefficients , , such that
The quotient at is interpreted by continuity. Both series, regarded as series in , have radius of convergence exactly . Their coefficients are determined by
In particular, is absolutely monotone on : every derivative is strictly positive there. The same is true of every derivative of positive order of . In addition,
with strict inequality as quadratic forms when . Here denotes the order on symmetric matrices.
Theorem 1.2 (Sharp transformed concavity). For every solution of (1.1),
For its Hessian is negative definite away from the center. At the center it is continuous and has a norm-type cusp; no differentiability assertion there is intended.
Theorem 1.3 (Positive power concavity). There exists a number such that, for ,
It is characterized by an attained interior minimum,
The formula is implicit. We do not assert a positive exponent uniform over all radii, or monotonicity of this optimal exponent as a function of the radius.
Theorem 1.4 (Instability under convex deformation). Let , , and let be the positive Dirichlet solution on . Define and
For all sufficiently small , these are smooth uniformly convex domains and there is a smooth branch of positive solutions , with . For each sufficiently small nonzero , the solution is strictly log-concave and has a unique maximum, but
is not concave in any neighborhood of that maximum. The domains converge to in every boundary norm.
The normalization in this statement is essential. The scale-invariant notion of -log-concavity permits a sufficiently small multiplicative factor in front of ; see [IST, Section 4.2]. Theorem 1.4 concerns the specific factor , where the transform has its cusp at the maximum. It does not disprove the weaker rescaled property. It answers the controlled-deformation question posed in the earlier version of Section 7 and leaves the ball theorem unchanged.
Theorem 1.5 (Ground-state uniqueness on convex domains). Let be nonempty, bounded, open and convex. The problem with zero Dirichlet boundary values has a unique positive ground state . Every positive log-concave solution is this ground state, and
Let . For every nonzero real ,
with equality exactly for the nonzero multiples of . The real ground states are . No regularity of is required. Here a ground state minimizes the action on its nonzero Nehari set; equivalently it is a nonzero solution of least action.
This theorem does not assert uniqueness among all positive solutions. It identifies the solution constructed in [GMS] as the ground state and proves its uniqueness at least energy. Its proof in Section 8 uses interior level sets and a local strict entropy remainder to avoid assumptions on boundary derivatives or global integrability of quotients of solutions.
1.1. Prior results and the scope of the contribution
The amplitude threshold, uniqueness in balls, and strict ordering of first zeros are prior results. For and , Ben Chrouda [BC, Theorem 1.2 and Corollaries 1.3 and 1.5] proves these facts for . The change gives the normalization (1.1). The equality case in the amplitude threshold is excluded by the Gaussian solution and uniqueness of the central initial-value problem. We give new proofs of these particular consequences because they fit the recurrence argument, but do not claim their first discovery.
Liu, Sun and Zou [LSZ] prove a broader shooting classification of whole-space nodal solutions; in particular, their argument establishes strict motion of zeros with the initial value. Gallo, Moraschi and Squassina [GMoS] develop parabolic concavity principles whose logarithmic example yields preservation of logarithmic concavity. Neither of these results supplies the positive radial series used here. For the ball results, our priority claim concerns that structure and the resulting stronger concavity, rather than existence, radial symmetry, whole-space Gaussian ground-state classification, or shooting uniqueness. Current versions of these sources were compared on September 15, 2026. The convex-domain ground-state extension was checked against the final [GMS] article and later whole-space uniqueness results on September 16, 2026. Its contribution is the Hessian strengthening and the resulting variational rigidity on arbitrary bounded convex domains. Caffarelli contraction, Gaussian logarithmic Sobolev inequalities and interior elliptic regularity are classical inputs, credited in Section 8.
The proof of Theorem 1.1 has three stages. An integrating factor puts the logarithmic radial derivative above its Gaussian value. Its equation then produces positive Taylor coefficients. Finally, positivity and a real-axis singularity argument show that the series represents the solution all the way to its first zero. Convexity of a Hilbert-space norm proves Theorem 1.2. A separate boundary argument, which accommodates the singular derivative of at zero, completes Theorem 1.3. For Theorem 1.4, the ball Hessian bound and the classical weighted Poincare inequality make the linearized Dirichlet operator invertible. A degree-three shape derivative then violates a necessary cubic-jet condition at the moving maximum.
2. The radial equation and the Gaussian threshold
We use radial symmetry and radial decrease from [GMS, Theorem 6.5]. These are nontrivial inputs: the derivative of the nonlinearity is unbounded below near zero, so an unqualified appeal to the usual locally Lipschitz moving-plane theorem would be inappropriate. Existence follows from [GMS, Theorem 1.1], with its one-dimensional result covering . The rest of the argument applies to every positive radial decreasing solution.
Put for , and . The radial equation and its integral form are
The central initial-value problem is unique as long as its solution stays positive. Indeed, integrating (2.1) once more gives a Volterra equation. On a sufficiently small interval the Lipschitz bound for near a prescribed positive central value makes its integral operator a contraction, since the norm of the double integral is bounded by a constant times . Ordinary uniqueness then continues the solution away from zero. The same identity, using smoothness of near a positive central value, gives the usual even expansions at the center.
Since has its maximum at zero, (2.1) implies , hence . If , central uniqueness would give , which violates the boundary condition. Thus .
In fact for . If vanished there, its nonpositivity would imply at that point. Equation (2.1) would give , and ordinary uniqueness with data would again force the constant solution.
In terms of (1.2), equation (2.1) becomes
Define for and extend it to the center. The central expansion gives
Differentiation of (2.2) yields
Equivalently,
Lemma 2.1. We have and for all .
Proof. Suppose first that . Integrating (2.5) from zero shows that while . The boundary term at zero vanishes by (2.3). The function cannot first reach , since it is decreasing. It cannot first reach zero either: the integral in (2.5) would then give a strictly negative derivative, whereas throughout the interior. Thus on , giving . This contradicts as .
If , the central solution is
by the uniqueness following (2.1). It has no finite zero. Hence . Now (2.5) gives as long as , and this strict increase prevents any first exit from that region. ∎
The eigenvalues of are in the radial direction and in the tangential directions. Lemma 2.1 proves (1.5), including its strict form away from the center. In dimension one only the radial eigenvalue is present.
3. A positive recurrence valid throughout the ball
Let and . Equation (2.4) becomes
Lemma 3.1 (Local construction). For each , recurrence (1.4) defines strictly positive coefficients satisfying
The resulting series solves (3.1) near zero and agrees with the transformed radial solution.
Proof. The coefficient of in the linear derivative terms of (3.1) is . In the nonlinear terms, symmetry of convolution gives
which proves (1.4). In particular . For , its numerator can be written
so positivity follows by induction. Dropping the negative term in (1.4) and using the inductive upper bounds gives
Consequently the series converges for , and termwise differentiation verifies (3.1).
Integrate with . The derivative of the difference between the two sides of (2.2) vanishes by (3.1). Its limiting value at zero is . Thus (2.2) holds. Setting recovers the radial equation with the prescribed central data. The uniqueness argument after (2.1) identifies this constructed solution with the given one. ∎
We next state the positive-series continuation principle, including its short proof.
Lemma 3.2 (Positive real singularity). Suppose has and finite positive radius of convergence . It cannot admit a holomorphic continuation to a neighborhood of the positive real point .
Proof. If such a continuation existed, continuity of its derivatives and monotone convergence as would give, for every ,
Choose a sufficiently small in the convergence disk of the Taylor series centered at . Its coefficients are nonnegative by (3.6). Interchanging nonnegative sums gives
contradicting the definition of . ∎
Proof of Theorem 1.1. By Lemma 3.1 the series for represents the actual solution near zero. Denote its convergence radius by . If , the actual solution is regular at the positive point . On an interval around that point, (3.1) is an ordinary differential equation with analytic right-hand side after solving for . It therefore has a locally holomorphic solution with the actual values of and there. Ordinary uniqueness identifies this solution with the continuation from zero. This contradicts Lemma 3.2. Thus .
Conversely, tends to infinity as : by Lemma 2.1 it is increasing, and a finite limit would make bounded. Therefore is impossible. We have proved . Integration gives the second series in (1.3). Dividing its coefficients by a factor linear in the index leaves the convergence radius unchanged. All derivative assertions now follow by differentiating convergent positive series on compact subintervals of . The Hessian assertion was proved after Lemma 2.1. ∎
The continuation step is essential: positivity of finitely many coefficients, or even a formal positive recurrence without convergence control, would not imply a global concavity theorem.
4. Convexity of the square root and sharpness
Put . Theorem 1.1 gives
Every coordinate on the right is nonnegative, nondecreasing and convex as a function of . Coordinatewise convexity, monotonicity of the norm on nonnegative sequences, and the triangle inequality show that is convex and nondecreasing. Equivalently one may first prove this for the finite partial sums and pass to the finite pointwise limit on .
For strictness it is useful to give an exact identity. At let
These are a probability distribution, its mean and its variance; all moments are finite inside the convergence interval. Direct differentiation gives
The strict inequality holds because , as . Thus for . Its tangential Hessian eigenvalue is .
Proof of Theorem 1.2. A convex nondecreasing function of the Euclidean norm is convex. Applying this fact to (4.1) proves convexity of across the center as well. The Hessian assertions away from the center follow from (4.3). For , the function is convex and nondecreasing on ; composing it with proves the forward implication of (1.6).
For necessity, the central expansion is
If , the restriction to a positive radial segment has strictly negative second derivative for sufficiently small . Indeed its leading term is , whose second derivative has the negative factor ; the differentiated remainder has higher order. It cannot be convex. Finally, (4.4) gives , which describes the stated cusp. ∎
The same series gives more than a Hessian bound. For every integer , the radial differential operator satisfies on , with a positive limit at zero. These infinitely many sign inequalities refer to differentiation with respect to squared radius, not arbitrary mixed spatial derivatives.
5. Boundary behavior and the optimal positive power
Because is bounded on the compact range of , (2.1) gives a finite limit of at . The differential equation then gives a continuous limit of there. The next argument rules out a zero boundary slope without assuming a finite value of .
Lemma 5.1 (Simple boundary zero). There is such that
and
Proof. Write . Near we have , and (2.1) gives . Thus is decreasing there. Set
The energy identity is . Suppose . Then , and for sufficiently close to ,
Choose this neighborhood so that . Equation (5.4) implies
Since is strictly decreasing, it can be used as the variable of integration on this interval. The remaining time to reach zero would satisfy
a contradiction. When , the integral in (5.4) is zero and the same argument applies. Thus . The bounded second derivative gives (5.1), and gives (5.2). ∎
Proof of Theorem 1.3. For ,
Thus the radial Hessian condition is precisely , where . The condition at the center holds automatically, because . By Lemma 2.1, . Equation (4.4) gives at zero, while Lemma 5.1 gives
Moreover near , so there. Choose any such point. The limits at both endpoints and continuity imply that the global infimum is achieved on a compact interior interval; its value is strictly between zero and one. Equations (5.7) prove (1.7)--(1.8), and continuity extends concavity to the closed ball. ∎
6. Coefficient comparison and two consequences
The recurrence contains information about variation with the central height. Fix and put . Write for its coefficients.
Proposition 6.1 (Positive parameter dependence). For every , is a polynomial in with nonnegative coefficients and zero constant term. If two solutions have and , then on their common radial interval,
and
Proof. We have . For , rewrite (1.4) as
The polynomial assertion follows by induction. Every monomial of positive degree satisfies . Apply this coefficientwise, including the constant-in- term , and sum the convergent series to obtain (6.1). Integrating gives (6.2). ∎
Corollary 6.2 (Known shooting order and uniqueness). If , their first zero radii satisfy . Consequently (1.1) has exactly one positive solution for every .
Proof. Suppose . By (6.1) and divergence of at , strict inequality is impossible. If , (5.2) gives , contradicting (6.1) and . This proves the strict shooting order. Two distinct positive solutions on the same ball are radial and have either different central values, contradicted by this order, or equal central values, contradicted by initial-value uniqueness. Existence is the prior input stated in Section 2. ∎
As emphasized in Section 1.1, this is an alternative proof of a prior theorem. The coefficientwise parameter comparison (6.1)--(6.2) is the additional conclusion of the present method.
Corollary 6.3 (Products of balls). Let be positive solutions of (1.1) on , for . Then
solves the same equation on the product domain, and is concave there.
Proof. Dividing the Laplacian of the product by the product gives . The negative normalized logarithm is . Its square root is the Euclidean norm of the vector . Each component is nonnegative and convex in its corresponding variables by Theorem 1.2. The argument of (4.1) proves convexity of this norm. ∎
This includes the interval-product construction already treated in [GMS]; the factorization itself is elementary and is not a separate novelty claim.
7. Instability under convex deformation
We prove Theorem 1.4. All boundary-distance and function-space statements below refer to a fixed smooth ball before pulling back the domain.
7.1. A necessary cubic-jet condition
Let near , with , and positive-definite
Hessian . If is convex in a neighborhood of ,
then
Indeed, on any unit direction , write
For , the second derivative of the square root tends to ;
for it tends to . To justify differentiating the remainder,
write with twice continuously differentiable and
, . Convexity on both half intervals forces .
A symmetric cubic tensor whose diagonal polynomial vanishes is zero.
Conversely, if the tensor is nonzero, one half of a line has strictly
negative square-root second derivative at points arbitrarily close to .
The vanishing condition is necessary, not sufficient.
7.2. The ball linearization is invertible
Use Theorem 1.1 and its estimate
The linearized Dirichlet operator is
It satisfies . Since , is its simple first eigenvalue.
For every perpendicular to in ,
Here has a strictly negative outward derivative and is comparable
to the boundary distance . Consequently the logarithmic potential is
form-bounded with relative bound zero on : use
and Hardy's inequality. The operator has
compact resolvent and its form domain is .
For a compactly supported smooth , set . Integration by
parts gives the ground-state identity
The classical strongly log-concave weighted Poincare inequality (see [KS, Theorem 2.1] for the Brascamp-Lieb form on a convex domain), applied
to the density , gives
The relevant boundary version can be obtained without any assumption on a
positive limiting density at . On each smaller concentric
ball, the density is smooth and positive. For its weighted Neumann operator
, integration of the Bochner identity
for an eigenfunction gives
The outward second fundamental form of the sphere is positive.
Equation (D2) therefore bounds each nonconstant weighted Neumann eigenvalue
below by . The spectral expansion yields (D6) on ; let .
First apply this to the smooth compactly supported quotient. Ground-state
identity and form density then extend the result to all functions.
Perpendicularity to is exactly , proving (D4). Thus has
no zero eigenvalue and has exactly one negative eigenvalue.
For use in a shape argument, fix and set
,
. Then
To see the functional-analytic point at the vanishing boundary, choose a
smooth positive defining function equal to near the boundary.
Division by maps continuously to , and
with on the closed ball. Hence multiplication by
maps to . It is compact: for , a bounded sequence in
has a subsequence converging in ; its quotients by
converge in , and multiplication by maps that
convergence to . Thus is a compact perturbation of the
Dirichlet Laplacian isomorphism. Its kernel is zero by (D4) and ;
the Fredholm alternative proves (D8).
7.3. A genuine branch on deformed domains
Let
and define
For all sufficiently small , this is a smooth uniformly convex
domain. Indeed the radial graph converges to the sphere in and its
principal curvatures converge uniformly to ; its outward normal
parametrization, or the usual support-plane criterion, gives convexity.
It converges in every norm because is smooth.
Choose a smooth radial cutoff zero on and one near the
boundary, and use
with the second term zero near the origin. For small parameter this is a
smooth diffeomorphism from the ball to (D9), equal to the identity near zero.
The pulled-back Laplacian has coefficients smooth in and real analytic
in .
Although is not differentiable at as a scalar function,
the Dirichlet nonlinear map is smooth in the spaces just specified near
. On the open positive-Hopf cone write , ; then
The first term is a bounded linear map of into , and the second is
real analytic for bounded away from zero in . Division by
is bounded on . Thus this is a real-analytic map from that open
cone in to . At parameter zero, its equation has derivative (D8).
The Banach implicit-function theorem produces a unique nearby branch
with satisfying the pulled-back equation.
It remains in the positive-Hopf cone. Set
on
; these are genuine positive Dirichlet solutions.
Interior elliptic regularity gives smooth dependence in every local
norm. This does not assert global uniqueness among all positive solutions
on the deformed domains.
7.4. The shape derivative has a nonzero cubic term
Let and let be the Eulerian
shape derivative, . Differentiation of the equation
in each fixed compact subset and of the transported zero boundary value
gives
This is a weak Dirichlet problem with a smooth nonzero boundary trace;
need not be up to the boundary, since the potential is logarithmic
there. It is up to the boundary and smooth in the interior.
Uniqueness follows from (D8), or from the form kernel exclusion. Rotation
invariance and spherical harmonic projection therefore give
where
Near zero, the regular solutions of this equation form a one-dimensional
space and have the expansion
This follows by inserting : satisfies a regular radial
integral equation in dimension , with smooth even potential near zero.
Its initial value determines the solution uniquely. In particular
, since would force , contradicting its boundary value.
In fact : test the radial equation with its negative part times the
same degree-three harmonic; (D4) is positive on this angular subspace, so
that negative part vanishes, and uniqueness excludes a zero leading value.
Thus
7.5. Move the maximum before computing its jet
On a small fixed ball at zero, is negative definite. The nearby
solutions have the same property there. Outside a smaller neighborhood of
zero, has a strict value gap below , which persists under the
uniform pullback convergence. Hence each nearby has a
unique global maximum near zero.
The implicit-function theorem applied to the gradient gives smooth
. Since , differentiation gives
, and therefore . Because the base
solution is radial, . Interior differentiability and (D15) imply
Set
At the maximum the logarithmic third derivative has no lower-derivative
cross terms, since . In particular,
for every sufficiently small nonzero parameter. Its Hessian at the maximum
is positive definite. Apply Section 7.1 to conclude that
is not convex in any neighborhood of the maximum.
Equivalently, is not concave.
The expansion centers at the actual maximum, rather than keeping the old
center fixed; this is essential for the obstruction to apply.
7.6. Ordinary strict log-concavity persists
This additional statement keeps the two concavity properties distinct.
On a fixed interior compact set, (D2) and convergence preserve
positive definiteness of . Near the boundary,
uniform convexity, uniform Hopf bounds and uniform bounds
supply a collar on which the same is true, uniformly for small parameter.
Here is the local estimate.
In normal coordinates , write ,
where is uniformly positive. At the boundary, for a unit tangent ,
.
The bounds and the positive lower curvature bound therefore
give, in the tangent/normal splitting at ,
for fixed positive constants and a sufficiently small common collar.
For the first estimate use that the tangential gradient term in
is nonnegative. For the mixed estimate
use and bounded ; for the normal estimate use the Hopf
bound and . The Schur complement is bounded below by
, hence is positive for smaller . Combining collar and
interior proves strict log-concavity throughout .
8. Ground-state uniqueness on every bounded convex domain
This section leaves the radial setting. Throughout, is nonempty, bounded, open and convex. By [GMS, Theorem 1.1], it admits a positive log-concave solution of
Interior elliptic regularity makes smooth wherever it is positive. We first strengthen its concavity, then compare its energy with every function in . In this section is unnormalized; this changes no Hessian.
8.1. A Gaussian-strength Hessian bound without boundary regularity
Lemma 8.1. Every positive log-concave solution as above satisfies
The strict inequality is pointwise. A uniform positive gap on all of is not part of the assertion.
Proof. Put and . Direct differentiation gives
Initially . For a fixed unit vector , the nonnegative function satisfies
On compact subsets the coefficients are bounded. The strong minimum principle for , with a sufficiently large nonnegative local constant , and propagation through overlapping balls show that a zero of forces . Along a maximal segment parallel to , the function would then be affine. The segment has finite boundary endpoints, while there because is continuous and zero on . This is impossible. Hence throughout .
For set
The strict convexity of gives a unique critical point, its minimum. Thus is a smooth regular level surface. Its second fundamental form is on tangent vectors, so is uniformly convex, and . Moreover,
On the boundary of this fixed level domain, has a nonzero normal derivative. In an inward distance collar write with smooth and positive. Then
Uniform convexity bounds the tangential block below by and the normal block below by for small , while the mixed block stays bounded. Thus at . In dimension one the normal estimate suffices. Its global minimum is consequently attained at an interior point .
The equation for is
Writing and , we calculate
Choose a unit eigenvector for at and keep it fixed when differentiating. The scalar has a minimum at . Differentiating (8.4) twice and using gives there
If , (8.5) gives and . Since , (8.6) implies .
Otherwise lies in the fixed compact set . If is empty, the preceding case suffices. If it is nonempty, set . Equation (8.3) gives at . On we have and uniform bounds on and , so
When , (8.6) therefore gives , and hence . The constants are independent of . Letting at each fixed interior point proves .
To make the inequality strict, set . From (8.2),
For this implies . The same strong minimum principle shows that a zero forces . Then has second derivative exactly one along every segment parallel to . It is quadratic along a maximal such segment and cannot tend to infinity at its finite endpoints. This contradiction proves (8.1).
8.2. A local strict remainder in the logarithmic Sobolev inequality
For a probability measure , write
Lemma 8.2. Put and . There are compact sets , whose union has full Lebesgue measure in , and constants such that, for every ,
Proof. Let . Lemma 8.1 gives . Thus , extended to outside , is proper, lower semicontinuous and convex. Caffarelli's contraction theorem [Caf; DF, Theorem 1.1], with the Gaussian normalization
gives a Brenier transport from to with
Here the theorem applies to an extended-valued convex perturbation. To see the extension explicitly, approximate by smooth convex functions using Moreau envelopes and mollification. Retain a common affine lower bound; the resulting densities have a common integrable Gaussian upper bound and converge almost everywhere to , including outside . Normalize the densities. Their first moments are bounded. For the corresponding contraction maps ,
They therefore have a locally uniformly convergent subsequence. Its limit is a gradient of a convex function, transports to and remains a contraction. Uniqueness of the Brenier map identifies the limit as . This argument uses no regularity of . The later formulation [DF] explicitly credits both the original theorem and its erratum [CafE].
We justify the interior differentiations used next. Set and . The density is globally bounded, say by . The almost-everywhere inverse property of Brenier maps and the area formula for the Lipschitz map give
on any source ball . All eigenvalues in (8.8) are at most one, so (8.9) bounds every eigenvalue below by the same positive constant on . Consequently is locally uniformly convex. Integrating its almost-everywhere Hessian bound, or first mollifying it, shows that is strictly monotone and injective, and has a locally Lipschitz inverse. Invariance of domain makes its image open. Continuity and the transport property put its image in , so in fact it lies in . Its complement there has -measure zero.
The image of each compact source ball is compactly inside . On that ball, the equation
is uniformly elliptic, and its right side is locally Lipschitz. The almost-everywhere equation for the convex potential also holds in the viscosity sense. Interior Evans--Krylov regularity and Schauder bootstrapping give ; see [CC] for the interior concave-equation regularity used here. For applying uniform ellipticity, one can extend from the compact interval of positive Hessians supplied by (8.8)--(8.9) to a uniformly elliptic concave operator. No boundary regularity or pointwise surjectivity of is required.
In fact everywhere. If for a unit eigenvector , the function has a maximum at . Put and differentiate (8.10) twice in direction :
At the left side is nonpositive. On the right, , , and , making it strictly positive. This contradiction proves the strict contraction. Let . These compact sets cover the image of , a full-measure subset of , and for some ,
The Gaussian logarithmic Sobolev inequality [Gross], in the chosen normalization, is
Its constant can be checked by the Ornstein--Uhlenbeck semigroup with generator . For smooth bounded positive bounded away from zero, invariance, integration by parts and the gradient commutation identity give
Integration yields . Applying this to and using approximation proves (8.13), including bounded Lipschitz . These are classical facts, recalled to fix the coefficient.
Apply (8.13) to and use change of variables. By the chain rule and (8.12),
The sets have full-measure union because in . This proves (8.7).
8.3. A sharp energy inequality and its equality case
For real define
with . Both integrals are finite on a bounded domain.
Proposition 8.3. For every nonzero real ,
with equality if and only if for some nonzero real constant .
Proof. First let and . Testing the weak equation of with gives the ground-state identity
Writing , the definition of entropy then gives exactly
Thus Lemma 8.2 proves, for every ,
Now take strongly in . Choose a small with strictly below the Sobolev exponent. The bound
and strong Sobolev convergence show uniform integrability and hence convergence of the entropy integrals. The same argument uses any suitable finite Sobolev exponent when . Consequently the left side of (8.18) converges to . On a neighborhood of each , division by the smooth positive is continuous in , so the right side converges as well. Formula (8.18) holds for every .
This passage does not assume that has finite global weighted Dirichlet energy or that is integrable. Only the compactly supported identity (8.16) and convergence on compact subsets are used. In particular (8.15) holds. If its left side vanishes, (8.18) forces on every , hence almost everywhere in . Since and is connected, .
Conversely testing the equation with gives . Direct scaling yields , so every nonzero multiple attains equality in (8.15).
8.4. Ground states and the remaining uniqueness question
The action corresponding to the equation is
It is on ; its critical points are the weak solutions. The nonzero Nehari set is . On this set . This agrees with the ground-state definition in [GMS, Section 4.3], restricted there to nonnegative functions.
By (8.15), every has . The existing solution attains and therefore minimizes the action on . If equality holds, Proposition 8.3 gives , and forces . Thus the real Nehari minimizers, and the real ground states, are precisely and . In particular the positive ground state is unique. This proves Theorem 1.5.
The same proposition gives the equivalent constrained characterization
Every positive log-concave solution satisfies Lemma 8.1 and the ensuing comparison, so every such solution is this unique ground state. In particular the solutions in Theorem 1.4 are ground states on their respective deformed domains. Their loss of normalized square-root-log concavity therefore occurs already among ground states.
This answers the ground-state uniqueness and ground-state identification questions in [GMS, Remark 1.6]. The existence theorem of [GMS] remains an essential input. Uniqueness among all positive solutions would additionally require excluding positive solutions of greater action; nothing above supplies that exclusion.
9. Further questions
Uniqueness among all positive solutions on an arbitrary bounded convex domain remains open here: Theorem 1.5 rules out additional ground states, but does not exclude positive solutions of higher energy. The optimal positive-power exponent on balls in higher dimensions remains implicit, and its radius dependence is unresolved. Theorem 1.4 settles the normalized square-root-log deformation question negatively, now for ground states by Theorem 1.5; the weaker property allowing sufficiently small rescaling remains a separate question.
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[DF] G. De Philippis and A. Figalli, Rigidity and stability of Caffarelli's log-concave perturbation theorem, Nonlinear Analysis 154 (2017), 59--70. DOI: 10.1016/j.na.2016.10.006. Theorem 1.1 restates the contraction theorem, citing both [Caf] and [CafE].
[Gross] L. Gross, Logarithmic Sobolev inequalities, American Journal of Mathematics 97 (1975), 1061--1083.
[CC] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications 43, 1995.
Preparation note
Henry Zweiman. Revision 1.2, September 16, 2026. This revision adds Theorem 1.5 and Section 8, proving ground-state uniqueness on every bounded convex domain and identifying every log-concave solution as that ground state. The ball and deformation proof bodies in Sections 2--7 are retained. The argument is by hand; no finite computation is used as a proof of a universal claim. Proof and source checks are internal and do not constitute independent expert review, exhaustive priority certification or journal acceptance.