An anisotropic Serrin's problem in general domains

Abstract: Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more generally rough) domains in the weak formulation has been a long-standing and challenging problem, recently resolved by the authors in [12]. In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex $C<sup>{2,γ}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $Δ_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $Ω$. Assuming the Ahlfors--David regularity of $\partial<sup>*Ω$ and a global $β$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $Ω$ is a translate and dilation of the Wulff shape, in which case the solution is unique and explicit. In particular, the result applies to Lipschitz domains. While our approach follows the rough-domain strategy of [12] at a high level, the key Laplacian-specific ingredients exploited there have no direct analog for $Δ_H$, necessitating the development of new ideas and techniques.

• The paper proves that the weak anisotropic overdetermined problem is solvable exactly when a bounded indecomposable finite-perimeter domain is a translate and dilation of the Wulff shape K.
• The authors combine blow-up analysis, Hessian localization, a dilation-based volume identity, Green functions, and an anisotropic P-function argument to overcome the lack of global W²,² regularity.
• The result applies to Lipschitz and uniformly rectifiable domains satisfying Ahlfors–David regularity and a Jones β-square-function bound, while leaving weaker rectifiability and degenerate anisotropies open.

Overview

This paper by Figalli and Zhang extends Serrin's overdetermined symmetry principle to the anisotropic setting for domains that are merely rough — bounded indecomposable sets of finite perimeter satisfying a weak uniform rectifiability condition. The operator in question is the anisotropic Laplacian $$\Delta_H u = \operatorname{div}(H(\nabla u)\,DH(\nabla u))$$, where $H$ is the Wulff potential associated with a uniformly convex $C^{2,\gamma}$ body $K$. The main result states that the overdetermined problem admits a weak solution if and only if $\Omega$ is homothetic to $K$, with the explicit solution $u(x) = (r^2 - H_*^2(x))/(2n)$.

The classical smooth theory is well developed: Serrin's original result [S1971] and Weinberger's alternative proof [W1971] treat the Euclidean ball, while Cianchi–Salani [CS2009] and Wang–Xia [WX2011] independently established the anisotropic rigidity for classical solutions on smooth domains. The genuinely new contribution here is the extension to rough domains. In the isotropic case this was recently achieved by the same authors [FZ2025], answering a question posed in [HLL2024]; alternative approaches via non-tangential limits and Alt–Caffarelli free boundary regularity appear in [DZ2025, DR2026]. All three of those proofs rely essentially on structural identities specific to the Laplacian, which have no direct analog for $\Delta_H$. Moreover, the harmonic-analysis strategy of [DZ2025] faces intrinsic obstructions: solvability of regularity problems for divergence-form operators in general Lipschitz domains remains open except under small-Lipschitz-constant assumptions [DPR2017]. This motivates the geometric-measure-theoretic route pursued here.

Main theorem

Let $\Omega \subset \mathbb{R}^n$ be a bounded indecomposable set of finite perimeter whose reduced boundary satisfies:

• Ahlfors–David regularity: $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$ for all $H$0, $H$1;
• Jones-type square-function bound: $H$2, where $H$3 measures the best $H$4 approximation of $H$5 at scale $H$6 by affine hyperplanes.

Then the weak overdetermined system

$H$7

with $H$8, is solvable if and only if $H$9 is a translate and dilation of $C^{2,\gamma}$0, in which case $C^{2,\gamma}$1.

Since Lipschitz domains, and more generally uniformly $C^{2,\gamma}$2-rectifiable sets, satisfy both hypotheses [DS1991, DS1993], the theorem yields the anisotropic Serrin theorem for Lipschitz domains. Two useful relaxations are noted: the lower Ahlfors bound is only used to control "bad" balls, so domains with finitely many cusps can also be handled by excising neighborhoods of the cusps; and the upper Ahlfors bound implies finite $C^{2,\gamma}$3-dimensional upper Minkowski content of $C^{2,\gamma}$4, which is used repeatedly.

Regularity of weak solutions

The first structural lemma establishes that any weak solution $C^{2,\gamma}$5 is nonnegative, globally Lipschitz, identifies $C^{2,\gamma}$6 with $C^{2,\gamma}$7 up to null sets, and exhibits linear boundary growth. The Lipschitz bound is obtained by contradiction: rescaled functions violating the growth estimate converge to nontrivial nonnegative $C^{2,\gamma}$8-harmonic functions vanishing at a point, contradicting the strong maximum principle; global Lipschitz continuity then follows from interior gradient estimates combined with the boundary growth.

Two further facts carry substantial weight downstream. First, the blow-up analysis at reduced-boundary points shows that

$C^{2,\gamma}$9

using nonlinear potential estimates [KM12, KM1994] for Hölder continuity and the anisotropic Liouville classification of one-phase solutions on half-spaces. Second, a small-scale Hessian vanishing property holds: for every interior ball $K$0 tangent to the boundary at scale $K$1, $K$2 for $K$3 small enough. This follows from uniform interior $K$4 estimates plus compactness against the affine blow-up limit. These two properties substitute for the global $K$5 regularity that would make the subsequent chain-rule arguments immediate but which is unavailable here even on Lipschitz domains.

The volume identity and the Hessian localization argument

The cornerstone is Weinberger's volume identity, proved here without Pohozaev-type tools:

$K$6

Testing the equation against the dilation difference quotient $K$7, the delicate term is

$K$8

If $K$9 globally, integration by parts transfers the difference quotient onto $\Omega$0 directly. Since only local $\Omega$1 regularity is available, the authors prove a fine localization lemma: along a suitable sequence $\Omega$2,

$\Omega$3

with $\Omega$4.

The proof exploits the Hölder continuity of $\Omega$5 on the sphere to reduce matters to controlling $\Omega$6, decomposed into three regions relative to the boundary layer. The interior region is handled by Besicovitch covering arguments using the finite Minkowski content; the near-boundary region by the Lipschitz bound; and the intermediate region — the crux — by combining the small-scale Hessian estimate with a covering lemma (Proposition prop:ADR below) that provides, at selected scales $\Omega$7, a cover of the boundary layer in which all but $\Omega$8 balls are simultaneously $\Omega$9-flat and density-correct. Iterating limits in $K$0, then $K$1, then $K$2 yields the identity. This is precisely where the square-function assumption enters the proof, and it constitutes the main technical innovation replacing Laplacian-specific structure.

Green function, harmonic measure, and the $K$3-function argument

On the open representative of $K$4, the matrix field $K$5 is bounded and uniformly elliptic, defining the linearized operator $K$6. The Green function $K$7 is constructed by exhaustion over smooth subdomains. It satisfies $K$8 away from the pole — hence linear decay toward $K$9 — and its distributional Laplacian is a Radon measure absolutely continuous with respect to $u(x) = (r^2 - H_*^2(x))/(2n)$0:

$u(x) = (r^2 - H_*^2(x))/(2n)$1

This gives a representation formula for $u(x) = (r^2 - H_*^2(x))/(2n)$2-harmonic functions in terms of approximate boundary traces, generalizing the corresponding Laplacian construction in [FZ2025].

Applying this to directional derivatives $u(x) = (r^2 - H_*^2(x))/(2n)$3 (which are $u(x) = (r^2 - H_*^2(x))/(2n)$4-harmonic by differentiating the equation) and using the boundary trace $u(x) = (r^2 - H_*^2(x))/(2n)$5 together with the support function inequality $u(x) = (r^2 - H_*^2(x))/(2n)$6 for $u(x) = (r^2 - H_*^2(x))/(2n)$7, one obtains the sharp maximum principle

$u(x) = (r^2 - H_*^2(x))/(2n)$8

This step is a genuine replacement for the isotropic argument, where the analogous bound follows from elementary means unavailable here.

Finally, the Weinberger $u(x) = (r^2 - H_*^2(x))/(2n)$9-function $\Delta_H$0 satisfies

$\Delta_H$1

the last inequality being Cauchy–Schwarz applied to $\Delta_H$2. Subharmonicity, the maximum principle on superlevel sets $\Delta_H$3, and the gradient bound give $\Delta_H$4 pointwise. Integrating and invoking the volume identity forces $\Delta_H$5, so equality holds in Cauchy–Schwarz everywhere:

$\Delta_H$6

As in [CS2009], this rigidity implies $\Delta_H$7 up to translation, and $\Delta_H$8 is homothetic to $\Delta_H$9.

Covering lemma under ADR with logarithmic gain

The appendix proves a covering proposition of independent interest. Under ADR and the global $\Omega \subset \mathbb{R}^n$0-square bound, for each $\Omega \subset \mathbb{R}^n$1 there exist an Egorov-type good set $\Omega \subset \mathbb{R}^n$2 (of measure deficit $\Omega \subset \mathbb{R}^n$3, where density ratios are within $\Omega \subset \mathbb{R}^n$4 at all small scales) and a sequence of scales $\Omega \subset \mathbb{R}^n$5 admitting Besicovitch covers of the boundary layer such that the number of centers failing either $\Omega \subset \mathbb{R}^n$6-flatness or density correctness is at most

$\Omega \subset \mathbb{R}^n$7

The $\Omega \subset \mathbb{R}^n$8 gain comes from a dyadic averaging argument: since $\Omega \subset \mathbb{R}^n$9 over dyadic annuli, some scale in $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$0 carries $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$1, and averaging the bad-center count in $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$2 over that interval produces the logarithmic improvement. Density-bad centers are controlled via a Lipschitz dependence of $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$3 on its center, reducing them to the complement of $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$4.

Limitations and open questions

Several restrictions are inherent to the framework. The anisotropy must come from a uniformly convex $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$5 Wulff shape with $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$6 small, ensuring $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$7 is uniformly elliptic and Hölder continuous on the sphere; degenerate or less smooth anisotropies fall outside the method. The domain hypotheses — ADR plus the global $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$8-square bound — exclude boundaries with infinitely many cusps or other non-ADR features, although finitely many cusps can be excised as noted above. The paper leaves open whether global $$A_2^{-1} r^{n-1} \le \mathscr{H}^{n-1}(B_r(x) \cap \partial^*\Omega) \le A_2\, r^{n-1}$$9 regularity holds for $H$00 on Lipschitz domains, which would substantially simplify the localization argument; it also does not address whether the harmonic-analysis approach of [DZ2025] can overcome the solvability obstruction for regularity problems of general divergence-form operators in Lipschitz domains. Whether the rigidity conclusion persists under weaker rectifiability assumptions than the $H$01-square bound is not settled here.

Conclusion

The paper establishes the anisotropic Serrin rigidity theorem for bounded indecomposable sets of finite perimeter satisfying Ahlfors–David regularity and a Jones square-function bound, thereby covering all Lipschitz domains. The proof adapts the geometric-measure-theoretic strategy of the isotropic case while developing the necessary substitutes for Laplacian-specific ingredients: a dilation-difference-quotient volume identity justified through a $H$02-number-driven localization of Hessian estimates, a Green-function representation for the linearized uniformly elliptic operator, and an anisotropic maximum principle for $H$03. Together these yield the sharp classification $H$04 with the explicit Wulff-shaped solution, confirming the anisotropic conjecture formulated in [DZ2025] under weak uniform rectifiability assumptions.