Sharp Planar Anisotropic Capillarity Outside Convex Obstacles and Two-Point Rigidity of Transported Boundary Images
Abstract
For a convex surface tension φ and the wall-energy density σ(n)=−λ φ(−n), we prove sharp capillary inequalities outside arbitrary convex planar obstacles for −1<λ≤0 and every finite-perimeter competitor of finite positive volume, including non-even and crystalline tensions. Equality holds precisely for translated dilates of Wulff caps in minimum-volume directions, with the entire truncating facet resting on a flat supporting face. A uniform quantitative estimate controls component concentration, shape up to translation, and directional cap-volume excess. The proof combines exact filling, planar completion, and quantitative Wulff stability. For the transport DΦ, Φ=φ2/2, the zero-adhesion boundary-image estimate characterizes centered ellipsoids among even anisotropies with Φ∈ C3 away from the origin and positive definite Hessian. Two-point slab tests with arbitrary boundary data suffice: their projection balances yield a cubic identity forcing Φ to be quadratic. Facets and smooth perturbations provide counterexamples to the transported estimate, including failures at prescribed adhesion levels in (−1,1). Strict finite deficits persist under Hausdorff perturbations of the Wulff body. The classification concerns an unrestricted boundary-data property, stronger than the image estimate needed only for traces of auxiliary Neumann solutions.
Related
- Generalized minimizers for the capillarity energy outside convex obstacles
- Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
- Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
- Wulff Rigidity for Overdetermined Anisotropic k-Hessian Equations
- The anisotropic Calderón problem: rigidity near the Euclidean metric
BibTeX
@misc{carrero2026sharp,
title = {{Sharp Planar Anisotropic Capillarity Outside Convex Obstacles and Two-Point Rigidity of Transported Boundary Images}},
author = {Lisbeth Carrero and Alexander Quaas and Andres Zuniga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37440},
eprint = {2610.37440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37440},
}
RIS
TY - UNPB
TI - Sharp Planar Anisotropic Capillarity Outside Convex Obstacles and Two-Point Rigidity of Transported Boundary Images
AU - Lisbeth Carrero
AU - Alexander Quaas
AU - Andres Zuniga
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37440
UR - https://arxiv.org/abs/2610.37440
AB - For a convex surface tension φ and the wall-energy density σ(n)=−λ φ(−n), we prove sharp capillary inequalities outside arbitrary convex planar obstacles for −1<λ≤0 and every finite-perimeter competitor of finite positive volume, including non-even and crystalline tensions. Equality holds precisely for translated dilates of Wulff caps in minimum-volume directions, with the entire truncating facet resting on a flat supporting face. A uniform quantitative estimate controls component concentration, shape up to translation, and directional cap-volume excess. The proof combines exact filling, planar completion, and quantitative Wulff stability. For the transport DΦ, Φ=φ^2/2, the zero-adhesion boundary-image estimate characterizes centered ellipsoids among even anisotropies with Φ∈ C^3 away from the origin and positive definite Hessian. Two-point slab tests with arbitrary boundary data suffice: their projection balances yield a cubic identity forcing Φ to be quadratic. Facets and smooth perturbations provide counterexamples to the transported estimate, including failures at prescribed adhesion levels in (−1,1). Strict finite deficits persist under Hausdorff perturbations of the Wulff body. The classification concerns an unrestricted boundary-data property, stronger than the image estimate needed only for traces of auxiliary Neumann solutions.
ER -