A rigidity result for the weighted p-Laplace equation in higher dimensions
Abstract
We study rigidity for the weighted p-Laplace equation on smooth bounded domains in dimensions three and higher. We show that a smooth positive weight sufficiently close to $1$ is identically $1$ if the derivative of its Dirichlet-to-Neumann map at one affine boundary value agrees with that for the unit weight. We also prove local rigidity at every smooth positive reference weight which is constant in one direction. The reference weight need not be close to $1$, and the unknown weight is only required to be sufficiently close to it, without a directional restriction. No convexity, analyticity, or agreement near the boundary is assumed. The proof combines linearization at a solution without critical points with recent rigidity results for the anisotropic Calderón problem. An identity for the linearized conductivity tensors removes the remaining diffeomorphism. This last step does not require smallness of either weight.
Related
- A Liouville type rigidity result for entire solutions of the k-Hessian equation
- Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data
- Unique continuation, nonexistence and bubbling for the critical p-Laplace equation II: the degenerate elliptic case and higher dimensions
- Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the p-Laplace Lane--Emden equation
- A rigidity result for a global vectorial free boundary problem with a linear datum
BibTeX
@misc{lin2026rigidity,
title = {{A rigidity result for the weighted p-Laplace equation in higher dimensions}},
author = {Ching-Lung Lin and Yi-Hsuan Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05866},
eprint = {2610.05866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05866},
}
RIS
TY - UNPB
TI - A rigidity result for the weighted p-Laplace equation in higher dimensions
AU - Ching-Lung Lin
AU - Yi-Hsuan Lin
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05866
UR - https://arxiv.org/abs/2610.05866
AB - We study rigidity for the weighted p-Laplace equation on smooth bounded domains in dimensions three and higher. We show that a smooth positive weight sufficiently close to $1$ is identically $1$ if the derivative of its Dirichlet-to-Neumann map at one affine boundary value agrees with that for the unit weight. We also prove local rigidity at every smooth positive reference weight which is constant in one direction. The reference weight need not be close to $1$, and the unknown weight is only required to be sufficiently close to it, without a directional restriction. No convexity, analyticity, or agreement near the boundary is assumed. The proof combines linearization at a solution without critical points with recent rigidity results for the anisotropic Calderón problem. An identity for the linearized conductivity tensors removes the remaining diffeomorphism. This last step does not require smallness of either weight.
ER -