Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data
Abstract
Let p>1 and Ω⊂ℝn be a bounded smooth domain. We study the following overdetermined problem for the p-Laplacian: { ll −Δp u=f(x)&x∈Ω, u=0&x∈∂Ω, (∂ u)/∂ν=g(x)&x∈∂Ω, . where ν is the inward unit normal to ∂Ω and both data may depend on position. A boundary-local version of the moving-plane argument is developed for the possibly degenerate p-Laplace operator. The advantage of this argument is that it avoids the analysis of the degenerate set of the p-Laplacian. Under some reasonable structural conditions, we prove that the solvability of the above mentioned problem forces the domain to be a ball, the source to be radial, and the Neumann datum to be a suitable constant. We also give an explicit formula for the solution. Further consequences include one directional domain symmetry, the conclusion that full rigidity follows from reflection information in any set of directions whose linear reflections generate a dense subgroup of the orthogonal group O(n), and a constant mean curvature characterization.
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BibTeX
@misc{dai2026rigidity,
title = {{Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data}},
author = {Xi Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16686},
eprint = {2610.16686},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16686},
}
RIS
TY - UNPB
TI - Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data
AU - Xi Dai
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16686
UR - https://arxiv.org/abs/2610.16686
AB - Let p>1 and Ω⊂ℝ^n be a bounded smooth domain. We study the following overdetermined problem for the p-Laplacian: { ll −Δ_p u=f(x)&x∈Ω, u=0&x∈∂Ω, (∂ u)/∂ν=g(x)&x∈∂Ω, . where ν is the inward unit normal to ∂Ω and both data may depend on position. A boundary-local version of the moving-plane argument is developed for the possibly degenerate p-Laplace operator. The advantage of this argument is that it avoids the analysis of the degenerate set of the p-Laplacian. Under some reasonable structural conditions, we prove that the solvability of the above mentioned problem forces the domain to be a ball, the source to be radial, and the Neumann datum to be a suitable constant. We also give an explicit formula for the solution. Further consequences include one directional domain symmetry, the conclusion that full rigidity follows from reflection information in any set of directions whose linear reflections generate a dense subgroup of the orthogonal group O(n), and a constant mean curvature characterization.
ER -