p-Laplace Serrin problem on planar annuli
Abstract
We prove a rigidity result for the problem Δp u=−2 defined on a ring-shaped domain in ℝ2, when homogeneous Dirichlet boundary conditions are imposed and p∈(1,2). This problem admits a natural physical interpretation, as it describes the steady laminar flow of a generalised Newtonian fluid exhibiting shear-thinning behaviour. Our symmetry result is obtained assuming that the set of maximum points, denoted by Σ, is a smooth closed curve, and there are no other critical points outside it; moreover, an appropriate asymptotic expansion of the solution near Σ is required. Although our approach is inspired by the comparison strategy of [arXiv:2505.04458], the main ingredients used there are intrinsically linear and have no direct counterparts for the p-Laplacian. This requires the introduction of new ideas and techniques.
Related
- Isolated singularities of weak solutions to the Serrin-critical p-Laplace equation
- Serrin-type overdetermined problem for the p-Laplacian with Robin boundary conditions
- Dynamical quasi-symmetry of the classical Kepler-Coulomb problem and the conservation of the Laplace-Runge-Lenz vector
- Hyperbolic scattering in the planar n-body problem
- C-networks and the planar Ising inverse problem
BibTeX
@misc{agostiniani2026plaplace,
title = {{$p$-Laplace Serrin problem on planar annuli}},
author = {Virginia Agostiniani and Chiara Bernardini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20861},
eprint = {2610.20861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20861},
}
RIS
TY - UNPB
TI - p-Laplace Serrin problem on planar annuli
AU - Virginia Agostiniani
AU - Chiara Bernardini
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20861
UR - https://arxiv.org/abs/2610.20861
AB - We prove a rigidity result for the problem Δ_p u=−2 defined on a ring-shaped domain in ℝ^2, when homogeneous Dirichlet boundary conditions are imposed and p∈(1,2). This problem admits a natural physical interpretation, as it describes the steady laminar flow of a generalised Newtonian fluid exhibiting shear-thinning behaviour. Our symmetry result is obtained assuming that the set of maximum points, denoted by Σ, is a smooth closed curve, and there are no other critical points outside it; moreover, an appropriate asymptotic expansion of the solution near Σ is required. Although our approach is inspired by the comparison strategy of [arXiv:2505.04458], the main ingredients used there are intrinsically linear and have no direct counterparts for the p-Laplacian. This requires the introduction of new ideas and techniques.
ER -