Sharp estimates for solutions to elliptic problems with mixed boundary conditions

2020/09/09 by Alvino, A., Chiacchio, F., Nitsch, C. +1 · 1 citation
#35B06 #35B45 #35J05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.04320

Abstract

We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on L¹ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set Ω with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as Ω. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.

Cited by

• Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian

Related

• PDE Comparison Principles for Robin Problems
• Mixed boundary value problems for fully nonlinear degenerate or singular equations
• A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions
• Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results
• Comparison results for semilinear elliptic equations using a new symmetrization method