An Overdetermined Neumann boundary value problem with a general driving force

2024/05/11 by Minlend, Ignace Aristide, Wu, Jing
#2020: 35J57 #35J25 #35J66 #35N25 #35R35 #58J55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.07063

Abstract

In this paper, we prove the existence of a family of non trivial compact subdomains Ø in the manifold M=\R^N× \R/2π\Z, N≥ 2 for which the overdetermined Neumann boundary value problem \ \beginaligned -\D wamp;=μg(w) amp;amp; in Ω, (∂ w)/(∂η) amp;=0 amp;amp; on ∂ Ω, wamp;=c≠ 0 amp;amp; on ∂ Ω, \endaligned . admits solutions for some μ> 0 and a C^{1, α} function g:\R → \R. The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions g. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \citeFall-MinlendI-Weth4, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.

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