Convexity for free boundaries with singular term (nonlinear elliptic case)

2022/11/18 by Seongmin Jeon, Henrik Shahgholian, Jeon, Seongmin +1
Computer Science · Mathematics · #35E10 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2211.10434

openalex publication_date 2022/11/18 · openalex created_date 2022/11/28 · openalex updated_date 2026/07/28

Abstract

We consider a free boundary problem in an exterior domain \begincasesLu=g(u) · amp; in Ω∖ K,
u=1 · amp; on ∂ K,
|∇ u|=0 · amp;on ∂ Ω,\endcases where K is a (given) convex and compact set in ℝⁿ (n≥2), Ω=\u>0\⊃ K is an unknown set, and L is either a fully nonlinear or the p-Laplace operator. Under suitable assumptions on K and g, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set K is contained in \x_n=0\, and obtain similar results.

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