Nonlinear boundary value problems relative to harmonic functions

We study the problem of finding a function u verifying −Δu=0 in Ω under the boundary condition ∂u∂n+g(u)=μ on ∂Ω where Ω⊂RN is a smooth domain, n is the normal unit outward vector to Ω, μ is a measure on ∂Ω and g a continuous nondecreasing function. We give sufficient condition on g for this problem...

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Vydané v:	Nonlinear analysis Ročník 201; s. 112090
Hlavní autori:	Boukarabila, Y. Oussama, Véron, Laurent
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elmsford Elsevier Ltd 01.12.2020 Elsevier BV Elsevier
Predmet:	Analysis of PDEs Boundary conditions Boundary value problems Continuity (mathematics) Dirichlet to Neumann operator Harmonic functions Laplace–Beltrami operator Limit set Mathematics Radon measure Singularities Singularity (mathematics) 35J65 Dirichlet to Neumann operator Radon measure 35L71 Laplace–Beltrami operator Singularities Limit set Radon Measure Sin- gularities Laplace-Beltrami operator
ISSN:	0362-546X, 1873-5215
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Popis
Shrnutí:	We study the problem of finding a function u verifying −Δu=0 in Ω under the boundary condition ∂u∂n+g(u)=μ on ∂Ω where Ω⊂RN is a smooth domain, n is the normal unit outward vector to Ω, μ is a measure on ∂Ω and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r)=\|r\|p−1r, p>1, we give conditions in order an isolated singularity on ∂Ω to be removable. We also give capacitary conditions on a measure μ in order the problem with g(r)=\|r\|p−1r to be solvable for some μ. We also study the isolated singularities of functions satisfying −Δu=0inΩ and ∂u∂n+g(u)=0 on ∂Ω∖{0}.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0362-546X 1873-5215
DOI:	10.1016/j.na.2020.112090