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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Veröffentlicht in:Nonlinear analysis Jg. 201; S. 112090
Hauptverfasser: Boukarabila, Y. Oussama, Véron, Laurent
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elmsford Elsevier Ltd 01.12.2020
Elsevier BV
Elsevier
Schlagworte:
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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Zusammenfassung: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}.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2020.112090