Nonlinear boundary value problems in connexion with harmonic functions

Abstract: We study the problem of finding a functionuverifying−∆u= 0 inΩ under the boundary condition∂u∂n+g(u) =μon∂Ω where Ω⊂RNis a smoothdomain,nis the normal unit outward vector to Ω,μis a measure on∂Ω andga continuous nondecreasing function. We give sufficient condition ongfor thisproblem to be solvable for any measure. Wheng(r) =|r|p−1r,p >1, we giveconditions in order an isolated singularity on∂Ω to be removable. We also givecapacitary conditions on a measureμin order the problem withg(r) =|r|p−1rto be solvable for someμ. We also study the isolated singularities of functionssatisfying−∆u= 0inΩ and∂u∂n+g(u) = 0 on∂Ω\{0}.

analysis of PDEsdynamical systemsfunctional analysisoptimization and controlspectral theory

Audience: advanced learners

Webinar on PDE and related areas

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