Nonlinear boundary value problems relative to harmonic functions

International audienceWe study the problem of finding a function u verifying −∆u = 0 in Ω under the boundary condition ∂u ∂n + g(u) = µ on ∂Ω where Ω ⊂ R N is a smooth domain, n 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−1 r, p > 1, we give conditions in order an isolated singularity on ∂Ω be removable. We also give capacitary conditions on a measure µ in order the problem with g(r) = |r| p−1 r to be solvable for some µ. We also study the isolated singularities of functions satisfying −∆u = 0 in Ω and ∂u ∂n + g(u) = 0 on ∂Ω \ {0}