The heat equation with nonlinear generalized Robin boundary conditions

Let be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H1-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))t 0 on L2(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v Lp(Ω), p 2 and every t>0, one has for some constants C1,C2 0. In the second part, we prove that if Ω is a bounded Lipschitz domain, one can also define a realization of the Laplacian with nonlinear Robin boundary conditions on and this operator generates a strongly continuous and contractive nonlinear semigroup.