The heat equation with nonlinear generalized Robin boundary conditions

Abstract

Let Ω ⊂ R N be a bounded domain and let μ be an admissible measure on ∂ Ω. We show in the first part that if Ω has the H 1 -extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by ∂ u ∂ ν d σ + β ( x , u ) d μ = 0 on ∂ Ω, generates a strongly continuous nonlinear submarkovian semigroup S B = ( S B ( t ) ) t ⩾ 0 on L 2 ( Ω ) . We also obtain that this semigroup is ultracontractive in the sense that for every u , v ∈ L p ( Ω ) , p ⩾ 2 and every t > 0 , one has ‖ S B ( t ) u − S B ( t ) v ‖ ∞ ⩽ C 1 e C 2 t t − N 2 p ‖ u − v ‖ p , for some constants C 1 , C 2 ⩾ 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 C ( Ω ¯ ) and this operator generates a strongly continuous and contractive nonlinear semigroup.