Quasilinear parabolic equations with nonlinear Wentzell–Robin type boundary conditions

Abstract

Let Ω ⊂ R N be a bounded domain with Lipschitz boundary, a ∈ C ( Ω ¯ ) with a > 0 on Ω ¯ . Let σ be the restriction to ∂ Ω of the ( N − 1 ) -dimensional Hausdorff measure and let B : ∂ Ω × R → [ 0 , + ∞ ] be σ-measurable in the first variable and assume that for σ-a.e. x ∈ ∂ Ω , B ( x , ⋅ ) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p ∈ ( 1 , ∞ ) , the operator A p : = div ( a | ∇ u | p − 2 ∇ u ) with nonlinear Wentzell–Robin type boundary conditions A p u + b | ∇ u | p − 2 ∂ u ∂ n + β ( ⋅ , u ) ∋ 0 on ∂ Ω , generates a nonlinear submarkovian C 0 -semigroup on suitable L 2 -space. Here n ( x ) denotes the unit outer normal at x and for σ-a.e. x ∈ ∂ Ω the maximal monotone graph β ( x , ⋅ ) denotes the subdifferential ∂ B ( x , ⋅ ) of the functional B ( x , ⋅ ) . We also assume that b ∈ L ∞ ( ∂ Ω ) and satisfies b ( x ) ⩾ b 0 > 0 σ-a.e. on ∂ Ω for some constant b 0 . As a consequence we obtain that there exist consistence nonexpansive, nonlinear semigroups on suitable L q -spaces for all q ∈ [ 1 , ∞ ) . In the second part we show some domination results.