Quasilinear parabolic equations with nonlinear Wentzell–Robin type boundary conditions

Let Ω⊂RNΩ⊂RN be a bounded domain with Lipschitz boundary, a∈C(Ω¯) with a>0a>0 on Ω¯. Let σ be the restriction to ∂Ω   of the (N−1)(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∈∂Ωx∈∂Ω, B(x,⋅)B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞)p∈(1,∞), the operator Ap:=div(a|∇u|p−2∇u)Ap:=div(a|∇u|p−2∇u) with nonlinear Wentzell–Robin type boundary conditionsApu+b|∇u|p−2∂u∂n+β(⋅,u)∋0on ∂Ω, generates a nonlinear submarkovian C0C0-semigroup on suitable L2L2-space. Here n(x)n(x) denotes the unit outer normal at x and for σ  -a.e. x∈∂Ωx∈∂Ω the maximal monotone graph β(x,⋅)β(x,⋅) denotes the subdifferential ∂B(x,⋅)∂B(x,⋅) of the functional B(x,⋅)B(x,⋅). We also assume that b∈L∞(∂Ω)b∈L∞(∂Ω) and satisfies b(x)⩾b0>0b(x)⩾b0>0σ-a.e. on ∂Ω   for some constant b0b0. As a consequence we obtain that there exist consistence nonexpansive, nonlinear semigroups on suitable LqLq-spaces for all q∈[1,∞)q∈[1,∞). In the second part we show some domination results.