A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p ∈ ( 1 , N ) , Ω ⊂ R N a bounded W 1 , p -extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d ∈ ( N − p , N ) . We show in the first part that for every p ∈ [ 2 N / ( N + 2 ) , N ) ∩ ( 1 , N ) , a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L 2 ( Ω ) , and hence, the associated first order Cauchy problem is well posed on L q ( Ω ) for every q ∈ [ 1 , ∞ ) . In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.