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 ∈ [2N/ (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.

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