Solutions entropiques des problèmes non locaux

The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

“…By [40, Corollary 2.1.9] and since is connected, P ϕ 1 = P ϕ 2 + c on for some c ∈ R. This means that P ϕ 1 |∂ = ϕ 1 = ϕ 2 + c on ∂ . However, ϕ 1 and ϕ 2 satisfy condition (3). This implies that c = 0 and thereby we have shown that ϕ 1 = ϕ 2 on ∂ , proving that is strictly monotone on the space W…”

“…Results about existence and uniqueness of entropy solution of elliptic and parabolic equations involving the operator in L 1 (∂ ) have been first announced in [2] and later established in the thesis [3] (see also [4,1]). …”

“…Further, let ψ ∈ L q (∂ ) satisfying (3). Then there are α ∈ (0, 1) and c α ≥ 0 such that every weak solution ϕ ∈ W 1−1/p,p (∂ ) of Eq.…”

“…But to the best of our knowledge, this was not known so far for general Dirichlet-to-Neumann operators associated with second order quasi-linear operators of Leray-Lions type. It was shown in [3] that the entropy solutions operator associated with 1 is m-completely accretive in L 1 (∂ ). In this article, we complement this result by establishing in Proposition 3.12 that the part c of in C(∂ ) is m-completely-accretive in C(∂ ).…”

“…An additional condition on ϕ ∈ W 1−1/p,p (∂ ) would be compatibility condition (3). As a result, the space W 1−1/p,p m (∂ ) has been introduced by us in Section 1.…”