Potential Analysis

, Volume 24, Issue 1, pp 15–45

# A Fully Nonlinear Nonhomogeneous Neumann Problem

Dedicated to the memory of the Professor Philippe Bénilan
Article

## Abstract

Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, fL 1(ℝN), gL 1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
$$\mathbf{(P)}\quad \left\{\begin{array}{l}-\mathrm{div}[a(x,\nabla u)]+\beta(u)=f\quad \mbox{in}\ \mathbb{R}^{N}\setminus \partial \Omega,\cr \bigl[\frac{\partial u}{\partial \nu_{a}}\big]+\gamma(\tau u)=g\quad \hbox{on}\ \partial \Omega,\cr [u]=0\end{array}\right.$$
in the sense that, if T k(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DT k u on the set {|u|<k}) and $$\mathcal{T}^{1,p}(\mathbb{R}^{N})=\{u:\mathbb{R}^{N}\rightarrow \mathbb{R}$$ , u measurable; DT k(u)∈L p(ℝN), k>0}, then $$u\in \mathcal{T}^{1,p}(\mathbb{R}^{N})$$ and u satisfies,
$$\int_{\mathbb{R}^{N}}\langle a(\cdot,Du),DT_{k}(u-\varphi)\rangle \leqslant \int_{\mathbb{R}^{N}}(f-\beta(u))T_{k}(u-\varphi)+\int_{\partial \Omega}(g-\gamma(u))T_{k}(\tau u-\tau\varphi),$$
for every k>0 and every $$\varphi\in \mathcal{T}^{1,p}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$$ .

## Keywords

nonlinear Neumann problem elliptic operator of Leray Lions type m-completely accretive operator

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