Renormalized solutions of elliptic equations with Robin boundary conditions

Abstract

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type {-div(B(x,u)∇u)=f in Ω,u=0 on ΓoB(x,u)∇u⋅n→+γ(x)h(u)=g on Γ1where f and g are the element of L1 (Ω) and L1 (Γ1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.