Positivity of weak solutions of non-uniformly elliptic equations

Abstract

Let A be a symmetric N × N real-matrix-valued function on a connected region Ω in RN, with A positive definite a.e. and A, A−1 locally integrable. Let b and c be locally integrable, non-negative, real-valued functions on Ω, with c positive, a.e. Put a(u, v)= = $$\mathop \smallint \limits_\Omega $$ ((A∇u, ∇v)+buv) dx. We consider the boundary value problem a(u, v)= $$\mathop \smallint \limits_\Omega $$ fvcdx, for all v ε C 0 ∞ (Ω), and the eigenvalue problem a(u, v)=λ $$\mathop \smallint \limits_\Omega $$ uvcdx, for all v ε C 0 ∞ (Ω). Positivity of the solution operator for the boundary value problem, as well as positivity of the dominant eigenfunction (if there is one) and simplicity of the corresponding eigenvalue are proved to hold in this context.