A Generalized Dirichlet Principle for Second Order Nonselfadjoint Elliptic Operators

Abstract

The classical Dirichlet principle states that if the spectrum of $L = \frac{1}{2}\Delta - V$ on a smooth bounded domain $D \subset R^n $ with the Dirichlet boundary condition on $\partial D$ is negative, then the unique solution $\phi _0 $ of $L\phi _0 = 0$ in D with $\phi _0 = f$ on $\partial D$, for a smooth function f, minimizes a certain energy integral. This may be easily extended to the operator $L = \frac{1}{2}\nabla \cdot a\nabla + a\nabla Q \cdot \nabla - V$. Now L is selfadjoint with respect to the density $e^{2Q} $. In this paper, we generalize this result to nonselfadjoint operators on bounded domains. We consider the solution $\phi _0 $ to $L\phi _0 = (\frac{1}{2}\nabla \cdot a\nabla + b \cdot \nabla - V)\phi _0 = 0$ in D and $\phi _0 = f \geqq 0$ on $\partial D$ under the assumption $\operatorname{Re} (\sigma (L)) < 0$ for L with the Dirichlet boundary condition on $\partial D$.