Inverse Monotonicity of Elliptic Operators in Variational Form

Abstract

AbstractThe inverse monotonicity of elliptic operators in classical formulation is a well known consequence of the maximum principle. This result is often formulated as a comparison theorem for solutions of linear elliptic PDEs as it derives order in the solution space from the order in the space of data. Variational formulation and the associated concept of weak solution is widely used in the theory, applications and numerical analysis of elliptic PDEs.The space of weak solutions as well as space of data are Sobolev spaces, which are wider than the respective spaces of solutions and data in the classical formulation. This paper proves inverse monotonicity, or equivalently comparison theorems, for this much more general formulation of the operators and respective equations. Since the maximum principle does not apply to weak solutions, the presented here theory is a useful set of tools that can be used in its place to derive order in the space of weak solutions from the order in the generalized data space. We specifically discuss the case of a single equation and the case of weakly coupled system of equations.KeywordsElliptic PDEsComparison theoremVariational formulationWeak solutionsInverse Monotone operatorsMathematics Subject Classification (2010)Primary 35J50; Secondary 47H07