Comparison theorems for elliptic and parabolic operators in variational form

Abstract

Order properties are normally derived from maximum principles associated with an operator, say $P$, in classical formulation. 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. The solutions of the classical formulation in variational form are called weak solutions. The variational formulation and the associated concept of weak solution is widely used in the theory, applications and numerical analysis of elliptic and parabolic PDEs. In practice, often the variational formulation is used in order to accommodate generalized/ weak solutions and also to prepare for the use of numerical schemes such as finite element methods.

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 dissertation proves inverse monotonicity, or equivalently comparison theorems, for this much more general formulation of the operators and respective equations. More precisely, we prove results regarding order/comparison for the solutions of the variational problem through the concept of inverse monotone operators, which put them in a more general framework. We specifically discuss the case of a single equation and the case of systems of PDEs for both elliptic and parabolic equations.