A Brief Overview of Partial Differential Equations

In this section, we introduce few key definitions and results from PDE literature. We note that the results in this section are standard and have been included in the Appendix for completeness. We refer the reader to classical texts on PDEs [11, 14] for more details. We will use the following Poincaré inequality throughout our proofs. A.1 Proof of Proposition 1 We first show that if u is the unique solution then it minimizes the variational norm.