Variational Principle for Elliptic Equations

PDEs Given the divergence form an Elliptic PDE, with , a weak solution satisfies the following variational formulation holds for all with in The Dual of Sobolev Space, with LHS .

Note: When , we replace by and by , to utilize a weak solution with zero Trace Operator.

General elliptic operators might not satisfy the Lax-Milgram Theorem but the regularized operator will satisfy the following energy estimates for some :