Direchlet principle: $ \Omega \subset R^n$ is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on $ \partial \Omega$. Perron lifting and barrier function We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary. The key point is if we define the variation energy $ E(u)=\int_{\Omega}|\nabla u|^2$, then it is easy to see for $ u_1,u_2$ is in perron set, $ E(sup (…
