Elliptic equation

I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition. Linear elliptic equation case First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e. MVP of laplace equation $\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$ I need to point out first, this property is not difficult to proof by standard integral by part meth…

In this note we discuss the Schauder theory for uniformly elliptic linear equations and Sobelov inequality. the three main topics ars a priori estimate in Holder norms,regularity of arbitrary solutions and the solvability of the Dirichlet problem.Among these topics,a priori estimates are the most fundamental and the basis of the follows two.we will discuss both the interior Schauder estimate and global Schauder estimate. -Schauder Theory- 1. Interior Schauder Theory $ {\Omega} $ be a domain in $…