Linear Elliptic Equations

摘要: The first major topic of this chapter is the Dirichlet problem for Laplace operator on a compact domain with boundary: $$\Delta u = 0\text{ }\Omega,\quad {u\bigr |}_{\partial\Omega } f.$$ (0.1) We also consider nonhomogeneous Δu g and allow lower-order terms. As in Chap. 2, Δ determined by Riemannian metric. In §1 we establish some basic results existence regularity solutions, using theory Sobolev spaces. §2 maximum principles, which are useful uniqueness theorems treating (0.1) f continuous, among other things.