The Variational Theory of Elliptic Boundary Value Problems

Abstract

Consider the following “model problem”:

$$\begin{gathered} - \Delta u + u = f\quad in\;Q, \hfill \\ u = 0\quad on\;\partial \Omega \backslash \Gamma ,\quad \left( {\nabla u} \right) \cdot v = 0\quad on\quad \Gamma , \hfill \\ \end{gathered}$$

((2.1))

where ∆ denotes, as is usual in the literature, the Laplacian $\sum\nolimits_{{i = 1}}^{N} {{\partial _{2}}} /\partial x_{i}^{2}$, and f is an arbitrarily fixed function from L ²(Ω). (As stipulated in the Glossary of Basic Notations, Ω is from now on supposed to be a bounded domain.) Let ∂Ω be of class C ¹ and let its open portion Γ be closed as well. With the help of Section 1.7.3 for what concerns boundary values, we see that (2.1) certainly makes sense in the function space H ²(Ω) and implies

$$\begin{gathered} u \in H_{0}^{1}\left( {\Omega \cup \Gamma } \right), \hfill \\ a\left( {u,v} \right) \equiv \int_{\Omega } {\left( {{u_{{{x_{i}}}}}{v_{{{x_{i}}}}} + uv} \right)dx = \int_{\Omega } {fvdx\quad for\;v \in H_{0}^{1}\left( {\Omega \cup \Gamma } \right)} } \hfill \\ \end{gathered} $$

((2.2))

by the divergence theorem: see Theorem 1.53. (From now on we adopt the summation convention: repeated dummy indices indicate summation from 1 to N.)