Weak Solutions of Elliptic Differential Equations

Abstract

In this chapter we consider Sobolev spaces in Section 1 and prove the Sobolev embedding theorem and the Rellich selection theorem in Section 2. Then we establish the existence of weak solutions in Section 3. With the aid of Moser’s iteration method we show the boundedness of weak solutions in Section 4. In the subsequent Section 5 we deduce Hölder continuity of weak solutions with the aid of the weak Harnack inequality by J. Moser. The necessary regularity theorem of John and Nirenberg will be derived in Section 6. Finally, we investigate the boundary regularity of weak solutions in Section 7. Then we apply our results to equations in divergence form (compare Section 8). At the end of this chapter we present Green’s function for elliptic operators with the aid of capacity methods, and we treat the eigenvalue problem for the Laplace-Beltrami operator.