ABSTRACT
In this chapter we continue our investigation of the methods and techniques used in the study of linear and nonlinear elliptic partial differential equations. First we analyze the spectrum of linear elliptic differential operators and then we pass to certain nonlinear ones, namely the p-Laplacian. Our analysis involves energy methods which are based on the critical point theory developed in the previous chapter. In the last two sections we develop the other distinct technique in the study of stationary partial differential equations which is based on maximum principles. This method leads to pointwise deductions and for this reason requires higher smoothness in the functions and it is different from the integral-based energy methods which are developed in the framework of Sobolev spaces. In Section 6.1 we study linear eigenvalue problems with weight for general
linear elliptic differential operators in divergence form. As a particular case we obtain a complete description of the spectrum of the negative Laplacian with Dirichlet or Neumann boundary conditions. We also obtain variational characterizations of the eigenvalues via minimax expressions (Courant’s theory). At the end of the section we present the basic maximum principles for linear elliptic partial differential equations (weak and strong (Hopf) maximum principles). In Section 6.2 we pass to nonlinear elliptic partial differential operators and
examine the spectrum of the p-Laplacian. The relevant eigenvalue problem is nonlinear and we use energy methods (critical point theory; see Chapter 5) to study it. We determine the beginning of the spectrum of the negative pLaplacian with Dirichlet or Neumann boundary conditions and establish the existence of a principal eigenvalue. Our investigations involve also some basic nonlinear regularity results. In Section 6.3 we examine the ordinary (one dimensional) p-Laplacian differ-
ential operator. We consider both the scalar and vector ordinary p-Laplacian differential operators under Dirichlet, Neumann or periodic boundary conditions. In Section 6.4 we develop versions of the maximum principle when the
differential operator is nonlinear and involves the p-Laplacian. We also state the nonlinear Picone’s identity and illustrate how it can be used in the study of nonlinear eigenvalue problems.
