An existence theorem for non-linear elliptic partial differential equations

Abstract

When a heat-producing reaction takes place within a confined region then, under certain circumstances, a thermal explosion will occur. When investigating from a theoretical viewpoint the conditions under which this happens, we are led to discuss a particular class of non-linear parabolic initial-boundary value problems. However, under fairly general conditions, it is possible to approach the problem indirectly and consider, instead, a class of closely related elliptic boundary value problems, the so-called steady-state problems. The object of this thesis is to prove the existence of solutions of these mildly non-linear elliptic partial differential equations by using variational techniques. Following a chapter of introductory material, Chapter II deals with the theory of Sobolev spaces, which play a fundamental rôle in the subsequent analysis. As is usually the case with work concerning partial differential equations, little more theory is developed than is required in the following sections. Chapter III contains the main results of the thesis, and gives a generalised form of the main theorem. This is proved by constructing a functional on a suitable Hilbert space whose stationary points coincide, in some sense, with the solutions of the elliptic boundary value problem under consideration. The underlying principles are borrowed from the calculus of variations and, indeed, the aforementioned functional is chosen to have the partial differential equation in question as its Euler-Lagrange equation. The following chapter then establishes the classical form of the theorem of Chapter III. The analysis here rests heavily on the theory of the linear Dirichlet problem which, for the purposes of this work, will be taken as known and consequently many of the crucial results are simply lifted from relevant texts. Finally, Chapter V briefly discusses various related topics and extensions of the foregoing analysis, finishing with an example illustrating the techniques employed throughout the thesis. At each stage of the work, an attempt is made to motivate the analysis. Often this results in touching lightly upon subjects which are separate from the topic at hand and, in such cases, a reference will be given where a more thorough treatment of the theory can be found.