Partial Differential Equations

Abstract

This is the third edition of my textbook intended for students who wish to obtain an introduction to the theory of partial differential equations (PDEs, for short). Why is there a new edition? The answer is simple: I wanted to improve my book. Over the years, I have received much positive feedback from readers from all over the world. Nevertheless, when looking at the book or using it for courses or lectures, I always find some topics that are important, but not yet contained in the book, or I see places where the presentation could be improved. In fact, I also found two errors in Sect. 6.2, and several other corrections have been brought to my attention by attentive and careful readers. So, what is new? I have completely reorganized and considerably extended Chap. 7 on hyperbolic equations. In particular, it now also contains a treatment of first-order hyperbolic equations. I have written a new Chap. 9 on the relations between different types of PDEs. I have inserted material on the regularity theory for semilinear elliptic equations and systems in various places. In particular, there is a new Sect. 14.3 that shows how to use the Harnack inequality to derive the continuity of bounded weak solutions of semilinear elliptic equations. Such equations play an important role in geometric analysis and elsewhere, and I therefore thought that such an addition should serve a useful purpose. I have also slightly rewritten, reorganized, or extendedmost other sections of the book,with additional results inserted here and there. But let me now describe the book in a more systematic manner. As an introduction to the modern theory of PDEs, it does not offer a comprehensive overview of the whole field of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding question is how one can find a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for finding a solution of a PDE; they can be informally characterized as follows: 0. Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this is possible only in rather particular and special cases. Also, such a formula may be rather complicated, so that it is not very helpful for detecting qualitative properties of a solution. Therefore, mathematical analysis has developed other, more powerful, approaches. 1. Solve a sequence of auxiliary problems that approximate the given one and show that their solutions converge to a solution of that original problem. Differential equations are posed in spaces of functions, and those spaces are of infinite dimension. The strength of this strategy lies in carefully choosing finitedimensional approximating problems that can be solved explicitly or numerically and that still share important crucial features with the original problem. Those features will allow us to control their solutions and to show their convergence. 2. Start anywhere, with the required constraints satisfied, and let things flow towards a solution. This is the diffusion method. It depends on characterizing a solution of the PDE under consideration as an asymptotic equilibrium state for a diffusion process. That diffusion process itself follows a PDE, with an additional independent variable. Thus, we are solving a PDE that is more complicated than the original one. The advantage lies in the fact that we can simply start anywhere and let the PDE control the evolution. 3. Solve an optimization problem and identify an optimal state as a solution of the PDE. This is a powerful method for a large class of elliptic PDEs, namely, for those that characterize the optima of variational problems. In fact, in applications in physics, engineering, or economics, most PDEs arise from such optimization problems. The method depends on two principles. First, one can demonstrate the existence of an optimal state for a variational problem under rather general conditions. Second, the optimality of a state is a powerful property that entails many detailed features: If the state is not very good at every point, it could be improved and therefore could not be optimal. 4. Connect what you want to know to what you know already. This is the continuity method. The idea is that if you can connect your given problem continuouslywith another, simpler, problem that you can already solve, then you can also solve the former. Of course, the continuation of solutions requires careful control.