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This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd 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 ?nding 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 pr- lem. Di?erential equations are posed in spaces of functions, and those spaces are of in?nite dimension.
Introduction * The Laplace equation as the prototype of an elliptic partial differential equation of 2nd order * The maximum principle * Existence techniques I: methods based on the maximum principle * Existence techniques II: Parabolic methods. The Head equation * The wave equation and its connections with the Laplace and heat equation * The heat equation, semigroups, and Brownian motion * The Dirichlet principle. Variational methods for the solution of PDE (Existence techniques III) * Sobolev spaces and L2 regularity theory * Strong solutions * The regularity theory of Schauder and the continuity method (Existence techniques IV) * The Moser iteration method and the reqularity theorem of de Giorgi and Nash * Banach and Hilbert spaces. The Lp-spaces * Bibliography