Boundary value problems are the subject of this book. All boundary value conditions for elliptic differential operators are given, using the Lopatinskiĭ–Šapiro condition (= covering condition), which lead to the normal solvability of a boundary value problem. The variational method is also presented in detail, and questions of its connections with general elliptic theory considered. Those parabolic and hyperbolic equations for which the right-hand side (derivatives with respect to x) is an elliptic differential operator are considered, and the knowledge about elliptic operators is used in order to obtain insight into the solvability and regularity properties of the solution for mixed problems.

I have chosen a form of the Lopatinskiĭ–Šapiro condition which allows us to test immediately, whether or not given boundary value conditions satisfy it. It appears that all classical boundary value problems satisfy it, the examples are worked through individually.

In order not to overexpand the compass of the book, and to maintain its introductory character, I have not considered pseudo-differential operators; all the same I have proved the main theorem for elliptic boundary value problems by means of pseudo-differential operators – without calling them such.

Before the discussion of differential equations there is an introductory chapter on distributions and Sobolev spaces; here I have proceeded in an elementary way, working with the Fourier transformation, and not using interpolation theorems. This is possible without further assumptions as long as one remains inside L2-theory.