The material of the present book has been used for graduate-level
courses at the University of Ia i during the past ten years. It is a
revised version of a book which appeared in Romanian in 1993 with the
Publishing House of the Romanian Academy. The book focuses on classical
boundary value problems for the principal equations of mathematical
physics: second order elliptic equations (the Poisson equations), heat
equations and wave equations. The existence theory of second order
elliptic boundary value problems was a great challenge for nineteenth
century mathematics and its development was marked by two decisive
steps. Undoubtedly, the first one was the Fredholm proof in 1900 of the
existence of solutions to Dirichlet and Neumann problems, which
represented a triumph of the classical theory of partial differential
equations. The second step is due to S. 1. Sobolev (1937) who introduced
the concept of weak solution in partial differential equations and
inaugurated the modern theory of boundary value problems. The classical
theory which is a product ofthe nineteenth century, is concerned with
smooth (continuously differentiable) sollutions and its methods rely on
classical analysis and in particular on potential theory. The modern
theory concerns distributional (weak) solutions and relies on analysis
of Sob ole v spaces and functional methods. The same distinction is
valid for the boundary value problems associated with heat and wave
equations. Both aspects of the theory are present in this book though it
is not exhaustive in any sense.
