T h e classical partial differential equations of mathematical physics, for­ mulated and intensively studied by the great mathematicians ofthe nineteenth century, remain the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. These equations, in the early twentieth century, prompted further mathematical researches, and in turn themselves benefited by the application of new methods in pure mathematics. The theories of sets and of Lebesgue integration enable us to state conditions and to characterize solutions in a much more precise fashion; a differential equation with the boundary conditions to be imposed on its solution can be absorbed into a single formulation as an integral equation; Green’s function permits a formal explicit solution; eigenvalues and eigenfunctions generalize Fourier’s analysis to a wide variety of problems.All these matters are dealt with in Sobolev’s book, without assumption of previous acquaintance. The reader has only to be familiar with element­ ary analysis; from there he is introduced to these more advanced concepts, which are developed in detail and with great precision as far as they are re­ quired for the main purposes of the book. Care has been taken to render the exposition suitable for a novice in this field: theorems are often approach­ ed through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems.