INTERPOLATION AND PARTIAL-DIFFERENTIAL EQUATIONS

One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations (PDEs). Nowadays interpolation theory has been developed in an almost unbelievable way {see the bibliography of Maligranda [Interpolation of Operators and Applications (1926-1990), 2nd ed. (Lulea University, Lulea, 1993), p. 154]}. In this article some model examples are presented which display how powerful this theory is when dealing with PDEs, One main aim is to point out when it suffices to use classical interpolation theory and also to give concrete examples of situations when nonlinear interpolation theory has to be applied. Some historical remarks are also included and the relations to similar results are pointed out.