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Regularity theory for nonlinear diffusion processes. (English) Zbl 1462.76176

The paper is a survey of results on diffusion equations and is an introductory material. The focus is on steady equations, which are elliptic. The author tells about the connection between elliptic equations and analytic functions, the mean properties, etc, and then passes to regularity results as a priori norm estimations. Then heterogeneous media are considered, with diffusivity being a tensor, and regularity results for this case are discussed. The rest of the paper is devoted to nonlinear elliptic PDEs, free boundary, and regularity of solutions of such problems.

MSC:

76R50 Diffusion
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
35R35 Free boundary problems for PDEs
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