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Nonlinear Mean Value Properties and PDE's

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Nonlinear Mean Value Properties and PDE's


Ángel Arroyo


It is a well-known fact in the classical theory of PDE's that harmonic functions in Euclidean domains can be characterized in terms of the mean value property. Namely, a continuous function u is harmonic if and only if the value of u at x coincides with its average over any ball centered at x. In recent years, similar mean value characterizations have been obtained for p-harmonic functions, which are defined as weak solutions of the so-called p-Laplace equation. This interplay between equations and mean value properties has turned out to be the cornerstone in the development of new approximation techniques for the study of certain properties of the p-Laplace equation, such as the regularity of solutions. In this talk we briefly review some of these results.



2020-06-09 at 3:00 pm