Local behavior of solutions of quasilinear elliptic equations with general structure

Authors:
J.-M. Rakotoson and William P. Ziemer

Journal:
Trans. Amer. Math. Soc. **319** (1990), 747-764

MSC:
Primary 35J60; Secondary 35B65, 35D10, 35J70

DOI:
https://doi.org/10.1090/S0002-9947-1990-0998128-9

MathSciNet review:
998128

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Abstract: This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form , where and are Borel measurable, are solutions to the equation for some nonnegative Radon measure . Among other things, it is shown that if is a Hölder continuous solution to this equation, then the measure satisfies the growth property for all balls in . Here depends on the Hölder exponent of while is given by the structure of the differential operator. Conversely, if is assumed to satisfy this growth condition, then it is shown that satisfies a Harnack-type inequality, thus proving that is locally bounded. Under the additional assumption that is strongly monotonic, it is shown that is Hölder continuous.

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0998128-9

Article copyright:
© Copyright 1990
American Mathematical Society