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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
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 $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u)$, where $ A$ and $ B$ are Borel measurable, are solutions to the equation $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu $ for some nonnegative Radon measure $ \mu $. Among other things, it is shown that if $ u$ is a Hölder continuous solution to this equation, then the measure $ \mu $ satisfies the growth property $ \mu [B(x,r)] \leqslant M{r^{n - p + \varepsilon }}$ for all balls $ B(x,r)$ in $ {{\mathbf{R}}^n}$. Here $ \varepsilon $ depends on the Hölder exponent of $ u$ while $ p > 1$ is given by the structure of the differential operator. Conversely, if $ \mu $ is assumed to satisfy this growth condition, then it is shown that $ u$ satisfies a Harnack-type inequality, thus proving that $ u$ is locally bounded. Under the additional assumption that $ A$ is strongly monotonic, it is shown that $ u$ is Hölder continuous.

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