Regularity for anisotropic solutions to some nonlinear elliptic system

Abstract

This paper deals with anisotropic solutions u\in W^{\mathrm{1},(pi)}(\Omega ,{ℝ}^N) to the nonlinear elliptic system

-\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}Di\left(a_i^{\alpha }(\chi ,Du\left(\chi \right))\right)=-\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}Di{F}_i^{\alpha }\left(\chi \right),\alpha =1,2,\mathrm{...},N,

We present a monotonicity inequality for the matrix a=({{\displaystyle a}}_i^{\alpha })\in {ℝ}^{N\times n},whichguarantees global pointwise bounds for anisotropic solutionsu.

Keywords

Regularity / anisotropic solution / nonlinear elliptic system

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Higher Education Press and Springer-Verlag Berlin Heidelberg