Quasilinear degenerate elliptic equations in divergence form

Research Report MRR96-016

Quasilinear degenerate elliptic equations in divergence form

Pengfei Guan

Abstract We consider the regularity for the solutions of degenerate elliptic quasilinear equations in the form

$\begin{displaymath}\sum_{i,j} \partial_j(a_{ij}(x,u)\partial_iu) = f,\tag1\end{displaymath}$

where $a_{ij}(x,u) \in C^\infty({\bold R}^n \times {\bold R})$ , semi-positive. We prove that any $C^{0,1}$ weak solution of the equation (1) is in fact $C^{\infty}$ if the linearized equation is subelliptic. The proof we produce here don't use elliptic theory, therefore, our proof is also applicable to elliptic case (without using Schauder theory). This generalies a previous result in [G].

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