The continuity of weak solutions of elliptic partial differential equations $$div \mathcal{A}(x,\nabla u) = 0$$ is considered under minimal structure assumptions. The main result guarantees the continuity at the point$x$_{0}for weakly monotone weak solutions if the structure of$A$is controlled in a sequence of annuli $$B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )$$ with uniformly bounded ratio$R$_{$j$}$/r$_{$j$}such that lim_{$j→∞$}$R$_{$j$}=0. As a consequence, we obtain a sufficient condition for the continuity of mappings of finite distortion.

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold of$T$^{*}$R$^{$n$+1}-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.

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