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In 1971 Fedi\u{i} proved the remarkable theorem that the linear second orderpartial differential operator in the plane with coefficients 1 and f^2 ishypoelliptic provided that f is smooth, vanishes at the origin and is positiveotherwise. Variants of this result, with hypoellipticity replaced by continuityof weak solutions, were recently given by the authors, together with CristianRios and Ruipeng Shen, to infinitely degenerate elliptic divergence formequations where the nonnegative matrix A(x,u) has bounded measurablecoefficients with trace roughly 1 and determinant comparable to f, and whereF=ln(1/f) is essentially doubling. However, in the plane, these variantsassumed additional geometric constraints on f, something not required inFedi\u{i}'s theorem. In this paper we in particular remove these additionalgeometric constraints in the plane for homogeneous equations with F essentiallydoubling.

Continuity of infinitely degenerate weak solutions via the trace method