We consider the simplest form of a second order, linear, degenerate, elliptic equation with divergence structure in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.
We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace by an arbitrary compact set of conformal matrices, bounded away from and invariant under , and rigid motions by Möbius transformations.
Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of -growth with exponents and show for the scalar case that locally bounded local minimizers are of class . Note that to our knowledge the only -results without imposing a relation between and concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.
We establish continuity, openness and discreteness, and the condition (N) for mappings of finite distortion under minimal integrability assumptions on the distortion.
We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually...
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