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Geometric rigidity of conformal matrices

Daniel FaracoXiao Zhong — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace SO ( n ) by an arbitrary compact set of conformal matrices, bounded away from 0 and invariant under SO ( n ) , and rigid motions by Möbius transformations.

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael BildhauerMartin FuchsXiao Zhong — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q < and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q 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.

On the Regularity of p-Harmonic Functions in the Heisenberg Group

Giuseppe MingioneZatorska-Goldstein AnnaXiao Zhong — 2008

Bollettino dell'Unione Matematica Italiana

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 p [ 2 , 4 ) 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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