1,721,025 research outputs found
Convex functions in Carnot Groups: old and new
No full textIn this talk we consider convex functions on general Carnot groups. We consider several definitions, prove their equivalence, and establish basic regularity properties. A key observation is the realization that convexity depends only on the horizontal distribution and not on the particular base chosen to represent it . This allows us to use potential theoretic representation formulas developed by Bonfiglioli and Lanconelli to approximate convex functions by smooth convex functions. Another new ingredient is Wang's extension to Carnot groups of Bieske's uniqueness result for ∞-harmonic functions in the Heisenberg group
Partial Regularity results for quasimonotone elliptic systems with general growth
No full textWe present a partial H"older regularity result for the gradient of solutions to { quasimonotone} systems: on bounded domains in the weak sense. Here certain continuity,{ uniformly strictly quasimonotonicity}, growth conditions are imposed on the coefficients, including an asymptotic Uhlenbeck behaviour close to the origin, while the inhomogeneous term satisfies controllable growth conditions. The result is achieved along a two-scale regime: degenerate and non-degenerate. In particular, we will use approximation lemmas, Diening et al.[ J.~Diff.~Equ. 253 (2012)(7), 1943–-1958; SIAM ~J.~Math.~ Anal. 44 (2012)(5), 3594-–3616], that simplify and unify the proof in the power growth case and allow us to consider also the general growth case
Homogenization of Hamilton-Jacobi equations in Carnot Groups
No full textWe present an homogenization for Hamilton-Jacobi equations in the geometryof a Carnot Group. The tiling and the corresponding notion of periodicity are compatible with the dilations and use the Lie bracket generating property
Everywhere regularity for functionals with general growth
No full textWe prove everywhere Holder regularity for the gradient of a local minimizers of functionals with general growth, giving also the decay estimate. In particular, we present a unified approach in the case of power-type functions
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