C1,αC^{1,\alpha}-Regularity of Quasilinear equations on the Heisenberg Group

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The conditions encompass a very wide class of equations with isotropic growth conditions, which are a generalization of the $p$-Laplace type equations in this respect; these also include all equations with polynomial or exponential type growth. In addition, some even more general conditions have also been explored.Comment: long versio

Regularity of inhomogeneous Quasi-linear equations on the Heisenberg Group

We establish Holder continuity of the horizontal gradient of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg Group.Comment: Errors in earlier versions are corrected and significant structural changes have been mad

On the Minkowski problem for p-harmonic measures

We study the Minkowski problem corresponding to the p-harmonic measures and obtain results previously known for harmonic measures due to Jerison (Invent Math 105(2):375–400, 1991). We show that a class of Borel measures on spheres can be prescribed by p-harmonic measures on convex domains

A Variational Characterisation of the Second Eigenvalue of the p-Laplacian on Quasi Open Sets

In this article, we prove a minimax characterization of the second eigenvalue of the p-Laplacian operator on p-quasi-open sets, using a construction based on minimizing movements. This leads also to an existence theorem for spectral functionals depending on the first two eigenvalues of the p-Laplacian.Comment: 34 page

Regularity of quasi-linear equations with H??rmander vector fields of step two

If the smooth vector fields X1,…,Xm and their commutators span the tangent space at every point in Ω⊆RN for any fixed m≤N, then we establish the full interior regularity theory of quasi-linear equations ∑i=1mXi⁎Ai(X1u,…,Xmu)=0 with p-Laplacian type growth condition. In other words, we show that a weak solution of the equation is locally C1,α

Sulla caratterizzazione di minmax in problemi di autovalori nonlineari

This is a note based on the paper [20] written in collaboration with N. Fusco and Y. Zhang. The main goal is to introduce minimax type variational characterization of non-linear eigenvalues of the p-Laplacian and other results related to shape and spectral optimization problems.Questa `e una nota basata sul documento [20] scritto in collaborazione con N. Fusco e Y. Zhang. L'obiettivo principale è introdurre la caratterizzazione variazionale di tipo minimax di autovalori non lineari del p-Laplaciano e altri risultati relativi a problemi di forma e ottimizzazione spettrale

C1,α-regularity of quasilinear equations on the Heisenberg group

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The considered cases encompass a very wide class of equations with isotropic growth conditions that are generalizations of the p-Laplacian and include equations with polynomial or exponential type growth. Some more general conditions have also been explored