Nicola Garofalo

An embedding theorem and the harnack inequality for nonlinear subelliptic equations

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CAPACITARY ESTIMATES AND THE LOCAL BEHAVIOR OF SOLUTIONS OF NONLINEAR SUBELLIPTIC EQUATIONS

We establish sharp capacitary estimates for Carnot-Carathéodory rings associated to a system of vector fields of Hörmander type. Such estimates are instrumental to the study of the local behavior of singular solutions of a wide class of nonlinear subelliptic equations. One of the main results is a generalization of fundamental estimates obtained independently by Sanchez-Calle and Nagel, Stein and Wainger.

A Symmetry Result Related to Some Overdetermined Boundary Value Problems

Soit ΩCR n un ensemble ouvert connexe borne et on suppose pour p fixe, 1 0, on suppose que |⊇u(x)|→a, u(x)→0 quand x→∂Ω au sens suivant: etant donne e>0, il existe un ensemble ouvert 0=O(e)⊃∂Ω tel que ∥⊇u(x)|−a|<e, u(x)<e, pour presque tout x∈0∩Ω par rapport a la mesure de Lebesgue n. Alors Ω est une boule et u est radialement symetrique autour du centre de la boule

Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type

Here, G is a stratified, nilpotent Lie group, in short a Carnot group, of arbitrary step, and Ω ⊂ G is a domain which can be bounded or unbounded. The second order differential operator L represents a given sub-Laplacian on G. If g = r ⊕ j=1 Vj is a stratification of the Lie algebra g of G, with [V1, Vj ] ⊂ Vj+1 for 1 ≤ j < r, [V1, Vr] = {0}, we assume that a scalar product < ·, · > is given on g for which the V ′ j s are mutually orthogonal. The stratification allows to define a natural family of non-isotropic dilations ∆λ : g → g as follows ∆λ(X1 + ... + Xr) = λX1 + ... + λXr. The exponential map exp : g → G is an analytic diffeomorphism. It induces a group of dilations on G via the formula

Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces

Introduction Carnot groups The characteristic set

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

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A Notable Family of Entire Intrinsic Minimal Graphs in the Heisenberg Group which are not Perimeter Minimizing

-variation,

Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces

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Second order parabolic equations in nonvariational form: Boundary Harnack principle and comparison theorems for nonnegative solutions

-perimeter and surface measure Geometric estimates from above on CC balls for the perimeter measure Geometric estimates from below on CC balls for the perimeter measure Fine differentiability properties of Sobolev functions Embedding a Sobolev space into a Besov space with respect to an upper Ahlfors measure The extension theorem for a Besov space with respect to a lower Ahlfors measure Traces on the boundary of

Wiener's criterion for parabolic equations with variable coefficients and its consequences

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