Juan J. Manfredi

On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation

We discuss and compare various notions of weak solution for the p-Laplace equation -\text{div}(|\nabla u|^{p-2}\nabla u)=0 and its parabolic counterpart u_t-\text{div}(|\nabla u|^{p-2}\nabla u)=0. In addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise derivatives (jets). Our main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.

Weakly monotone functions

The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spacesW1,p,p >n − 1. It is proven that such weakly monotone functions are continuous except in a singular set ofp-capacity zero that is empty in the casep =n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.

An asymptotic mean value characterization for p-harmonic functions

We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to Δ p u = div(|∇u| p-2 ∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansion u(x) = α/2 {max/B e (x) u + min/B e (x) u} + β/|B e (x)| ∫B e (x) u dy + o(e 2 ) holds as e → 0 for x ∈ Ω in a weak sense, which we call the viscosity sense. Here the coefficients α, β are determined by α + β = 1 and α/β = (p - 2)/(N + 2).

Mappings with integrable dilatation in higher dimensions

Let F ∈ W 1 , n loc (Ω ; R ) be a mapping with nonnegative Jacobian JF (x) = detDF (x) ≥ 0 for a.e. x in a domain Ω ⊂ R n . The dilatation of F is defined (almost everywhere in Ω) by the formula K(x) = |DF (x)| JF (x) · Iwaniec and Sverak [IS] have conjectured that if p ≥ n − 1 and K ∈ Lploc(Ω) then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2 . In this article, we verify it in the higher-dimensional case n ≥ 2 whenever p > n − 1 .

An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games

We characterize solutions to the homogeneous parabolic p-Laplace equation

Regularity theory and traces of -harmonic functions

u_{t}=|\nabla u|^{2-p}\Delta_{p}u=(p-2)\Delta_{\infty}u+\Delta u

Fundamental solution for the Q-Laplacian and sharp Moser–Trudinger inequality in Carnot groups

in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these games approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.

A VERSION OF THE HOPF-LAX FORMULA IN THE HEISENBERG GROUP

In this paper we discuss two different topics concerning Aharmonic functions. These are weak solutions of the partial differential equation div(A(x,∇u)) = 0, where α(x)|ξ|p−1 ≤ 〈A(x, ξ), ξ〉 ≤ β(x)|ξ|p−1 for some fixed p ∈ (1,∞), the function β is bounded and α(x) > 0 for a.e. x. First, we present a new approach to the regularity of A-harmonic functions for p > n−1. Secondly, we establish results on the existence of nontangential limits for A-harmonic functions in the Sobolev space W 1,q(B), for some q > 1, where B is the unit ball in Rn. Here q is allowed to be different from p.

Convex functions on Carnot groups

Abstract For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q -Laplacian u Q and a constant a Q >0 such that exp(− a Q u Q ) is a homogeneous norm on G . This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser–Trudinger inequality.

Subelliptic Cordes estimates

ABSTRACT We consider Hamilton-Jacobi equations in the , where is the Heisenberg group and denotes the horizontal gradient of u. We establish uniqueness of bounded viscosity solutions with continuous initial data . When the hamiltonian H is radial, convex and superlinear the solution is given by the Hopf-Lax formula where the Lagrangian L is the horizontal Legendre transform of H lifted to by requiring it to be radial with respect to the Carnot-Carathéodory metric.