Search results for "PSC"

Evolution Problems Associated to Linear Growth Functionals: The Dirichlet Problem

2003

Let Ω be a bounded set inIR N with Lipschitz continuous boundary ∂Ω. We are interested in the problem

Absolutely continuous functions with values in a Banach space

2017

Abstract Let Ω be an open subset of R n , n > 1 , and let X be a Banach space. We prove that α-absolutely continuous functions f : Ω → X are continuous and differentiable (in some sense) almost everywhere in Ω.

Lipschitz conditions,b-arcwise connectedness and conformal mappings

1982

Equivalence of AMLE, strong AMLE, and comparison with cones in metric measure spaces

2006

MSC (2000) Primary: 31C35; Secondary: 31C45, 30C65 In this paper, we study the relationship between p-harmonic functions and absolutely minimizing Lipschitz extensions in the setting of a metric measure space (X, d, µ). In particular, we show that limits of p-harmonic functions (as p →∞ ) are necessarily the ∞-energy minimizers among the class of all Lipschitz functions with the same boundary data. Our research is motivated by the observation that while the p-harmonic functions in general depend on the underlying measure µ, in many cases their asymptotic limit as p →∞ turns out have a characterization that is independent of the measure. c

POINTS OF $\varepsilon$ -DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS FROM ${\bb R}^n$ TO ${\bb R}^{n-1}$

2002

This paper proves that for every Lipschitz function $f:{\bb R}^n\longrightarrow {\bb R}^m,\;m < n$ , there exists at least one point of $\varepsilon$ -differentiability of $f$ which is in the union of all $m$ -dimensional affine subspaces of the form $q_0+{\rm span}\{q_1,q_2,\ldots,q_m\},\;{\rm where}\;q_j(j=0,1,\ldots,m)$ are points in ${\bb R}^n$ with rational coordinates.

On the best Lipschitz extension problem for a discrete distance and the discrete ∞-Laplacian

2012

Abstract This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞-Laplacian problem, which arises as the dynamic programming formula for the value function of some e -tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p -Laplacian problem.

Radon–Nikodym Property and Area Formula for Banach Homogeneous Group Targets

2013

We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable Lipschitz mappings defined on a Carnot group. peerReviewed

2-SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS

2008

AbstractIn this paper, some min–max theorems for even andC1functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

Rademacher Theorem for Fréchet spaces

2010

Abstract Let X be a separable Frechet space. In this paper we define a class A of null sets in X that is properly contained in the class of Aronszajn null sets, and we prove that a Lipschitz map from an open subset of X into a Gelfand-Frechet space is Gateaux differentiable outside a set belonging to A. This is an extension to Frechet spaces of a result (see [PZ]) due to D. Preiss and L. Zajicek.

On the solutions to 1-Laplacian equation with L1 data

2009

AbstractIn the present paper we study the behaviour, as p goes to 1, of the renormalized solutions to the problems(0.1){−div(|∇up|p−2∇up)=finΩ,up=0on∂Ω, where p>1, Ω is a bounded open set of RN (N⩾2) with Lipschitz boundary and f belongs to L1(Ω). We prove that these renormalized solutions pointwise converge, up to “subsequences,” to a function u. With a suitable definition of solution we also prove that u is a solution to a “limit problem.” Moreover we analyze the situation occurring when more regular data f are considered.