Enrique Villamor

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 .

Regularity theory and traces of -harmonic functions

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.

Traces of monotone Sobolev functions

AbstractIn this paper we prove that ifu: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFbcVqdaahaaWcbeqa% aGqaciaa+5gaaaGccqGHsgIRcqWFDeIuaaa!45FB!

Analytic properties of monotone sobolev functions

{\mathbb{B}}^n \to {\mathbb{R}}

Positive results on discreteness and openness for mappings of finite distortion

, where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFbcVqdaahaaWcbeqa% aGqaciaa+5gaaaaaaa!42EF!

AN EXTENSION OF RESHETNYAK'S THEOREM

{\mathbb{B}}^n