Search

Talk

Weak convergence of minors

  • Irene Fonseca (CMU + MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)

Abstract

Using calculus of variations techniques, it is shown that if $u_n \in w^{1,N} (\Omega ; R^N)$ converge to a function $u$ in $L^1 (\Omega ; R^N)$, where $\Omega$; is an open, bounded subset of $R^N$, if the sequence of all minors $\{ M (\nabla u_n)\}$ is equi-bounded in $L^1$, and if $det \nabla u_n$ converge to a function $f$ weakly in $L^1 (\Omega)$, then $$f=det \nabla u$$ a.e. $$x\in \Omega$$.This result was previously obtained by Giaquinta, Modica and Soucek using tools from Geometric Measure Theory, and it is sharp. In particular, for all $q \geq 1$, $1 \leq p < N-1$, for all $f \in L^q (\Omega)$, and for every $u \in W^{1,p} (\Omega ; R^N)$ $$\inf\limits_{\{ u_n\}} \left\{ \liminf\limits_{n \to 0} \int |\det \nabla u_n -f|^q dx : u_n \in W^{1,N} \left(\Omega ; R^N \right),u_n \rightharpoonup u \ in \ W^{1,p} \left( \Omega; R^N \right) \right\} = 0$$This work in collaboration with Jan Malý was initiated during his visit to the Max Planck Institute in March-April 1998.

seminar
26.11.96 30.01.25

Oberseminar Analysis

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E2 10 (Leon-Lichtenstein) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Augusteum - A314