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UID:news514@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190104T233639
DTSTART;TZID=Europe/Zurich:20120321T151500
SUMMARY:Seminar Analysis: Lisa Beck (Bonn)
DESCRIPTION:In this seminar we will give a survey on some aspects of the cl
 assical regularity theory for W1\,p-solutions to elliptic problems (convex
  variational integral or elliptic systems)\, restricting ourselves to simp
 le model cases and explaining the challenges behind proving such results. 
 For scalar valued solutions full regularity (continuous or even better) ca
 n be established under very mild assumptions\, which is nowadays known as 
 the De Giorgi-Nash-Moser theory. In the vectorial case instead\, the vario
 us component functions and their partial derivative can interact in such a
  way that the system or variational integral under consideration allows di
 scontinuous or even unbounded solutions\, and in fact various counterexamp
 les to full regularity have been constructed. As a consequence\, only part
 ial regularity can be expected\, in the sense that the solution (or its gr
 adient) is locally continuous outside of a negligible set (the singular se
 t). We will give some heuristics on the generalapproach to partial regular
 ity results and then we briefly discuss how in some particular situations 
 (small space dimensions\, special structure conditions) an upper bound on 
 the Hausdorff dimension of the singular set can be obtained.
X-ALT-DESC:\nIn this seminar we will give a survey on some aspects of the c
 lassical regularity theory for W<sup>1\,p</sup>-solutions to elliptic prob
 lems (convex variational integral or elliptic systems)\, restricting ourse
 lves to simple model cases and explaining the challenges behind proving su
 ch results. For scalar valued solutions full regularity (continuous or eve
 n better) can be established under very mild assumptions\, which is nowada
 ys known as the De Giorgi-Nash-Moser theory. In the vectorial case instead
 \, the various component functions and their partial derivative can intera
 ct in such a way that the system or variational integral under considerati
 on allows discontinuous or even unbounded solutions\, and in fact various 
 counterexamples to full regularity have been constructed. As a consequence
 \, only partial regularity can be expected\, in the sense that the solutio
 n (or its gradient) is locally continuous outside of a negligible set (the
  singular set). We will give some heuristics on the general<br />approach 
 to partial regularity results and then we briefly discuss how in some part
 icular situations (small space dimensions\, special structure conditions) 
 an upper bound on the Hausdorff dimension of the singular set can be obtai
 ned.
DTEND;TZID=Europe/Zurich:20120321T161500
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