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SUMMARY:Lower semicontinuity and relaxation of nonlocal $L^\\infty$ functi
 onals. - Elvira Zappale (Università degli Studi di Salerno)
DTSTART:20190221T150000Z
DTEND:20190221T160000Z
UID:TALK120205@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<br>We consider variational problems involving nonlocal suprem
 al functionals\, i.e.<br>L1(\;Rm) 3 u 7! esssup(x\;y)2&nbsp\; W(u(x)\; u(y
 ))\; with Rn a bounded\, open set and a suitable function W : Rm &#2\; Rm 
 ! R.<br>Using the direct methods of the Calculus of Variations it is shown
  for m = 1 that weak&#3\; lower semi-continuity holds if and only if the l
 evel sets of a symmetrized and suitably diagonalized version of W are<br>s
 eparately convex. Moreover the supremal structure of the functionals is pr
 eserved in the process of relaxation\, a question which is still open in t
 he related context of double-integral functionals. In our proofs we<br>str
 ongly exploit the connection between supremal and indicator functionals\, 
 thus reformulating the relaxation problem into studying weak&#3\; closures
  of a class of nonlocal inclusions. Some special assumptions on<br>W allow
  us to generalize the results to the vectorial setting m &gt\; 1.<br>Joint
  work with Carolin Kreisbeck (Utrecht University)<br><br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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