University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Lower semicontinuity and relaxation of nonlocal $L^\infty$ functionals.

Lower semicontinuity and relaxation of nonlocal $L^\infty$ functionals.

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DNM - The mathematical design of new materials


We consider variational problems involving nonlocal supremal functionals, i.e.
L1(;Rm) 3 u 7! esssup(x;y)2  W(u(x); u(y)); with Rn a bounded, open set and a suitable function W : Rm  Rm ! R.
Using the direct methods of the Calculus of Variations it is shown for m = 1 that weak lower semi-continuity holds if and only if the level sets of a symmetrized and suitably diagonalized version of W are
separately convex. Moreover the supremal structure of the functionals is preserved in the process of relaxation, a question which is still open in the related context of double-integral functionals. In our proofs we
strongly exploit the connection between supremal and indicator functionals, thus reformulating the relaxation problem into studying weak closures of a class of nonlocal inclusions. Some special assumptions on
W allow us to generalize the results to the vectorial setting m > 1.
Joint work with Carolin Kreisbeck (Utrecht University)




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