Noticias en español
Nonlocal truncated Laplacians: representation formulas and Liouville results.
Abstract: We consider some nonlinear extremal integral operators that approximate the, so called, degenerate truncated Laplacians. For these operators we obtain representation formulas that lead to the construction of “fundamental solutions” and to Liouville type results. Differences with respect to both the local case and the uniformly elliptic framework will be emphasized.
Read MoreOptimal design problems for a degenerate operator in Orlicz-Sobolev spaces.
Abstract: An optimization problem with volume constraint involving the Φ-Laplacian in Orlicz-Sobolev spaces is considered for the case where Φ does not satisfy the natural condition introduced by Lieberman. A minimizer uΦ having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of uΦ along the free boundary, that the set {uΦ>0} has uniform positive density, that the free boundary is porous with porosity δ>0 and has finite (N−δ)-Hausdorff measure. Under a geometric compatibility condition set up by Rossi...
Read MoreIntegrability, gravitation and field theories.
Abstract: The fruitful relationship between physics and integrability has been widely tested by different approaches. They appear, for example, in areas like gravitation and condensed matter physics using holographic or generating solution techniques. In this talk I will review some recent developments connecting integrable models, gravitation, non-Hermitian physics and field theories. The relationship between the AKNS hierarchy and anti-de Sitter three dimensional gravity will be discussed as well as the stability of complex solitons having real conserved quantities. Based on: – M....
Read MoreBranch points for (almost-)minimizers of two-phase free boundary problems.
Abstract: In this talk, we will discuss minimizers and almost-minimizers of Alt-Caffarelli-Friedman type functionals. In particular, we will consider branch points in their free boundary. This is based on recent joint work with Guy David, Max Engelstein, and Tatiana Toro.
Read MoreExtremals in Hardy-Littlewood-Sobolev inequalities for stable processes.
Abstract: In this talk, we prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated with an stable operator, via a concentration-compactness principle for stable processes. This result can be seen as the first step to study existence of positive solutions of the corresponding nonlocal equations with critical nonlinearities perturbed by lower order terms.
Read MoreFinite point blowup for the critical generalized Korteweg-de Vries equation.
Abstract: In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de~Vries (gKdV) equation, including the determination of sufficient conditions for blowup,the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which...
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