Abstracts Spring Semester 2026

Title: The Caffarelli-Silvestre extension phenomenon for complete Bernstein functions of the Laplacian 

Abstract: In their renowed paper, Caffarelli and Silvestre showed that, for 0<s<1, the fractional Laplacian (-\Delta)^s is the Dirichlet-to-Neumann operator for the differential operator div(t^{1-2s}\nabla u(x,t)) on the upper half-space.

In this talk we present a generalization of the extension techniques due to M. Kwaśnicki and J. Mucha, to the case of operators that are complete Bernstein functions of the Laplacian.

Title: Diffeomorphism-based convex integration schemes for incompressible fluid flows 

Abstract: We will review how one can design convex integration schemes using diffeomorphisms to address problems that are typically not amenable to more traditional schemes. To illustrate this philosophy, we will consider the construction of rough steady Euler flows with certain prescribed topological properties and of Hölder continuous dissipative solutions to ideal MHD. The talk is based on joint work with Daniel Peralta-Salas and Javier Peñafiel-Tomás. 

Title: C^{1+alpha} regularity for fractional p-harmonic functions

Abstract: The fractional p-Laplacian is the natural nonlocal counterpart to the classical p-Laplacian, arising as the first variation of the fractional Gagliardo seminorm. In this talk, we present a recent result, obtained in collaboration with David Jesus (KAUST) and Luis Silvestre (University of Chicago), establishing that solutions to the homogeneous equation enjoy Hölder differentiability, extending the well-known regularity theory from the local to the nonlocal setting.

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