Applied Analysis and PDEs Seminar

November 24, 2023, and December 1, 2023
Jacopo Schino (North Carolina State University)
Mini course: A novel and simple approach to normalised solutions to Schrödinger equations ( Abstract).

March 29, 2024
Vitali Vougalter (University of Toronto)
Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions.
Abstract. The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in R^d, d=4,5. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used.
Webex link (3:00 PM - 4:00 PM, Friday, March 29, 2024) Meeting password: 2Cr5tmRxzu4

April 5, 2024
Spyridon Filippas (University of Helsinki)
On unique continuation for waves in singular media.
Abstract. The problem of unique continuation consists in recovering the whole wave from a partial observation and has applications to control theory and inverse problems. After presenting some fundamental results of the theory we will explain how one can prove a logarithmic stability result for wave operators whose metric exhibits a jump discontinuity across an interface. We make no assumption about the sign of the jump or the geometry of the interface. The key ingredient of our proof is a local Carleman inequality near the interface. Using a propagation argument, we derive then a global stability estimate.

April 12, 2024
Orestis Vantzos (Vantzos Research SMPC, Athens, Greece)
Pattern formation in Ginzburg-Landau potentials with hard obstacles.
Abstract. We present a new class of vector-valued phase field models, where the values of the phase parameter are constrained to a convex set. Like the classic Ginzburg-Landau functional, these models favour functions that partition the domain into subdomains, where the function takes one of a number of distinct values corresponding to distinct phases, separated by interfaces of small thickness. We characterise the phases and interfaces of the proposed generalized Ginzburg-Landau functional, in particular with respect to their dependency on the geometry of the convex constraint set. Furthermore, we introduce an efficient proximal gradient solver to study numerically their L2- gradient flow, i.e. the associated generalized Allen-Cahn equation. We look at different choices for the shape of the convex constraint set, leading to the formation of a number of distinct patterns.

April 19, 2024
Nicholas Alikakos (University of Athens, EKPA, Greece)
Multi-phase Minimizers for the Allen-Cahn System on the plane.
Abstract. In this talk we investigate multi-phase minimizers for the Allen-Cahn system on the plane. Our emphasis in on distinct surface tension coefficients. The proofs do not rely on symmetry. Coexistence of an arbitrary number of phases is related to the existence of the relevant minimizing cones for the minimal partition problem. For example, the orthogonal cross with four phases is minimizing for certain class of surface tension coefficients. We focus on two examples: the entire solution for the triple junction, and a four-phase minimizer with three-phase Dirichlet data (the triangle). The results presented in the talk are based on joint work with Zhiyuan Geng (Triple Junction), and with Dimitrios Gazoulis (The Triangle).

April 26, 2024
Nikolaos Roidos (University of Patras)
The heat asymptotics near cones and the spectrum of the Laplacian on the cross-sections.
Abstract. We consider the heat equation on manifolds with isolated conical singularities and investigate the full asymptotic behaviour of the solutions near the conical tips in terms of the spectrum of the Laplacian on the cross-sections.
Webex link (3:00 PM - 4:00 PM, Friday, April 26, 2024) Meeting password: 2Cr5tmRxzu4