Variational methods and PDEs
Organizers: dr hab. Jarosław Mederski, prof. IM PAN; dr Bartosz BieganowskiMonday, room 408, 12:15 - 13:45, usually on the 3rd Monday of the month. We are starting again in November 2020 (only online). Please contact us if you would like to join the zoom meeting.
Jacopo Schino (IMPAN): Normalized solutions to an elliptic Schrödinger equation in the infinite-mass case
Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties: one of them is the conservation of mass and gives rise to the search for normalized solutions. In this talk, I will explain a possible approach in the so-called strongly sublinear case (also referred to as the infinite-mass case), i.e., when the ratio between the nonlinearity and its argument diverges negatively at zero, which makes the usual approach impossible because the energy functional is not well defined. In the proposed approach, a family of approximating problems is considered so that the energy functional is well defined and a corresponding family of solutions is obtained, which eventually converge to a solution to the original problem.This is joint work in progress with Jarosław Mederski.
January 25, 2021
Simone Secchi (University of Milano-Bicocca): Concentration phenomena for the Schrödinger-Poisson system in R^2
We present some recent results obtain in collaboration with Denis Bonheure (Bruxelles, Belgium) and Silvia Cingolani (Bari, Italy) about a semiclassical analysis for a planar Schrödinger-Poisson system with potential functions.
December 21, 2020
Panayotis Smyrnelis (IMPAN): Equivariant vortices of the anisotropic Ginzburg-Landau system
The Ginzburg-Landau system has been extensively studied due to its applications in the theory of superconductors and superfluids. On the other hand, the anisotropic equation (with an additional Curl*curl term) models the formation of vortices in anisotropic liquid crystals (cf. the recent articles of Colbert-Kelly & Phillips (2013), and Golovaty, Sternberg & Venkatraman (2019)). In the isotropic case, vortex solutions of arbitrary degree are easily obtained by separating the variables, and reducing the Ginzburg-Landau equation to an O.D.E. I will explain how the anisotropic vortices can be constructed, as minimizers of the anisotropic energy in an appropriate class of equivariant maps. The equivariant constraint is instrumental in our construction, since the anisotropic vortices of degree |d|>1 are unstable and can only be observed indirectly in the experiments.
November 23, 2020
Bartosz Bieganowski (IMPAN, UMK): Solutions to a nonlinear Maxwell equation with two competing nonlinearities
We are interested in the curl-curl problem with an external potential and general sign-changing nonlinearities, in particular we consider the nonlinearity consisting of two competing powers. Under appropriate assumptions, we show that weak, cylindrically equivariant, divergence-free solutions are in one-to-one correspondence with weak solutions to a Schrödinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant, divergence-free solutions to the Maxwell equation, as well as to the Schrödinger equation.
May 13, 2019
Piotr Kozarzewski (Military University of Technology): On the condition of tetrahedral polyconvexity, arising from calculus of variations.
April 08, 2019
Jakub Siemianowski (UMK): Systems of elliptic PDEs on R^N
Abstract: We consider the system of second-order elliptic equations with the nonlinear term depending on the solution and its first-order derivatives on whole R^N. In order to solve it, firstly, we investigate auxiliary systems on large balls B_n. Then, by suitable estimates and the so-called tail estimate technique we show the convergence of auxiliary solutions to the solution of the initial problem.
March 25, 2019
Michał Gaczkowski (PW): Multiplicity of elliptic equations in critical case
Abstract: I will present results from the Ding's paper, about the multiplicity problem for the elliptic equations. The main idea of the work is to construct a corresponding equation on the sphere. Then, using the symmetries and the compactness, we will show that there are infinitely many solutions with different energy.
March 11, 2019
Jarosław Mederski (IMPAN): A new approach to normalized solutions of nonlinear Schrödinger equations.
Abstract: I will present a direct minimization method on the intersection od the unit L^2 sphere and the Pohozaev manifold. This approach allows to find normalized solutions to nonlinear Schrödinger equations.
February 18, 2019
Bartosz Bieganowski (UMK): The semirelativistic Choquard equation with a local nonlinear term
Abstract: We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity and a close-to-periodic potential in R^N. The result is proved by variational methods applied to an auxiliary problem in the half-space.
February 11, 2019
Jacopo Schino (IM PAN): From Schrödinger's to the curl-curl equation
Abstract: We show how we build a weak, cylindrically-equivariant and divergnce-free solution to the curl-curl problem from a weak and cylindrically-invariant solution to Schrödinger's equation in R^3 with a singular potential.
January 21, 2019
Jacopo Schino (IM PAN): On the Schrödinger equation with singular potential and double-behaviour nonlinearity.
Abstract: We provide a simple proof for the existence of a nontrivial solution to the Schrödinger equation with singular potential and where the nonlinearity has subcritical behaviour at infinity but supercritical behaviour at zero. We also provide motivations which lead to such an equation.
January 14, 2019
Jarosław Mederski (IMPAN): General class of optimal Sobolev inequalities and nonlinear scalar field equations.
Abstract: We find a class of optimal Sobolev inequalities with a general nonlinearity, and in particular we provide a new proof of the logarithmic Sobolev inequality. The optimizers of the inequalities satisfy nonlinear scalar field equations.
November 5, 2018
Panayotis Smyrnelis (IMPAN): Existence and properties of vortices in the Ginzburg-Landau model of liquid crystals.
Abstract: In the theory of light-matter interaction in nematic liquid crystals, the vector field of the molecules is described by a singular problem involving a Ginzburg-Landau type system of PDE. I will establish the existence of vortices and discuss some of their properties.