Variational methods and PDEs
dr hab. Jarosław Mederski, prof. IM PANMonday, room 408, 12:15 - 13:45 on the 3rd Monday of the month. We start again in October 2020.
October 19, 2020
Next seminar up to 5 persons due to Corona restrictions, possibly also online.
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.