Special Session 80: Functional inequalities and PDEs

Geometric Hardy inequalities on the Heisenberg group

Gerassimos Barbatis National and Kapodistrian University of Athens Greece Co-Author(s):    Marianna Chatzakou, Achilles Tertikas

Abstract: We present some new $L^p$ Hardy inequalities on the Heisenberg group which involve the distance to the boundary. We adopt an abstract approach which allows us to obtain inequalities with various distances. Some new Hardy inequalities for Carnot groups of step two are also established. The constants obtained are sharp. Joint work with M. Chatzakou and A. Tertikas

Fujita exponent for Hormander`s vector fields

Marianna Chatzakou Ghent University Belgium Co-Author(s):    Aidyn Kassymov, Michael Ruzhansky

Abstract: In the first part of the talk, on unimodular Lie groups, we study the global well-posedness of the heat equation with a time-dependent nonlinearity of the form $S_p(t)f(u)$. We obtain distinct nonexistence criteria according to the volume growth of the group (compact, polynomial, or exponential). In the specific case of the Heisenberg group $\mathbb{H}^n$, we also provide sufficient conditions; for the case of the first Heisenberg group $\mathbb{H}^1$ these coincide with the necessary ones. In the second part, we consider the same equation after dropping the group structure: the underlying space is $\mathbb{R}^n$ endowed with vector fields defining the operator and satisfying, among other assumptions, H\ormander's condition. In both settings, when $f(u)=u^p$, we compute the so-called critical Fujita exponent.

Functional inequalities in weighted fractional Sobolev spaces

Michal Kijaczko Wroclaw University of Science and Technology Poland Co-Author(s):

Abstract: In this talk, I will present recent results concerning weighted inequalities in fractional Sobolev-type spaces. The weights considered are powers of the distance to the boundary of the domain or any flat submanifold. This will include, among others, weighted Hardy inequalities with sharp constants for Gagliardo and Triebel-Lizorkin seminorms and related inequalities with remainder terms, such as fractional Hardy-Sobolev-Maz`ya inequality. Criticality of some Hardy weights will be discussed.

Liouville theorems for left-invariant PDEs via right-invariant derivatives

Alessia Kogoj University of Urbino Italy Co-Author(s):

Abstract: It is well known that a harmonic function with finite energy must be constant. We show how this result can be extended to partial differential operators that are left-invariant with respect to a Lie group law on $\mathbb{R}^n$.

Stability with explicit constants for reverse Sobolev inequalities on the sphere

Tobias K\"onig Goethe University Frankfurt Germany Co-Author(s):

Abstract: For $s - \frac{n}{2} \in (0,1) \cup (1,2)$, a reverse-type Sobolev inequality of order $2s$ holds on the sphere $\mathbb S^n$. In my talk, I will discuss recent results on the quantitative stability of this inequality. Implementing the classical proof strategy by Bianchi and Egnell is non-trivial here because the underlying operator $A_{2s}$ is not positive definite when $s > \frac{n}{2}$. Remarkably, the case $s - \frac{n}{2} \in (1,2)$ constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

On the superposition and the Phragm\`{e}n-Lindel\{o}f principles for the $p$-Laplacian

Pier Domenico Lamberti University of Padova Italy Co-Author(s):    Vitaly Moroz

Abstract: We prove a Phragm\`{e}n-Lindel\{o}f type comparison principle for equations driven by the $p$-Laplacian on exterior domains, in the presence of a negative potential and for $p \geq 2$. As a consequence, we obtain upper and lower bounds for subsolutions and supersolutions in the setting of Hardy-type potentials. The argument is based on a new superposition principle for the $p$-Laplacian. The results are part of a joint work with Vitaly Moroz.

Vectorial Kato inequality for p-harmonic maps

Katarzyna Mazowiecka University of Warsaw Poland Co-Author(s):    Andreas Gastel, Michal Miskiewicz, Patryk Tokarczuk

Abstract: We derive the sharp vectorial Kato inequality for $p$-harmonic mappings. Surprisingly, the optimal constant differs from the one obtained for scalar valued $p$-harmonic functions. As an application we show how Kato-type inequalities can be used to deduce regularity for $p$-harmonic maps from a three dimensional ball to a three dimensional sphere. Based on joint works with Andreas Gastel, Micha\l{} Mi\`skiewicz, and Patryk Tokarczuk.

Lieb-Thirring interpolation inequality and applications to systems of nonlinear Schr\{o}dinger equations

Phuoc Tai Nguyen Masaryk University Czech Rep Co-Author(s):    Phuoc-Tai Nguyen

Abstract: In this talk, I will discuss the existence of finite-rank operators for an interpolation version of the Lieb-Thirring inequality in the mass-supercritical case. Then I will apply this result to study global existence and finite-time blow-up for systems of nonlinear Schr\{o}dinger equations.

Symmetry of fractional Neumann eigenfunctions in the ball

Enea Parini Aix Marseille Universite France Co-Author(s):    Vladimir Bobkov

Abstract: We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian $(-\Delta)^s$, where $s \in (0,1)$, in an $N$-dimensional ball with nonlocal Neumann boundary conditions. By means of a spectral stability result, we prove that, when $s$ is sufficiently close to $1$, the eigenspace associated to the first nontrivial eigenvalue is generated by $N$ antisymmetric eigenfunctions with exactly two nodal domains in the ball. This is a joint work with Vladimir Bobkov (Ufa, Russia).

Semilinear Schr\odinger equations with critical Hardy potentials

Miltiadis Paschalis National and Kapodistrian University of Athens Greece Co-Author(s):    Konstantinos Gkikas; Miltiadis Paschalis

Abstract: Let $\Omega\subset\R^N$ ($N\geq 3$) be a bounded $C^2$ domain and $\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study positive solutions of the equation $(*)\, -\Delta u -\mu u/d_\Sigma^2 = g(u,|\nabla u|)$ in $\Omega$, where $\mu\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g:\R\times\R_+ \rightarrow \R_+$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data $u=\nu$ in the case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions, provided that the total mass of $\nu$ is small. This presentation is based on a joint work with Konstantinos Gkikas.

Davies-type Hardy inequalities on the Heisenberg group

Durvudkhan Suragan Nazarbayev University Kazakhstan Co-Author(s):

Abstract: We present Hardy inequalities on the Heisenberg group in a form reminiscent of Davies' inequalities in the Euclidean setting. Our results have two notable features. First, they improve previously known inequalities, yielding sharper estimates. Second, we prove them without imposing any boundary regularity assumptions. We also discuss how these Davies-type Hardy inequalities can be converted into sub-Laplacian eigenvalue lower bounds. This talk is based on recent joint work with Rupert L. Frank (LMU Munich and Caltech) and Ari Laptev (Imperial College London).

The FitzHugh-Nagumo System on cylindrical surfaces: symmetrization and effective system

Konstantinos Tzirakis Department of Mathematics and Applied Mathematics, University of Crete Greece Co-Author(s):    Georgia Karali and Israel Michael Sigal

Abstract: In this talk, I will present recent results on the FitzHugh-Nagumo (FHN) system of partial differential equations. We consider the FHN system in a more realistic geometric setting: on cylindrical surfaces of variable radii, rather than straight lines without internal geometric structure, as it has been extensively done. We show that, under some reasonable conditions, the solutions of the system are exponentially approximated by their radial averages. We also show that the radial averages are close, for very long times, to solutions of a 1-spatial-dimensional system involving the cylindrical profile: it is obtained from the standard FHN system by replacing the second-order derivative by the radial surface Laplace-Beltrami operator. This system is considerably simpler for mathematical analysis and numerical simulations. It offers an effective system for the pulse propagation. I will outline the analytical approach to obtain the above results, and, time permitting, I will discuss interesting related open problems and natural extensions of these results. The talk is based on a recent joint work with Israel Michael Sigal (University of Toronto, Canada) and Georgia Karali (NKUA, Greece).

Sharp remainder terms and stability of Lp-Poincare and Heisenberg-Pauli-Weyl inequalities with applications to spectral gaps

Nurgissa Yessirkegenov KIMEP University and Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):

Abstract: This talk is devoted to our recent results on sharp remainder terms for weighted Hardy-Poincare type inequalities, including the case of general non-radial weights and explicit sharp constants. As an application of this framework, we derive sharp remainder formulae for the Lp-Poincare inequality and the Lp-Heisenberg-Pauli-Weyl inequality. If time permits, we will further discuss applications to stability, spectral gap problems, and nonlinear parabolic partial differential equations.