Recent webinars

Europe/Lisbon
Room 6.2.52, Faculty of Sciences, University of Lisbon — Online

Delia Schiera
Delia Schiera, Instituto Superior Técnico, Universidade de Lisboa

On a pure Neumann Lane-Emden system: existence, convergence, and related results

We will consider a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and (sub)critical nonlinearities. In the critical regime, we show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. Moreover, through a suitable nonlinear eigenvalue problem, we prove convergence of solutions in dependence of the exponents of the nonlinearities in the (sub)critical range. Finally, I will briefly discuss related results on multiplicity, symmetry breaking, and regularity.

Based on joint works with A. Pistoia, A. Saldaña and H. Tavares.

Europe/Lisbon
Room P3.10, Mathematics Building — Online

Dario Mazzoleni
Dario Mazzoleni, University of Pavia

Existence (and nonexistence) for functionals with competing attractive and repulsive interactions

In the last few years the Gamow problem, namely

\[ \min\Big\{P(\Omega)+\varepsilon \int_{\Omega}\int_\Omega \frac{1}{|x-y|}\,dx\,dy : \Omega\subset \mathbb{R}^3, |\Omega|=1\Big\}, \]

for $\varepsilon>0$, has attracted a lot of attention from mathematicians. Nowadays it is well understood that for small $\varepsilon$ there exist a minimizer and it is a ball, while for very large $\varepsilon$ there is no minimizer.

Although it is very easy to formulate, there are still several open problems about it (mostly concerning nonexistence of minimizers for large $\varepsilon$ in a generalized $N$-dimensional setting).

A variation of this model, which could be called spectral Gamow problem consists in using the first eigenvalue of the Dirichlet Laplacian instead of the Perimeter, namely to consider

\[ \min\Big\{\lambda_1(\Omega)+\varepsilon \int_{\Omega}\int_\Omega \frac{1}{|x-y|}\,dx\,dy : \Omega\subset \mathbb{R}^3, |\Omega|=1\Big\}, \]

and in this talk we will provide some new results on this case.

Moreover, we will consider a different problem but with a similar structure, which can be seen as the minimization of a Hartree functional settled in a box, namely

\[ \min\Big\{\min_{u\in H^1_0(\Omega),\;\int u^2=1}\Big\{\int_\Omega|\nabla u|^2+q \int_\Omega\int_\Omega\frac{u^2(x)u^2(y)}{|x-y|}\,dx\,dy\Big\} : \Omega\subset \mathbb{R}^3,\;|\Omega|=1\Big\}, \]

for $q>0$.

The study of this functional arises when describing the ground state of a superconducting charge qubit.

We show that there is a threshold $q_1>0$ such that for all $q\leq q_1$ existence of minimizers occurs and minimizers are $C^{2,\gamma}$ nearly spherical.

We will also give some ideas (although nonconclusive) on how to treat the nonexistence issue for this functional.

The techniques and tools needed in the proofs are very broad. We employ spectral quantitative inequalities, the regularity of free boundaries, spectral surgery arguments and shape variations.

This is a joint project with Cyrill Muratov (Pisa) and Berardo Ruffini (Bologna).

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Makson S. Santos
Makson S. Santos, Universidade de Lisboa

Optimal regularity for general normalized $p$-Laplacian and applications

We study the regularity properties of viscosity solutions to a class of degenerate normalized $p$-laplacian equations. In particular, we prove that the gradient of viscosity solutions are Hölder continuous, and we give the optimal exponent. Moreover, we also show that viscosity solutions to equations with very general degeneracy laws are differentiable.

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Juraj Földes
Juraj Földes, University of Virginia

Global solutions and invariant measures for equations of fluid dynamics

Using a fluctuation dissipation method, we construct an invariant measure for the surface quasi-geostrophic equation (SQG) and 3D Euler equation. Since the support of the measure contains entire solutions, we obtain a manifold containing solutions that do not blow-up. This complements results in which a blow-up solutions for SQG and grow up solutions for Euler are constructed. The method of the proof relies on an addition of a stochastic forcing and a small dissipation to the equation. For such stochastic equation, one can construct an invariant measure and by passing the strength of the forcing and the dissipation to zero, we obtain the desired invariant measure. We also discuss the size of the support of the measure, which relies on the number of conservation laws for the particular equation.

This is a joint project with Mouhamadou Sy.

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Giuseppe Negro
Giuseppe Negro, Instituto Superior Técnico, Universidade de Lisboa

Sharp constants for Fourier restriction to the sphere

One of the most successful lemmas of modern Harmonic Analysis is that "curvature induces decay of the Fourier transform". For example, if sources of electromagnetic waves are distributed on a sphere, the resulting waves will point in all possible directions and thus interact destructively; the same would not happen if those sources were distributed on a plane. In the 1970s, leading analysts such as E.M.Stein proposed to investigate and quantify this kind of phenomena. Since the main mathematical tool involved is the Fourier transform, this gave birth to a field of Harmonic Analysis known as "Fourier restriction theory".

This talk is aimed at non-specialists. We will study the following problem: how to optimally distribute wave sources on the sphere, maximizing the size of the corresponding superposition of waves? We will give a fully detailed solution of the 3-d case, then give a brief explanation of why this problem is still open in higher dimension.