We consider the Keller-Segel system in the plane with an initial condition with suitable decay and mass \(8 \pi\), which corresponds to the threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0\) with mass \(8 \pi\) such that for any initial condition sufficiently close to \(u_0\) and the same mass, the solution is globally defined and blows-up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is a joint work with M. del Pino, M. Musso and J. Wei.

We consider sinh-Poisson type equations (either Liouville or mean field form) with variable intensities and Dirichlet boundary condition on a pierced domain \(\Omega_\varepsilon:=\Omega\setminus \cup_{i=1}^m B(\xi_i,\varepsilon_i)\), where \(\Omega\) is a smooth bounded domain in \(\text{I\!R}^2\) which contains the points \(\xi_i\), \(i=1,\dots,m\), \(\varepsilon_i=\varepsilon_i(\varepsilon)>0\), \(i=1,\dots,m\), depending on some parameter \(\varepsilon>0\) small enough. Given \(m\) different points \(\xi_i\), \(i=1,\dots,m\), we have found suitable radii \(\varepsilon_i\), \(i=1,\dots,m\) such that for \(\varepsilon>0\) small enough our problem (either Liouville or mean field form) has a solution \(u_\varepsilon\) blowing up positively around each \(\xi_1,\dots,\xi_{m_1}\) and negatively around \(\xi_{m_1+1},\dots,\xi_m\) respectively, as \(\varepsilon\to0\). We have used a family of solutions of the singular Liouville equation \[ \Delta u+|x -\xi|^{\alpha-2}e^{u}=0 \quad\text{ in $\text{I\!R}^2$, }\quad\text{satisfying}\qquad \displaystyle\int_{\text{I\!R}^2} |x -\xi |^{\alpha-2} e^u <+\infty \] to construct an approximation of the solution depending on some parameters, suitable projected and scaled in order to make the error small enough in a suitable norm. Hence, we have found an actual solution as a small additive perturbation of this initial approximation by using a fixed point argument.

Part of these results have been obtained in collaboration with Pierpaolo Esposito (Università di Roma Tre, Italia) and Angela Pistoia (Università di Roma ``La Sapienza'', Italia).

In this talk, we establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \(\mathbb{R}^N\) which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \(-\infty\). We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \(\mathbb{R}^N\). Moreover, we give a detailed description of the shape of these domains.

This is a joint work with Mónica Clapp and Alberto Saldaña.

Transmission problems originally arose in elasticity theory, and they are nowadays of great interest due to its many applications in different areas in science. Typically, in these problems, there is a fixed interface where solutions may change abruptly, and the primary focus is to study their behavior across this surface. In this talk, we will discuss existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha}\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

These results are part of my PhD dissertation, and they are joint work with Luis A. Caffarelli and Pablo R. Stinga.

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized \(p-\)Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game.

Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation.

Joint work with P. Blanc (Jyväskylä), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).

We will discuss a variant of a classical geometric minimization problem in arbitrary dimensions \(d\geq 2\), known as nonlocal isoperimetric problem, which arises from studies in Nuclear Physics by Gamow in the 1930’s: \[ \inf_{\substack{\Omega\subset\mathbb{R}^d\\|\Omega|=m}}E_{a}^{\gamma}(\Omega):=\int_{\partial^*\Omega}a(x)d\mathcal{H}^{d-1}+\gamma\iint_{\Omega\times\Omega}\frac{1}{|x-y|^{\alpha}}dxdy \] for any \(\alpha\in(0,d)\) and \(\gamma>0\). By introducing a density \(a:\mathbb{R}^d\to[0,\infty)\) in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. Firstly, we get existence of minimizing sets for a general class of coercive densities \(a(x)\) and all values of ``mass". In addition, we prove essential boundedness of such minimizers. Finally, in the regime of “small” non-local contribution we completely characterize the minimizer, for densities that are monomial radial weights. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION No. 11201259.

In this talk we present recent results regarding the existence of infinitely many radially symmetric solutions to \[\label{eq main problem} \left\{ \begin{aligned} - \Delta_p u =&W(| x|)g(u) &\hbox{in} &\quad \,\,\,B_1 (0)\subset \mathbb{R} ^N, \,\\ u =& 0 & \hbox{in} & \quad \partial B_1 (0), \end{aligned} \right. \] where \(N\geq 2\), \(p>1\), \(\Delta _p u= \text{div} (|\nabla u|^{p-2} \nabla u)\) is the p-Laplacian operator, \(B_1 (0)\) stands for the unit ball in \(\mathbb{R} ^N\) centered at the origin, \(g: \mathbb{R} \longrightarrow \mathbb{R} \) is a nonlinear function which satisfies p-superlinearity conditions and the weight \(W: [0,1]\longrightarrow \mathbb{R}\) is a \(C^1 -\)function that changes sign.

Since the weight \(W\) takes negative values, the solutions to initial value problems associated to eq main problem may blow up at some \(r \in (X,1)\) and this fact represents a major difficulty when using phase-plane analysis arguments. We will discuss how this difficulty can be surpassed.

This is a recent work in collaboration with Alfonso Castro, Jorge Cossio and Sigifredo Herrón.

We extend the Krylov-Safonov theory and the corresponding H\"older estimates for the viscosity solutions of a type porous medium equation with non-variational structure. This work is in collaboration with Makson Santos (CIMAT-Guanajuato).