Francesca Bianchi (University of Parma)

A sharp Hardy inequality in fractional Sobolev spaces

We consider the fractional Hardy inequality and we look for the sharp constant on open subsets of the Euclidean space. In particular we compute such a constant in the class of convex sets under the restrictions $sp\ge 1$ or $p=2$.

Some of the results presented have been obtained in collaboration with Lorenzo Brasco (University of Ferrara), Firoj Sk (Okinawa Institute of Science and Technology) and Anna Chiara Zagati (University of Parma).

Renzo Bruera (Universitat Politècnica de Catalunya)

An interior regularity result for the MEMS problem

In this talk we present an interior regularity result for the class of stable solutions to a semilinear elliptic equation with a singular nonlinearity. More precisely, given a bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we consider the problem $$\{\begin{array}{rlrl}-\mathrm{\Delta }u& =f(u)& & \text{in }\mathrm{\Omega }\\ 0<& u<1& & \text{in }\mathrm{\Omega }\\ u& =0& & \text{on }\partial \mathrm{\Omega },\end{array}$$ where the nonlinearity $f\in C^1((0,1))$ is assumed to be positive, nondecreasing, and to satisfy $f(1)=+\mathrm{\infty }$ and ${\int }_0^{1}f(t)dt=+\mathrm{\infty }$. The model nonlinearities for this problem are the powers, i.e., $f(t)=(1-t{)}^{-p}$ for $p>1$. For $p=2$ and $n=2$ this equation models the deflection of a dielectric elastic membrane in a microelectromechanical system (MEMS).

A solution $u$ to the problem above is said to be stable if the following inequality holds: $$\int f^{\prime }(u){\xi }^2\le \int |\mathrm{\nabla }\xi {|}^2,\mathrm{\forall }\xi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }).$$ Under a Crandall-Rabinowitz type assumption on $f$, we are able to prove that $u<1$in $\mathrm{\Omega }$ up to the optimal dimension $n=6$, and as a consequence, $u$ is smooth in $\mathrm{\Omega }$ (in contrast to singular solutions, which attain the value 1 somewhere in $\mathrm{\Omega }$).

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Ángel Crespo-Blanco (Technische Universität Berlin)

A new class of double phase variable exponent problems and a Nehari manifold approach

During the last decade the so-called double phase operator has drawn attention from researchers. Originally it was introduced by Zhikov in the context of homogenization and elasticity theory (see [Zhikov, Kozlov and Oleinik, Springer Berlin, Heidelberg (1994)]) and as an example for the Lavrentiev phenomenon (see [Zhikov, Russ. J. Math. Phys. 3 (1995)]). It regained popularity after some novel regularity results for local minimizers of the corresponding functional (see [Baroni, Colombo and Mingione, Nonlinear Anal. 121 (2015)], [Baroni, Colombo and Mingione, St. Petersburg Math. J. 27 (2016)],[Baroni, Colombo and Mingione, Calc. Var. Partial Differential Equations 57 (2018)]).

In this talk we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. This part of the talk is based on [Crespo-Blanco, Gasinski, Harjulehto and Winkert, J. Differential Equations 323 (2022)], where we prove useful properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular, density of smooth functions) and also properties of this new double phase operator (continuity, strict monotonicity, (S+)-property). In contrast to the previously known constant exponent case we are able to weaken the assumptions on the data.

After that we consider a problem with superlinear right-hand side. This last part of the talk is based on [Crespo-Blanco and Winkert, arXiv preprint (2022)], in which under very general assumptions on the nonlinearity we prove a multiplicity result for such problems, whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains. Furthermore, we derive a priori estimates on the solutions in the $L^{\mathrm{\infty }}$-norm under the very general setting used above.

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Iñigo U. Erneta (Universitat Politècnica de Catalunya)

Global regularity of stable solutions to semilinear equations

In this talk, I will announce some recent results on the global regularity of stable solutions to semilinear equations. These are a follow-up to the interior estimates in [Erneta, Commun. Pure Appl. Anal. (2022)].

Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ of class $C^{1,1}$, we consider stable solutions to $a_{ij}(x)u_{ij}+b_i(x)u_i+f(u)=0$ in $\mathrm{\Omega }$ vanishing on the boundary. Here, stability amounts to the nonnegativity of the principal eigenvalue of the linearized equation. We will show that stable solutions are globally Hölder continuous when $n\le 9$. This dimension is optimal, since there are examples of unbounded stable solutions for $n\ge 10$. Before our work, the only optimal result [Cabré, Figalli, Ros-Oton, and Serra, Acta Math.224 (2020)] involved the model operator, the Laplacian, and needed a $C^3$ regularity assumption on the domain.

To prove the global result, flattening out the boundary, it will suffice to establish a priori estimates on half-balls. Our bounds there will be independent of the nonlinearity $f\in C^1$, which we assume to be nonnegative, nondecreasing, and convex. For the extension of these estimates to $C^{1,1}$ domains, it is essential to make the constants in our bounds depend on specific norms of the coefficients, namely, the $C^{0,1}$ norm of $a_{ij}$ and the $L^{\mathrm{\infty }}$ norm of $b_i$.

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Antonio J. Fernández (ICMAT)

Multiple solutions for the nonlocal Liouville equation in $\mathbb{R}$

We construct multiple solutions to the Liouville type equation $$(-\mathrm{\Delta }{)}^{\frac{1}{2}}u=K(x)e^u\text{ in }\mathbb{R}.$$ More precisely, for $K$ of the form $K(x)=1+\epsilon \kappa (x)$ with $\epsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha }(\mathbb{R})\cap L^{\mathrm{\infty }}(\mathbb{R})$ for some $\alpha >0$, we prove existence of multiple solutions to the above equation bifurcating from the so-called Aubin-Talenti bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.

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Alejandro Fernández-Jiménez (University of Oxford)

Concentration phenomena arising in Aggregation Fast-Diffusion equations

Over the last two decades, a lot of attention has been given to the Aggregation-Diffusion equation $${\partial }_t\rho =\mathrm{div}(\rho \mathrm{\nabla }(U^{\prime }(\rho )+V+W\ast \rho )).$$ This family of equations has become popular because, amongst others, it models the mean-field limit of systems with a large number of interacting particles arising in biology. Because of that, there has been a lot of interest concerning the study of the asymptotic behaviour as $t\to \mathrm{\infty }$, discussing the existence or not of Dirac deltas, which are usually described as concentration phenomena.

We can understand this problem as a gradient flow and this approach has lead to several satisfactory results [Carrillo, McCann, and Villani, Rev. Mat. Iber. 19(3) (2003)]. However, the case in which we are interested, $V$ and $W$ regular enough and Fast Diffusion, $U(\rho )={\rho }^m/(m-1)$ with $0<m<1$, presents a challenge because the free energy associated is no longer convex and a gradient flow approach does not guarantee convergence to the minimiser. Therefore, in order to overcome these difficulties we follow a similar approach to the one in [Carrillo, Gómez-Castro, and Vázquez, J. Math. Pures et Appl. 157 (2022)], where they study the case $W=0$, $V$ regular enough and Fast Diffusion. Then, we rely on a combination of gradient flows, viscosity solutions and compactness arguments to study the asymptotic behaviour of the problem. We will review some of the literature and we will show how to use these techniques to obtain this type of results.

The talk presents ongoing joint work with Prof. J. A. Carrillo and Prof. D. Gómez-Castro.

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Claudia García (Universidad Autónoma de Madrid)

Self-similar solutions for the generalized surface quasi-geostrophic equation

In this talk, we will construct a large class of non-trivial (nonradial) self-similar solutions of the generalized surface quasigeostrophic equation. To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. Moreover, the solutions are of spiral type, locally integrable and may have a change of sign.

This is a joint work with Javier Gómez-Serrano.

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David Gómez-Castro (Universidad Complutense de Madrid)

Concentration versus Simplification in Aggregation-Diffusion Equations

Over the last two decades, intense work has been devoted to the Aggregation-Diffusion equation $${\partial }_t\rho =\mathrm{div}(\rho \mathrm{\nabla }(U^{\prime }(\rho )+V+W\ast \rho )).$$ This family of problem model, amongst other phenomena, the mean-field limit of systems with a large number of interacting particles arising in biology. It includes, for example, the famous model by Keller-Segel for chemotaxis.

From the mathematical point of view, they benefit from having a gradient flow structure. There is a long literature discussing the existence of gradient-flow solutions, characterising the existence/non-existence of minimisers of the associated free-energy functional and, in particular, discussing the existence or not of delta Deltas. The presence of a Delta is usually described as a concentration phenomena. In the absence of candidate stationary state, the solutions diffuse. We will discuss some of the key elements of the theory.

The aim of this talk is to present two works at opposite ends of the spectrum. On the one hand, the asymptotic formation of Dirac deltas in the case of Fast Diffusion [ J. A. Carrillo, G-C, and J. L. Vázquez, J. Math. Pures Appl. 157 (2022) ]. On the other, a general result of asymptotic simplification to the heat kernel when $W$ is bounded, with suitably integrable derivatives [ J. A. Carrillo, G-C, Y. Yao, and C. Zeng, Arch. Ration. Mech. Anal. To appear.].

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Irene Gonzálvez (Universidad Autónoma de Madrid)

The KPP diffusion-reaction problem in the hyperbolic space

In this talk we study the Cauchy problem in the hyperbolic space for the heat equation with KPP type forcing term. Considering the isometries and the geodesics of this space, we work with some initial data trapped between two invariant isometric sets. This allows us to prove how the solution converges asymptotically to some specific travelling wave in a moving frame with a logarithmic correction. Our results translate the known asymptotic theory of the KPP problem in the Euclidean space to the Hyperbolic space paying special attention to the geometry of this space.

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Salvador López-Martínez (Universidad Autónoma de Madrid)

Blowing up solutions to elliptic problems with natural growth in the gradient

In the talk we will discuss the following model: $$\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda u+\mu (x)|\mathrm{\nabla }u{|}^q+f(x),& x\in \mathrm{\Omega },\\ u=0,& x\in \partial \mathrm{\Omega }.\end{array}$$ Here, $\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^N$ with smooth boundary, $\lambda \in \mathbb{R}$, $1<q\le 2$, $\mu$ is a non-negative bounded function and $f$ is a non-negative $L^p(\mathrm{\Omega })$ function for $p>N$. In this general framework, the problem is nowadays well-understood for $\lambda <0$. More precisely, there exists a unique bounded weak solution for every $\lambda <0$, and the family of solutions in terms of $\lambda$ constitutes a continuum (see \cite{ArcoyaEtAl} and references therein). Moreover, if the problem does not admit a solution for $\lambda =0$, then the continuum blows up as $\lambda \to 0^{-}$ and the asymptotic behavior is characterized in terms of the so-called ergodic problem (see \cite{Porretta}).

If, on the contrary, there is a solution for $\lambda =0$, then the behavior of the branch of solutions for $\lambda \ge 0$ is not completely understood. It is conjectured that the branch bifurcates from infinity still at zero, meaning that there is a continuum of blowing up solutions as $\lambda \to 0^{+}$. However, this conjecture has been proven only in particular cases where, typically, restrictions depending on $q$, $N$ or $\mu$ appear (see \cite{LopezMartinez} and references therein).

In the talk we will give an overview of the results that are known for the range $\lambda >0$. We will also present some new results in this direction as part of the joint work \cite{CarmonaEtAl} with J. Carmona and P.J. Martínez--Aparicio.

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Rafael López-Soriano (Universidad de Granada)

Bound and ground states for a family of singular systems in ${\mathbb{R}}^N$

In this talk we will focus on the solvability of a family of nonlinear elliptic systems defined in ${\mathbb{R}}^N$. Such equations contain Hardy type potentials and criticalities in Sobolev or Hardy-Sobolev sense coupled by a possible critical term. By means of variational techniques, we shall find ground and bound states in terms of the coupling parameter $\nu$ and the order of the different parameters and exponents. Actually, for a wide range of parameters we find solutions as minimzers or Mountain-Pass critical points of the energy functional on the underlying Nehari manifold.

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María Medina (Universidad Autónoma de Madrid)

Positive solutions to critical competitive systems in low dimension

We will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimension 3 and 4.

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Alejandro Ortega (Universidad Carlos III de Madrid)

Nonlocal Concave-Convex Critical problems with Mixed Boundary Conditions

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Félix del Teso (Universidad Autónoma de Madrid)

Evolution driven by the infinity fractional Laplacian

We consider the evolution problem associated to the following infinity fractional Laplacian operator ${\mathrm{\Delta }}_{\mathrm{\infty }}^s$ introduced by Bjorland, Caffarelli and Figalli as the infinitesimal generator of a non-Brownian tug-of-war game: $$\underset{|y|=1}{sup}\underset{|\tilde{y}|=1}{inf}{\int }_0^{\mathrm{\infty }}(\varphi (x+\eta y)+\varphi (x-\eta \tilde{y})-2\varphi (x))\frac{\text{d}\eta }{{\eta }^{1+2s}}.$$ We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions. Finally, we propose a fully discrete and monotone finite-difference scheme, and support our theoretical results with numerical evidence.

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Damià Torres-Latorre (Universitat de Barcelona)

Generic regularity of free boundaries for the thin obstacle problem

The Signorini problem is a free boundary problem arising from elastostatics, and can also be seen as the counterpart to the obstacle problem when the obstacle has codimension 1. The main questions are understanding the regularity of solutions and the regularity of the free boundary. In the first part of the talk, I will present the problem and discuss some relevant known results. Then, I will explain the concept of generic regularity in free boundary problems. Finally, I will expose my recent result with Fernández-Real about generic regularity in the Signorini problem, and comment briefly on the recent developments that have made it possible.

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Marvin Weidner (Universitat de Barcelona)

Bernstein method for nonlocal operators

The Bernstein method allows to prove derivative estimates for solutions to a large class of elliptic equations by application of the maximum principle to certain suitable auxiliary functions. In this talk, we explain how the Bernstein method can be extended to a large class of integro-differential equations driven by nonlocal operators that are comparable to the fractional Laplacian. Moreover, we discuss several applications of this technique to nonlocal obstacle problems in bounded domains, as well as to fully nonlinear integro-differential equations. This talk is based on a joint work with Xavier Ros-Oton and Damià Torres-Latorre.

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