# Talks

Gabriel Acosta (Universidad de Buenos Aires and IMAS-CONICET, Argentina)

Abstract: Weighted and fractional Sobolev spaces associated with seminorms of the type \[\int_\Omega \int_\Omega \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, \delta^{\beta}(x,y) \, dy \,dx,\]where $\delta(x,y)=\min\{d(x), d(y)\}$, have shown to be useful for establishing regularity results and developing efficient numerical methods for nonlocal equations involving the fractional Laplace operator [Acosta, G., Borthagaray, J. P., A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. (2017)].

In the first part of this talk [Acosta G., Drelichman, I., Durán, R. G., Weighted fractional Sobolev spaces as interpolation spaces in bounded domains, to appear in Math. Nach.], we characterize the real interpolation space between $L^p$ and a weighted Sobolev space involving weights that are certain positive powers of the distance to the boundary. In particular,\[(L^p(\Omega), W^{1,p}(\Omega,1,d^{ p}))_{s,p},\]is characterized by means of the seminorm \[\int_\Omega {\int_{|x-y|<\frac{d(x)}{2}} } \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, dy \, d(x)^{sp} \,dx.\]We discuss some related works and implications and generalizations of this result and its connection with the unweighted case treated in [Drelichman, I., Durán, R. G., On the interpolation space (Lp(Ω),W1,p(Ω))s,p in non-smooth domains. J. Math. Anal. Appl. (2019)].

In the second part of the talk, we explore an iterative method for solving systems arising from couplings between local and nonlocal models of the kind given in [Acosta G., Bersetche F., Rossi J., Local and nonlocal energy-based couplings models, to appear in SIAM J. Math.]. The proposed technique resembles the so-called Schwarz method and can be successfully treated within the framework developed by P.L. Lions.

Francisco Bersetche (Universidad Técnica Federico Santamaría, Chile)

Abstract: We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models, we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.

Andrea N. Ceretani (Universidad de Buenos Aires and IMAS-CONICET, Argentina)

Abstract: We present a definition for the fractional Laplacian $(-\Delta)^{s(\cdot)}$ with spatially variable order, $s(\cdot):\Omega\to [0,1]$; and discuss the well-posedness of the associated Poisson problem in a bounded domain $\Omega$. The approach is motivated by the extension results by Caffarelli-Silvestre; however, this theory does not apply directly to our case. For example, since we allow the order $s(\cdot)$ to take the values $0$ and $1$, we have to deal with Sobolev spaces with (in general) non-Muckenhoupt weights. In particular, this requires to investigate the existence of corresponding trace operators, in the new framework. We also present some results in this direction.

This is a joint work with Carlos N. Rautenberg, George Mason University, USA.

Patrick Ciarlet (ENSTA Paris, France)

Abstract: Some physical models may lead to mathematically ill-posed problems. For instance, in electromagnetism, one can consider a classical material and a metamaterial that are separated by an interface with a corner: using models derived from effective response measurements, it can happen that the permittivities (or permeabilities) which describe these materials are of opposite sign. Then, when solving the (local) interface models derived from electromagnetics theory, the overall permittivity is sign-changing: as a result, strong singularities may appear. In particular, the scalar problem may be ill-posed in $H^1$, even for two-dimensional models. More precisely, if one considers a piecewise constant permittivity $\varepsilon$, equal to $\varepsilon_c>0$ in the classical material, and $\varepsilon_m<0$ in the metamaterial, one finds that there exists an interval $I_{\ell}$ of $\mathbb{R}^-_*$ with $\{-1\}\in I_{\ell}$, called the critical interval, such that the problem is ill-posed if the contrast $\varepsilon_m/\varepsilon_c$ belongs to $I_{\ell}$, and well-posed (in the Fredholm sense) else.

To address this difficulty, at least two approaches are possible. One may use homogenization theory to change the local interface model. Or, one can propose nonlocal interface models with sign-changing coefficients, because for these models the requested regularity of the solution is weaker.

Regarding the second approach, that is the use of nonlocal interface models for scalar problems, it can be shown that, by using the same technique as for the local models, the so-called $T$-coercivity approach, the nonlocal critical interval $I_{n\ell}$ is a subset of the local interval $I_{\ell}$. In addition, numerical results suggest that the proposed nonlocal model has some advantages over the local one for values of the contrast that belong to $I_{\ell}$. On the practical side, it is well-known that nonlocal discrete models are more costly to solve than the local discrete models, so this issue must be addressed: for instance, by using high-performance storage schemes and linear algebra tools. Or, one may consider the nonlocal model near the interface only, and use the classical local models elsewhere.

In this talk, a brief overview of the local and nonlocal models will be proposed: for the local models, this is joint work with A.S. Bonnet-Ben Dhia, C.M. Zwölf, L. Chesnel, C. Carvalho, C. Scheid, and M. Rihani. For the nonlocal models, this is joint work with J.P. Borthagaray, H. Shourick, and P. Marchand.

João Vitor da Silva (Universidade Estadual de Campinas, Brazil)

Abstract: In this Lecture, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $\lambda_p$ and its corresponding eigenvector $(u_p,v_p)$, for the local/nonlocal PDE system \left\{-\Delta_p u + (-\Delta)^r_p u = \frac{2\alpha}{\alpha+\beta} \lambda|u|^{\alpha-2}|v|^{\beta}u in \Omega -\Delta_p v + (-\Delta)^s_p v = \frac{2\beta}{\alpha+\beta}\lambda |u|^{\alpha}|v|^{\beta-2}v in \Omega u = 0 on R^N \setminus \Omega v = 0 on R^N \setminus \Omega,where $\Omega\subset \R^N$ is a bounded open domain, $0<r, s<1$ and $\alpha(p)+\beta(p) = p$. Furthermore, we address the asymptotic limit as $p \to \infty$, proving the explicit geometric characterization of the corresponding first $\infty-$eigenvalue, namely $\lambda_{\infty}$, and the uniformly convergence of the pair $(u_p,v_p)$ to the $\infty-$eigenvector $(u_{\infty},v_{\infty})$. Finally, the triple $(u_{\infty},v_{\infty},\lambda_{\infty})$ verifies, in the viscosity sense, a limiting PDE system.

Leandro Del Pezzo (Universidad de Buenos Aires and CONICET, Argentina)

Abstract: In this talk we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary conditions is well posed for this operator in the framework of viscosity solutions. In addition, we show that solutions decay to zero exponentially fast as $t\to \infty$ with a bound that is given by the first eigenvalue for this problem that we also study. These results are obtained in a work performed in collaboration with Begoña Barrios, Julio Rossi and Alexander Quaas.

Aurelia Deshayes (Université Paris-Est Créteil, France)

Abstract: It is well-known that some partial differential equations, such as those involving the Laplace operator, can be interpreted using probabilistic objects such as random walks or Brownian motion. In a more general framework, the solutions of some nonlinear problems can be approximated by means of values of two-player games (with randomness).

In this talk, we will give an overview of these relationships at the interface between probability theory and PDEs and focus on tug-of-war games played on a random board, such as a geometric random graph. This talk is based on a work in progress in collaboration with Nicolás Frevenza, Alfredo Miranda and Julio Rossi.

Raúl Ferreira (Universidad Complutense de Madrid, Spain)

Abstract: We present some aspects about the blow-up phenomena for the numerical approximation (by finite differences) of the problem\[\left\lbrace \begin{array}u_t + \mathcal{L} u = u^p, & x \in \mathbb{R}^d, t>0, \\u(x,0) = g(x) ,\end{array} \right.\]where the diffusion operator $\mathcal{L}$ can be either, local or non-local. Joint work with F. del Teso.

Moritz Kaßmann (Universität Bielefeld, Germany)

Abstract: We explain recent results on the regularity of solutions to elliptic and parabolic nonlocal equations related to nonlocal quadratic forms. The driving operator can be seen as an integro-differential operator with fractional order of differentiability and measurable coefficients. We focus on pointwise bounds of the solutions, in particular of the fundamental solution and the Green function. The talk is based on recent results obtained together with Minhyun Kim and Marvin Weidner.

Minhyun Kim (Universität Bielefeld, Germany)

Abstract: In this talk, we study the boundary behavior of solutions to the Dirichlet problems for nonlocal nonlinear operators. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory. This talk is based on a joint work with Ki-Ahm Lee and Se-Chan Lee.

Alfredo Miranda (Universidad de Buenos Aires, Argentina)

Abstract: This talk deals with the interplay between probability and partial differential equations (PDEs). We will present two different obstacle-type systems of nonlinear PDEs. In both cases, we prove that there exists a two-player zero-sum game played in two different boards with different rules in each board whose value functions approximate the corresponding solutions to the PDE systems as a parameter that controls the size of the steps goes to zero. In the first game, in the first board one of the players decides to play a round of a Tug-of-War game, or to change boards and in the second board we play a random walk with the possibility of changing boards with a positive (but small) probability and a running payoff. In the second game, at each turn both players have the choice of playing a Tug-of-War game with noise, without changing boards, or to change to the other board (and then play one round of the other game). In collaboration with Julio D. Rossi.

Facundo Oliú (Universidad de la República, Uruguay)

Abstract: In the framework of singular control problems for linear diffusions, the problem of finding optimal long-run policies has ecological and economical applications. In this work, we prove that the reflecting barriers reach the infimum cost. Then a market affecting the cost is added and a mean field game is proposed. We give conditions for a mean field equilibrium and we show that the problem affected by the market is an asymptotic approximation of a flux of players.

Marcone C. Pereira (Universidade de São Paulo, Brazil)

Abstract: In this talk we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard formula is computed allowing domain perturbations given by embeddings of bounded sets while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.

Hernán Vivas (Universidad Nacional de Mar del Plata and CONICET, Argentina)

Abstract: Orlicz-Sobolev spaces are the natural setting for the study of variational problems with nonstandard growth, meaning that the energy under consideration is given by a potential whose behavior is different from a power. Such problems are typical, for instance, of statistical physics, where the exponential and entropic functions play a crucial role. Integro-differential equations, on the other hand, appear in the study of Lévy processes with jumps in which the infinitesimal generator of a stable pure jump process is given, through the Lévy-Khintchine formula, by an integro-differential operator. These have proven to be accurate models to describe phenomena in physics, biology, meteorology, and finance among many other fields.

In this talk we will discuss some recent results for integro-differential equations posed in fractional Orlicz-Sobolev spaces, ranging from eigenvalue problems to regularity and qualitative issues, and present some open problems and questions which we consider of interest. These are joint works with Julián Fernández Bonder, Ariel Salort and Sandra Molina.

Marvin Weidner (Universitat de Barcelona, Spain)

Abstract: 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.