# Geometric Properties of Local and non-Local PDEs (17w5047)

Arriving in Oaxaca, Mexico Sunday, May 21 and departing Friday May 26, 2017

## Organizers

Manuel del Pino (University of Chile)

Jeffrey Case (The Pennsylvania State University)

Juncheng Wei (University of British Columbia)

Maria del Mar Gonzalez (Universidad Autonoma de Madrid)

## Objectives

In the last decade, numerous publications have been devoted to the study of nonlinear non-local equations in connection with various areas of nonlinear analysis and PDEs; e.g.\ fluid dynamics, solid mechanics and the dynamics of dislocations, regularity theory of integro-differential equations, conformal geometry, and minimal surfaces. We describe below several of these topics and their motivations.

Non-local equations have led to several major advances, in both applied and purely theoretical aspects, in the last years. The prototypical operator is the fractional Laplacian $(-\Delta)^s$ for $s \in (0,1)$. Besides its own interest in harmonic analysis, the fractional Laplacian is of great importance in probability theory, since the fractional Laplacian is the basic example of infinitesimal generator of Levy processes. In general, one needs to deal with integro-differential equations, for which new analysis tools need to be developed.

From the physical point of view, nonlocal interactions involving long-range effects, particularly of Coulombic-type, arise in many physical systems. These effects result in systems with non-local energies and the surface tension associated with phase segregation often becomes non-local. For example, sharp-interface descriptions can result in interface evolution driven by a nonlocal curvature, and minimizers of the energies are a new type of minimal surfaces associated with non-local surface-area functionals for which a whole new geometrical theory needs to be developed.

We would like also to point out that in many models coming from physics (for instance, fluid dynamics or combustion theory), the fractional Laplacian as an operator of diffusion is much closer to the reality that is observed experimentally than the Brownian diffusion coming from the Laplacian. Physicists have given this the name of anomalous diffusion, in the sense that the scaling between time and space is not given by the one of a Brownian diffusion process but a different one.

The opportunity offered by BIRS to exchange ideas will lead to significant advances in tackling and modeling these problems and to broad impact in the many related application fields. Special attention will be given to the inclusion of young researchers, minorities and women.\\

Below we detail the scientific content of this proposal:

In this area of research, the goal is to describe the regularity properties of equations of the type $$\mathcal I u=f$$ where $\mathcal I$ is an integral operator given by the formula $$\int_{\mathbb R^n} (u(x+y)+u(x-y)-2u(y))K(x,y)\,dy,$$ for a suitable (singular) kernel $K(x,y)$. The fractional Laplacian is the canonical example of this type of integral operators. Schauder-type regularity theory has been developed, but there are still many open questions on regularity. A hot topic now is the study of fully non-linear non-local operators.

In addition, the regularity theory for parabolic non-local equations is at its initial stage. More specifically, it is very interesting to study free boundary problems, where one needs to deal with regularity aspects of thin two-phase free boundary models and obstacle problems.

Many curvature prescription problems from conformal geometry are closely related to various aspects of critical nonlinear PDE. For example, the study of the scalar and $\sigma_k$-curvatures introduced many new techniques to the study of semilinear and fully nonlinear second-order PDE. In the past year, many new techniques have been introduced to solve a semilinear fourth-order PDE, the $Q$-curvature prescription problem. Similar problems arise in the study of renormalized volumes and curvature prescription problems on manifolds with boundary.

Quite surprisingly, there is a connection between fractional Laplacians and conformal geometry. This happens in the framework of scattering theory. Indeed, on a Riemannian manifold $(M^n,g)$ which is the boundary of an $(n+1)$-dimensional manifold, one can construct conformally covariant operators of fractional order of the form $$P_s^g =(-\Delta_g)^s+ \Psi^g_s$$ where $\Psi^g_s$ is a pseudo-differential operator of lower order.

One can then introduce a notion of fractional curvature $Q_s^g$ and study the prescription of it on the compact manifold $M$; i.e.\ the Yamabe and Nirenberg problems for fractional curvature. The equation one would like to solve is $$ P_s^g u=Q_s^g u^{\frac{n+2s}{n-2s}},\quad u>0, $$ which is a non-local, semilinear equation with critical exponent. Some of the techniques for the 4th-order $Q$-curvature have been used to understand nonlocal PDE of fractional order in $(2,4)$. It is also very interesting to investigate the singular versions of these problems, in which the metric is allowed to blow-up on a closed set of the manifold.

A long term goal is to understand the geometric and topological information contained in the fractional curvature. This workshop will provide a timely opportunity for experts to discuss extending existing techniques for nonlinear curvature equations to these and other similar problems.

An innovative notion of perimeter has been recently introduced. Roughly speaking, if $E$ is an open set in $\mathbb R^n$, then its fractional perimeter of order $s\in (0,1/2)$ is given by $$\text{Per}_s(E)=\int_E \int_{E^c} \frac{dxdy}{|x-y|^{n+2s}}. $$ Up to a renormalization, it converges as $s\to\frac12$ to the standard perimeter in De Giorgi sense, and we can say that this nonlocal perimeter is an interpolation between the classical perimeter at one end and the volume at the other end.

As a consequence, one obtains the notion of fractional (or non-local) minimal curvature. So far very few examples of nonlocal minimal surfaces are known. Analogue to classical minimal surfaces there are many open questions regarding the critical dimension for regularity and Bernstein type theorems for nonlocal minimal graphs.

Finally in connection with De Giorgi conjecture and more generally the geometric theory of elliptic PDEs, it is of great interest to investigate the analogous De Giorgi conjecture for fractional Allen-Cahn equation $$(-\Delta)^s u=u-u^3\,\,\,x \in \mathbb R^n.$$ This conjecture asserts that monotone solutions are one-dimensional. There is a deep link between the De Giorgi conjecture and minimal surfaces given by entire graphs.

While the De Giorgi conjecture is almost completely solved in the standard diffusion case, in the fractional case, De Giorgi-type results have been only proved in some cases for low dimensions. The remaining cases are completely open.

Non-local equations have led to several major advances, in both applied and purely theoretical aspects, in the last years. The prototypical operator is the fractional Laplacian $(-\Delta)^s$ for $s \in (0,1)$. Besides its own interest in harmonic analysis, the fractional Laplacian is of great importance in probability theory, since the fractional Laplacian is the basic example of infinitesimal generator of Levy processes. In general, one needs to deal with integro-differential equations, for which new analysis tools need to be developed.

From the physical point of view, nonlocal interactions involving long-range effects, particularly of Coulombic-type, arise in many physical systems. These effects result in systems with non-local energies and the surface tension associated with phase segregation often becomes non-local. For example, sharp-interface descriptions can result in interface evolution driven by a nonlocal curvature, and minimizers of the energies are a new type of minimal surfaces associated with non-local surface-area functionals for which a whole new geometrical theory needs to be developed.

We would like also to point out that in many models coming from physics (for instance, fluid dynamics or combustion theory), the fractional Laplacian as an operator of diffusion is much closer to the reality that is observed experimentally than the Brownian diffusion coming from the Laplacian. Physicists have given this the name of anomalous diffusion, in the sense that the scaling between time and space is not given by the one of a Brownian diffusion process but a different one.

The opportunity offered by BIRS to exchange ideas will lead to significant advances in tackling and modeling these problems and to broad impact in the many related application fields. Special attention will be given to the inclusion of young researchers, minorities and women.\\

Below we detail the scientific content of this proposal:

#### Analysis of PDEs

**Existence and regularity of integro-differential equations**In this area of research, the goal is to describe the regularity properties of equations of the type $$\mathcal I u=f$$ where $\mathcal I$ is an integral operator given by the formula $$\int_{\mathbb R^n} (u(x+y)+u(x-y)-2u(y))K(x,y)\,dy,$$ for a suitable (singular) kernel $K(x,y)$. The fractional Laplacian is the canonical example of this type of integral operators. Schauder-type regularity theory has been developed, but there are still many open questions on regularity. A hot topic now is the study of fully non-linear non-local operators.

**Parabolic non-local equations**In addition, the regularity theory for parabolic non-local equations is at its initial stage. More specifically, it is very interesting to study free boundary problems, where one needs to deal with regularity aspects of thin two-phase free boundary models and obstacle problems.

#### Geometry

**Conformal geometry**Many curvature prescription problems from conformal geometry are closely related to various aspects of critical nonlinear PDE. For example, the study of the scalar and $\sigma_k$-curvatures introduced many new techniques to the study of semilinear and fully nonlinear second-order PDE. In the past year, many new techniques have been introduced to solve a semilinear fourth-order PDE, the $Q$-curvature prescription problem. Similar problems arise in the study of renormalized volumes and curvature prescription problems on manifolds with boundary.

Quite surprisingly, there is a connection between fractional Laplacians and conformal geometry. This happens in the framework of scattering theory. Indeed, on a Riemannian manifold $(M^n,g)$ which is the boundary of an $(n+1)$-dimensional manifold, one can construct conformally covariant operators of fractional order of the form $$P_s^g =(-\Delta_g)^s+ \Psi^g_s$$ where $\Psi^g_s$ is a pseudo-differential operator of lower order.

One can then introduce a notion of fractional curvature $Q_s^g$ and study the prescription of it on the compact manifold $M$; i.e.\ the Yamabe and Nirenberg problems for fractional curvature. The equation one would like to solve is $$ P_s^g u=Q_s^g u^{\frac{n+2s}{n-2s}},\quad u>0, $$ which is a non-local, semilinear equation with critical exponent. Some of the techniques for the 4th-order $Q$-curvature have been used to understand nonlocal PDE of fractional order in $(2,4)$. It is also very interesting to investigate the singular versions of these problems, in which the metric is allowed to blow-up on a closed set of the manifold.

A long term goal is to understand the geometric and topological information contained in the fractional curvature. This workshop will provide a timely opportunity for experts to discuss extending existing techniques for nonlinear curvature equations to these and other similar problems.

**Non-local minimal surfaces**An innovative notion of perimeter has been recently introduced. Roughly speaking, if $E$ is an open set in $\mathbb R^n$, then its fractional perimeter of order $s\in (0,1/2)$ is given by $$\text{Per}_s(E)=\int_E \int_{E^c} \frac{dxdy}{|x-y|^{n+2s}}. $$ Up to a renormalization, it converges as $s\to\frac12$ to the standard perimeter in De Giorgi sense, and we can say that this nonlocal perimeter is an interpolation between the classical perimeter at one end and the volume at the other end.

As a consequence, one obtains the notion of fractional (or non-local) minimal curvature. So far very few examples of nonlocal minimal surfaces are known. Analogue to classical minimal surfaces there are many open questions regarding the critical dimension for regularity and Bernstein type theorems for nonlocal minimal graphs.

**De Giorgi conjecture for non-local equations**Finally in connection with De Giorgi conjecture and more generally the geometric theory of elliptic PDEs, it is of great interest to investigate the analogous De Giorgi conjecture for fractional Allen-Cahn equation $$(-\Delta)^s u=u-u^3\,\,\,x \in \mathbb R^n.$$ This conjecture asserts that monotone solutions are one-dimensional. There is a deep link between the De Giorgi conjecture and minimal surfaces given by entire graphs.

While the De Giorgi conjecture is almost completely solved in the standard diffusion case, in the fractional case, De Giorgi-type results have been only proved in some cases for low dimensions. The remaining cases are completely open.