We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Amp\`ere that are given by infimums of integro-differential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem, prescribing data at infinity, and prove existence, uniqueness, and $C^{1,1}$-regularity of solutions in the full space.

We develop an analytic theory of existence and regularity of surfaces arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in $\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is the tangential gradient along $\mathcal{M}$ of the mean curvature $H$ of $\mathcal{M}$ and $dA$ is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results. These are the first analytic results available on the literature for this problem.

In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. We prove existence of a $W^{2,p}$-viscosity solution and regularity up to $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha<1$(we improve this regularity near the boundary). The difficulty of this problem lays on a right hand side which involves the measure of the superlevel sets, making the problem nonlocal.

A horizontal $N$-dimensional plane, having a diffusion of its own, exchanges with the lower half space. There, a reaction-diffusion process, modelled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane. The origin of this problem is a two-dimensional reaction diffusion model proposed some time ago by the second author, in collaboration with H. Berestycki and L. Rossi, to model how biological invasions can be enhanced by a line of fast diffusion. Some counter-intuitive numerical simulations of this model, due to A.-C. Coulon, have been explained by the first two authors by transforming the model into a free boundary interacting with a line, and a careful study of the free boundary. At this occasion, it was noticed that the free boundary very much like that of the obstacle problem. The goal of the paper is to explain how this analogy with the obstacle problem can be pushed further in higher space dimensions.

In this paper we establish the global $$C^{1,\alpha }$$ regularity for solutions to the Dirichlet problem of the Monge–Ampère equation $$\det D^2 u=f$$ . By examples we show that our conditions are optimal. Our proof allows the degenerate case $$f\ge 0$$ , including the special case $$f\equiv 0$$ . We also prove the global $$C^{1,\alpha }$$ regularity for the convex envelope of a given function under optimal conditions.

The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.

We construct merely Lipschitz and C-1,C-alpha with rational alpha is an element of ( 0, 1 - 2/n] viscosity solutions to the Monge-Amp`ere equation with constant right hand side.

We complete the description, initiated in [6], of a free boundary travelling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dicatated by the velocity of the wave imposed by the line. This helps understanding some rather counter-intuitive numerical simulations of [8].

We study the 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.

Our goal in this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half-plane in which reaction-diffusion phenomena occur exchanges mass with a line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi, and the second author as a model of enhancement of biological invasions by a line of fast diffusion. It was observed numerically by A.-C. Coulon that the leading edge of the front, rather than being located on the line, was in the lower half-plane. We explain this behavior for a closely related free boundary problem. We construct travelling waves for this problem, and the analysis of their free boundary near the line confirms the predictions of the numerical simulations.

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are C1,1/2 , and that near non-degenerate points the fracture set is C1,α , for some α∈(0,1) .

We develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Amp\`{e}re equation. We prove an ABP estimate and a Harnack inequality and derive Hölder and $C^{1,\alpha}$ regularity results for solutions.

In this manuscript we consider a porous medium equation with non-local diffusion effects given by a fractional heat operator $\partial_t + (-\Delta)^s$ in two space dimensions. Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality, a higher order integral estimate, and a generalized version of the Div-Curl lemma.

Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form , with f a coercive function and lambda a constant, in the subquadratic case, that is, , appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that and for some , essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for f. Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive f which satisfies for some positive constant kappa. Since , this assumption imposes very mild restrictions on the growth of the potential f. We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide optimality results for the associated ergodic control problem, even in a pathwise sense, and without resorting to the parabolic problem.

Free boundaries occur in a lot of physical phenomena and are of major interest both mathematically and physically. The aim of this contribution is to describe new ideas and results developed in the last 20 years or so that deal with some nonlocal (sometimes called anomalous) free boundary problems. Actually, such free boundary problems have been known for several decades, one of the main instances being the thin obstacle problem, the so-called (scalar) Signorini free boundary problem. We will describe in this survey some new techniques that allow to deal with long-range interactions. We will not try to be exhaustive since the literature on this type of problem has been flourishing substantially, but rather we give an overview of the main current directions of research. In particular, we want to emphasize the link, very much well-known in the community, between minimal surfaces, their “approximation” by the Allen–Cahn equation and free boundary problems.

We show the existence of a Lipschitz viscosity solution u in Ω to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface ∂{u>0}∩Ω and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition uν++=uν−−, and hence u is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in [5], [6], and generalized by Wang [20], [21] and Feldman [11] to fully nonlinear operators, to conclude that the regular points in ∂{u>0}∩Ω form an open set of class C1,α. A novelty in our problem is that we have different operators, F+ and F−, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators M+ and M−, our results provide an alternative approach to obtain the stationary limit of a segregation model of populations with nonlinear diffusion in [19].

In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.