A note on non lower semicontinuous perimeter functionals on partitions
Annibale Magni Matteo Novaga
We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.
keywords: regularity of interfaces. Semicontinuity minimal partitions
Closed curves of prescribed curvature and a pinning effect
Matteo Novaga Enrico Valdinoci
We prove that for any $H: R^2 \to R$ which is $Z^2$-periodic, there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close to $H$ in $L^1$, with $L^\infty$-norm controlled by the one of $H$, and with the same average of $H$, for which there exists a smooth closed curve $\gamma_\varepsilon$ whose curvature is $H_\varepsilon$. A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.
keywords: heterogeneous media. Prescribed curvature pinning phenomena
The geometry of mesoscopic phase transition interfaces
Matteo Novaga Enrico Valdinoci
We consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane.
keywords: plane-like solutions. Ginzburg-Landau-Allen-Cahn equation density estimates
The isoperimetric problem for nonlocal perimeters
Annalisa Cesaroni Matteo Novaga

We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.

keywords: Fractional perimeter nonlocal isoperimetric inequality Poincaré inequality
Volume constrained minimizers of the fractional perimeter with a potential energy
Annalisa Cesaroni Matteo Novaga

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume integral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.

keywords: Fractional perimeter nonlocal isoperimetric problem periodic medium prescribed curvature problem
A symmetry result for the Ornstein-Uhlenbeck operator
Annalisa Cesaroni Matteo Novaga Enrico Valdinoci
In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation $\Delta u=F'(u)$, which are monotone in some direction. In this paper we prove the analogous statement for the equation $\Delta u- \langle x,\nabla u\rangle u=F'(u)$, where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.
keywords: geometric Poincaré inequalities. Symmetry results Ornstein-Uhlenbeck operator
A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications
Matteo Novaga Diego Pallara Yannick Sire
The purpose of this paper is to study a boundary reaction problem on the space $X \times {\mathbb R}$, where $X$ is an abstract Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation $$ (-\Delta_\gamma)^s u= f(u), $$ with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$ monotone solutions are one-dimensional.
keywords: Wiener spaces. Fractional Ornstein-Uhlenbeck operator
Eventual regularity for the parabolic minimal surface equation
Giovanni Bellettini Matteo Novaga Giandomenico Orlandi
We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a strictly positive finite time.
keywords: eventual regularity of solutions. Parabolic minimal surface equation
One-dimensional symmetry for semilinear equations with unbounded drift
Annalisa Cesaroni Matteo Novaga Andrea Pinamonti
We consider semilinear equations with unbounded drift in the whole of $R^n$ and we show that monotone solutions with finite energy are one-dimensional.
keywords: semilinear elliptic PDEs Symmetry and monotonicity properties of solutions Ornstein-Uhlenbeck type operators Liouville theorems energy estimates Poincaré-type inequality.
Minimizers of anisotropic perimeters with cylindrical norms
Giovanni Bellettini Matteo Novaga Shokhrukh Yusufovich Kholmatov

We study various regularity properties of minimizers of the $\Phi$–perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

keywords: Non parametric minimal surfaces anisotropy sets of finite perimeter minimal cones anisotropic Bernstein problem

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