Related papers: The Allen-Cahn Action functional in higher dimensi…
We develop the hypothesis that the dynamics of a given system may lead to the activity being constricted to a subset of space, characterized by a fractal dimension smaller than the space dimension. We also address how the response function…
We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For…
We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric…
We present several asymptotic results concerning the non-local Massari Problem for sets with prescribed mean curvature. In particular, we show that the fractional Massari functional $\Gamma$-converges to the classical one, and this…
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work…
We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are…
Fractional Brownian motion, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze fractional Brownian motion in the presence of a reflecting wall by means of…
We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our…
We present a survey on multiplicity results for the Allen--Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through $\Gamma$-convergence and isoperimetric theory. In the scalar case,…
A novel general framework for the study of $\Gamma$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $\Gamma$-limit of these kind of functionals by knowing…
We consider a system of two PDEs arising in modeling of motility of eukariotic cells on substrates. This system consists of the Allen-Cahn equation for the scalar phase field function coupled with another vectorial parabolic equation for…
We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional…
We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order…
In this set of notes, we present some recent developments on the fractional Allen-Cahn equation $$ (-\Delta)^s u = u-u^3,$$ with special attention to $\Gamma$-convergence results, energy and density estimates, convergence of level sets,…
We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the…
The so called Ahlfors function is the solution to an extremal problem. In this thesis, we study several properties of this function in a most general domain Omega. We consider the Ahlfors function in the context of analytic capacity, and…
We consider a subdivision invariant action for dynamically triangulated random surfaces that was recently proposed (R.V. Ambartzumian et. al., Phys. Lett. B 275 (1992) 99) and show that it is unphysical: The grand canonical partition…
A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its derivatives are established in the $L_1$ norm. The dependence…
This paper presents a new partial differential equation to build Brakke's motion, which is a weak notion of mean curvature flow. We call the equation a modified Allen-Cahn equation abbreviated to MAC. After that we introduce its benefit: An…
We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in constantly eliminating the…