Related papers: Second-Order $\Gamma$-Limit for the Cahn-Hilliard …

In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed…

An existence result is shown for the asymptotic Dirichlet problem for harmonic maps from the product of the hyperbolic planes to the hyperbolic space, where the Dirichlet data is given on the distinguished boundary (the product of the…

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$ and let $d_\Gamma$ denote the Euclidean distance to $\Gamma$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous…

Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div}…

In this study we consider the $\Gamma$-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either $\{1,\infty\}$ or $\{1,\beta \varepsilon^{-p}\}$ where…

The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation…

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the…

In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,\alpha}$ or convex domains in two dimensions. We establish the optimal…

The Cahn-Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been…

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two disjoint open subsets of…

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary…

We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the…

We describe the asymptotic behaviour of the minimal heterogeneous $d$-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set $\Omega\subseteq \mathbb{R}^d$, with $d\geq2$. Two parameters are…

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian…

The formal sharp-interface asymptotics in a degenerate Cahn-Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical…

We consider a boundary-value problem for the second order elliptic differential operator with rapidly oscillating coefficients in a domain $\Omega_{\epsilon}$ that is $\epsilon-$periodically perforated by small holes. The holes are divided…

This paper explores large sample properties of the two-parameter $(\alpha,\theta)$ Poisson--Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension…