Related papers: Stochastic homogenization of fractional obstacle p…
We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional…
The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative…
We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the…
We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…
We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via…
We study two types of asymptotic problems whose common feature - and difficulty- is to exhibit oscillating Dirichlet boundary conditions : the main contribution of this article is to show how to recover the Dirichlet boundary condition for…
We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a…
We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the…
We study random walks in a balanced, i.i.d. random environment in $\mathbb Z^d$ for $d\geq 3$. We establish improved convergence rates for the homogenization of the Dirichlet problem associated with the corresponding non-divergence form…
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…
We consider a model of fractional diffusion involving the natural nonlocal version of the $p$-Laplacian operator. We study the Dirichlet problem posed in a bounded domain $\Omega$ of ${\mathbb{R}}^N$ with zero data outside of $\Omega$, for…
We investigate the asymptotic behavior of the solutions to the Neumann sieve problem for the Poisson equation in a thin, randomly perforated domain. The perforations (sieve-holes) are generated by a stationary marked point process.…
We study sums of locally dependent scores associated with general marked (i.e., labeled) Euclidean point processes. We introduce geometric mixing conditions on the underlying point process and a Lipschitz-"localization" condition on the…
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove…
In this paper we study the convergence of integral functionals with $q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$. Under the assumption that the perforations are small balls whose centres and radii are generated…
We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of…
We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to…
We present quantitative results for the homogenization of uniformly convex integral functionals with random coefficients under independence assumptions. The main result is an error estimate for the Dirichlet problem which is algebraic (but…
We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance and transmission boundary conditions. In particular, we aim to quantify diffracted fields…