Related papers: On the $L^p$-Poisson semigroup associated with ell…

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the…

We study the interrelation between the limit $L_p(\Omega)$-Sobolev regularity $\overline{s}_p$ of (classes of) functions on bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^d$, $d\geq 2$, and the limit regularity $\overline{\alpha}_p$…

In the present paper, we generalize the theory of quantitative homogenization for second-order elliptic systems with rapidly oscillating coefficients in $APW^2(\mathbb{R}^d)$, which is the space of almost-periodic functions in the sense of…

The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we…

In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: \[ \left\{{llll} \mathcal{A}^s u= v^p &…

In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda…

We study the eigenvalues for infinitesimal generators of semigroups of composition operators acting on Hardy spaces, Bergman spaces, and the Dirichlet space. Such semigroups are induced by semigroups of holomorphic functions. Depending on…

Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some…

In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for…

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its…

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain…

We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-\Delta)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the…

In this paper we study the asymptotic behavior of the solutions of a class of nonlinear elliptic problems posed in a 2-dimensional domain that degenerates into a line segment (a thin domain) when a positive parameter $\varepsilon$ goes to…

We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the…