Related papers: Regularity of a $\bar\partial$-solution operator f…

Let $\Omega$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also…

In this paper, we provide some characterizations of strong pseudoconvexity by the boundary behavior of intrinsic invariants for smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$. As a consequence, if such domain is…

In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are…

This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$,…

This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the…

In the paper we show the existence of different types of peak functions in classes of $\mathbb C$-convex domains. As one of tools used in this context is a result on preserving the regularity of $\mathbb C$-convex domains under projection.

We consider the incompressible Navier-Stokes equations in a moving domain whose boundary is prescribed by a function $\eta=\eta(t,y)$ (with $y\in\mathbb R^2$) of low regularity. This is motivated by problems from fluid-structure…

We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the…

We show that Property $(P)$ of $\partial\Omega$, compactness of the $\bar{\partial}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $\Omega={\{(z, w_1, w_2,\dots, w_n) \in…

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…

This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions. It is shown…

Let $(-\Delta)_c^s$ be the realization of the fractional Laplace operator on the space of continuous functions $C_0(\mathbb{R})$, and let $(-\Delta_h)^s$ denote the discrete fractional Laplacian on $C_0(\mathbb{Z}_h)$, where $0<s<1$ and…

We study maximal regularity with respect to continuous functions for strongly continuous semigroups on locally convex spaces as well as its relation to the notion of admissible operators. This extends several results for classical strongly…

In this paper, we prove the existence of solutions of the Poincar\'e-Lelong equation $\sqrt{-1}\partial\bar{\partial}u=f$ on a strictly convex bounded domain $\Omega\subset\mathbb{C}^n$ $(n\geq1)$, where $f$ is a $d$-closed $(1,1)$ form and…

Regularity theorems \`a la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general…

We give simple new proofs of two well-known results for the Schr\"odinger operator: first, the Brunn--Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first…

For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $\Omega$ we show that solutions of the corresponding elliptic problem with Robin and thus in…

We give a sufficient condition for subelliptic estimates for the d-bar-Neumann operator on smoothly bounded, pseudoconvex domains in $\mathbb{C}^n$. This condition is a quantified version of McNeal's condition ($\tilde{P}$) for compactness…

Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $\Omega$. Furthermore, there exists a $t_0>0$ such that the operators $N$,…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…