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The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside…

We prove various equivalent characterisations of the Hardy space $H^p_{\mathcal{L}}(\mathbb{C}^n)$ for $0<p<1$ associated with the twisted Laplacian $\mathcal{L}$ which generalises the result of [MPR81] for the case $p=1$. Using the atomic…

Let $\Omega\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partial\Omega$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(\Omega)$ with zero mean value and zero Neumann…

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times…

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain $\Omega$. On the boundary, strong tangential anchoring is imposed. We prove that…

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we…

We prove partial regularity of suitable weak solutions to the Navier--Stokes equations at the boundary in irregular domains. In particular, we provide a criterion which yields continuity of the velocity field in a boundary point and obtain…

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let…

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value…

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad…

We analyze a family of singular Schr\"odinger operators with local singular interactions supported by a hypersurface $\Sigma \subset \mathbb{R}^n, n \ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily…

We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the…

We consider truncated Toeplitz operator on nearly invariant subspaces of the Hardy space $H^2$. Of some importance in this context is the boundary behavior of the functions in these spaces which we will discuss in some detail.

We devise and analyse a novel $\boldsymbol{H}(\textbf{curl})$-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the…

To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be…

We study the Schr\"odinger operator $L=-\Delta+V$ on a star-shaped domain $\Omega$ in $\mathbb{R}^d$ with Lipschitz boundary $\partial\Omega$. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced…

An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve $\Gamma$ infinitesimally looks like a line at almost all of its points in the sense that at $\mathcal{H}^1$-almost every point of $\Gamma$, the…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness…

Let $\Omega$ be a domain in $\mathbf R^d$ and $h(\varphi)=\sum^d_{k,l=1}(\partial_k\varphi, c_{kl}\partial_l\varphi)$ a quadratic form on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the $c_{kl}$ are real symmetric…

We extend to multilinear Hankel operators the fact that truncation of bounded Hankel operators is bounded. We prove and use a continuity property of a kind of bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces.