arXiv subjects

Julia Lenczewska

Publications and source records attributed to Julia Lenczewska.

5 recordsLinked to original sources

arXiv2024

Asymptotics and geometric flows for a class of nonlocal curvatures

We consider a family of nonlocal curvatures determined through a kernel which is symmetric and bounded from above by a radial and radially non-increasing profile satisfying an integrability condition. It turns out that such definition encompasses various variants of nonlocal curvatures that have already appeared in the literature, including fractional curvature and anisotropic fractional curvature. The main task undertaken in the article is to study the limit behaviour of the introduced nonlocal curvatures under an appropriate limiting procedure. This enables us to recover known asymptotic results e.g. for the fractional curvature and for the anisotropic fractional curvature. For the convergence of anisotropic fractional curvatures we identify the limit object as the nonlocal curvature being the first variation of the related anisotropic fractional perimeter. We also prove existence, uniqueness and stability of viscosity solutions to the corresponding level-set parabolic Cauchy problem formulated in terms of the investigated nonlocal curvature.

math.AP↗

arXiv2024

Hardy inequalities and nonlocal capacity

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.

math.AP↗

arXiv2023

Sharp Hardy inequalities for Sobolev-Bregman forms

We obtain sharp fractional Hardy inequalities for the half-space and for convex domains. We extend the results of Bogdan and Dyda and of Loss and Sloane to the setting of Sobolev-Bregman forms.

math.AP↗

arXiv2023

Asymptotic expansion of the nonlocal heat content

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_Ω \mathbb{P}^x (X_t\inΩ^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $α$-stable Lévy processes and more general Lévy processes, under mild assumptions on the characteristic exponent.

math.PR↗

arXiv2021

Optimal Hardy inequality for the fractional Laplacian on $L^p$

For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.

math.AP↗