arXiv subjects

Denis Brazke

Publications and source records attributed to Denis Brazke.

6 recordsLinked to original sources

arXiv2026

A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies

We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 \Delta$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_\Omega$ associated to a measurable set $\Omega\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $\Omega$ has finite perimeter if and only if $c_\Omega$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $\Omega$. We show that on the round sphere $E_{\gamma,\varepsilon}$ admits a reformulation in terms of $c_\Omega$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $\Gamma$-convergence.

math.AP↗

arXiv2025

The Wiener Criterion at $\infty$ for Degenerate Elliptic Equations

This paper establishes a Wiener criterion at $\infty$ to characterise the unique solvability of the Dirichlet problem for degenerate elliptic equations with power-like weights in arbitrary open sets. In the measure-theoretical context, the criterion determines whether the $\A$-harmonic measure of $\infty$ is null or positive. From the topological point of view, it presents a test for the thinness of the exterior set at $\infty$ in the $\A$-fine topology.

math.AP↗

arXiv2025

Gamma-Convergence of Higher-Order Phase Transition Models

We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{\lambda,n}[u] := \int_I \frac 1\varepsilon W(u) -\lambda\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $\lambda >0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $\Gamma$-limit of $G_\varepsilon^{\lambda,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.

math.AP↗

arXiv2022

$\Gamma$-limit for a sharp interface model related to pattern formation on biomembranes

We derive a macroscopic limit for a sharp interface version of a model proposed in [29] to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercrital parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.

math.AP↗

arXiv2021

Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces

We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.

math.CA↗

arXiv2019

On BMO and Carleson measures on Riemannian manifolds

Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $\sigma$-harmonic extension $U: \mathcal{M} \times (0,\infty) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application we show that the famous theorem of Coifman--Lions--Meyer--Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the second-named author using only harmonic extensions, integration by parts, and trace space characterizations.

math.AP↗