arXiv subjects

Leon Happ

Publications and source records attributed to Leon Happ.

4 recordsLinked to original sources

arXiv2025

Homogenization of nonlocal exchange energies in micromagnetics

We study the homogenization of nonlocal micromagnetic functionals incorporating both symmetric and antisymmetric exchange contributions under the physical constraint that the magnetization field takes values in the unit sphere. Assuming that the nonlocal interaction range and the scale of heterogeneities vanish simultaneously, we capture the asymptotic behavior of the nonlocal energies by identifying their $\Gamma$-limit, leading to an effective local functional expressed through a tangentially constrained nonlocal cell problem. Our proof builds upon a tailored notion of two-scale convergence, which takes into account oscillations only in specific directions. It enables us to describe the two-scale limit of suitable nonlocal difference quotients, yielding a nonlocal analog of the classical limit decomposition result for gradient fields. To deal with the manifold constraint of the magnetization, we additionally prove that the microscopic oscillations in the two-scale limit are constrained to lie in the tangent space of the sphere.

math.AP↗

arXiv2024

Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel-Zheng theory

In this paper we prove a strong two-scale approximation result for sphere-valued maps in $L^2(\Omega;W^{1,2}_0(Q_0;\mathbb{S}^2))$, where $\Omega\subset \mathbb{R}^3$ is an open domain and $Q_0\subset Q$ an open subset of the unit cube $Q=(0,1)^3$. The proof relies on a generalization of the seminal argument by F. Bethuel and X.M. Zheng to the two-scale setting. We then present an application to a variational problem in high-contrast micromagnetics.

math.AP↗

arXiv2024

On the existence of extensions for manifold-valued Sobolev maps on perforated domains

Motivated by manifold-constrained homogenization problems, we construct suitable extensions for Sobolev functions defined on a perforated domain and taking values in a compact, connected $C^2$-manifold without boundary. The proof combines a by now classical extension result for the unconstrained case with a retraction argument that heavily relies on the topological properties of the manifold. With the ultimate goal of providing necessary conditions for the existence of extensions for Sobolev maps between manifolds, we additionally investigate the relationship between this problem and the surjectivity of the trace operator for such functions.

math.AP↗

arXiv2023

A modular Poincar\'e-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents

In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*} \int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x} \leqslant C \int_\Omega|\nabla f(x)|^{p(x)}{\mathrm{d} x}, \end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincar\'e- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincar\'e--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if $\Omega\subset \mathbb{R}^n$ is a bounded Lipschitz domain, and if $p\in L^\infty(\Omega)$, $p \geq 1$, then for every $f\in C^\infty(\bar\Omega)$ the following generalized Poincar\'e--Wirtinger inequality holds \begin{equation*} \int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x} \leq C \int_\Omega\int_\Omega \frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x}, \end{equation*} where $\langle f\rangle_{\Omega}$ denotes the mean of $f$ over $\Omega$, and $C>0$ is a positive constant depending only on $\Omega$ and $\|p\|_{L^\infty(\Omega)}$. Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincar\'e--Wirtinger constant on Lipschitz domains.

math.AP↗