arXiv subjects

Marius Müller

Publications and source records attributed to Marius Müller.

At least 19 recordsLinked to original sources

arXiv2026

Global minimizers for a two-sided biharmonic Alt-Caffarelli problem

We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we identify a further class of global minimizers, all of which have constant Laplacian. Recent work by J. Lamboley and M. Nahon reduces potential global minimizers in dimension two to four possible categories. Our work shows that three of these categories persist in any dimension and are in fact global minimizers. Moreover, we show that minimizers of the two-sided biharmonic Alt-Caffarelli problem do in general not satisfy a partial differential equation, not even with a signed measure as right-hand-side. This is in sharp contrast to the corresponding one-sided problem.

math.AP↗

arXiv2025

Lipschitz regularity for $p$-harmonic interface transmission problems

We prove optimal Lipschitz regularity for weak solutions of the measure-valued $p$-Poisson equation $-\Delta_p u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$. Here $p \in (1,2)$, $\Gamma$ is a compact and connected $C^2$-hypersurface without boundary, and $Q$ is a positive $W^{2,\infty}$-density. This equation can be understood as a nonlinear interface transmission problem. Our main result extends previous studies of the linear case and provides further insights on a delicate limit case of (linear and nonlinear) potential theory.

math.AP↗

arXiv2025

An anisotropic Alt-Caffarelli problem of higher order

We study a higher order version of the Alt-Caffarelli problem in two dimensions, where the Dirichlet energy is replaced by an anisotropic bending energy. This extends a previous study of the isotropic case in [41]. It turns out that smooth anisotropies do not affect the optimal $C^{2,1}$-regularity of minimizers. The proof requires an anisotropic version of an estimate by Frehse for the fundamental solution of the bilaplacian. This generalization paves the way for further studies of various free boundary problems of higher order.

math.AP↗

arXiv2025

A nongraphical obstacle problem for elastic curves

We study an obstacle problem for the length-penalized elastic bending energy for open planar curves pinned at the boundary. We first consider the case without length penalization and investigate the role of global minimizers among graph curves in our minimization problem for planar curves. In addition, for large values of the length-penalization parameter $\lambda>0$, we expose an explicit threshold parameter above which minimizers touch the obstacle, regardless of its shape. On contrary, for small values of $\lambda>0$ we show that the minimizers do not touch the obstacle, and they are given by an explicit elastica.

math.AP↗

arXiv2025

Dimension reduction for Willmore flows of tori: fixed conformal class and analysis of singularities

This work studies Willmore flows of tori and their singularities via a dimension reduction approach. We introduce a Willmore flow that preserves the degenerate constraint of prescribed conformal class and, for rotationally symmetric initial data, we establish a strong relation with the length-preserving elastic flow in the hyperbolic plane. We provide a necessary condition for singularities and a criterion for the initial datum that allows to exclude them. Our results allow for initial data with arbitrarily large energy, in particular exceeding the usual Li-Yau threshold of $8\pi$. As an application, we obtain existence of a new class of conformally constrained Willmore tori. Moreover, we investigate singularities of the classical Willmore flow. For a class of tori, we identify a non-smooth object, the inverted catenoid, as the limit shape and we show that the flow can be restarted at this singular surface and converges to a round sphere.

math.AP↗

arXiv2024

Classification and stability of penalized pinned elasticae

This paper considers critical points of the length-penalized elastic bending energy among planar curves whose endpoints are fixed. We classify all critical points with an explicit parametrization. The classification strongly depends on a special penalization parameter $\hat{\lambda}\simeq 0.70107$. Stability of all the critical points is also investigated, and again the threshold $\hat{\lambda}$ plays a decisive role. In addition, our explicit parametrization is applied to compare the energy of critical points, leading to uniqueness of minimal nontrivial critical points. As an application we obtain eventual embeddedness of elastic flows.

math.AP↗

arXiv2024

Precision spectroscopy on $^9$Be overcomes limitations from nuclear structure

Many powerful tests of the Standard Model of particle physics and searches for new physics with precision atomic spectroscopy are plagued by our lack of knowledge of nuclear properties. Ideally, such properties may be derived from precise measurements of the most sensitive and theoretically best-understood observables, often found in hydrogen-like systems. While these measurements are abundant for the electric properties of nuclei, they are scarce for the magnetic properties, and precise experimental results are limited to the lightest of nuclei. Here, we focus on $^9$Be which offers the unique possibility to utilize comparisons between different charge states available for high-precision spectroscopy in Penning traps to test theoretical calculations typically obscured by nuclear structure. In particular, we perform the first high-precision spectroscopy of the $1s$ hyperfine and Zeeman structure in hydrogen-like $^9$Be$^{3+}$. We determine its effective Zemach radius with an uncertainty of $500$ ppm, and its bare nuclear magnetic moment with an uncertainty of $0.6$ parts-per-billion (ppb) - uncertainties unmatched beyond hydrogen. Moreover, we compare to measurements conducted on the three-electron charge state $^9$Be$^{+}$, which, for the first time, enables testing the calculation of multi-electron diamagnetic shielding effects of the nuclear magnetic moment at the ppb level. In addition, we test quantum electrodynamics (QED) methods used for the calculation of the hyperfine splitting. Our results serve as a crucial benchmark essential for transferring high-precision results of nuclear magnetic properties across different electronic configurations.

physics.atom-ph↗

arXiv2024

Curvature Varifolds with Orthogonal Boundary

We consider the class $S^m_\perp(Ω)$ of $m$-dimensional surfaces in $\barΩ \subset {\mathbb R}^n$ which intersect $S = \partial Ω$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(Ω)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$-integral of the second fundamental form in three cases: first when $Ω$ admits no orthogonal slices, second for $m = p = 2$ if all orthogonal slices are topological disks, and finally for all $Ω$ if the surfaces are confined to a neighborhood of $S$. The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any $Ω$ the existence of an orthogonal $2$-varifold which minimizes the $L^2$ curvature in the integer rectifiable class.

math.DG↗

arXiv2024

Grokking the Sequent Calculus (Functional Pearl)

The sequent calculus is a proof system which was designed as a more symmetric alternative to natural deduction. The {\lambda}{\mu}{\mu}-calculus is a term assignment system for the sequent calculus and a great foundation for compiler intermediate languages due to its first-class representation of evaluation contexts. Unfortunately, only experts of the sequent calculus can appreciate its beauty. To remedy this, we present the first introduction to the {\lambda}{\mu}{\mu}-calculus which is not directed at type theorists or logicians but at compiler hackers and programming-language enthusiasts. We do this by writing a compiler from a small but interesting surface language to the {\lambda}{\mu}{\mu}-calculus as a compiler intermediate language.

cs.PL↗

arXiv2024

An obstacle problem for the p-elastic energy

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler--Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

math.AP↗

arXiv2024

A biharmonic analogue of the Alt-Caffarelli problem

We study a natural biharmonic analogue of the classical Alt-Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and $C^{1,α}$-regularity of minimisers. For the Navier problem we also obtain a symmetry result in case that the boundary data are radial. We find this remarkable because the problem under investigation is of higher order. Computing radial minimisers explicitly we find that the obtained regularity is optimal.

math.AP↗

arXiv2023

A Li-Yau inequality for the 1-dimensional Willmore energy

By the classical Li-Yau inequality, an immersion of a closed surface in $\mathbb{R}^n$ with Willmore energy below $8π$ has to be embedded. We discuss analogous results for curves in $\mathbb{R}^2$, involving Euler's elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

math.DG↗

arXiv2023

On elliptic equations involving surface measures

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner Γ$ in a smooth domain $Ω\subset \mathbb{R}^n$. Here $Γ$ is a $C^{1,α}$-regular hypersurface, $Q\in C^{0,α}$ is a density on $Γ$, and the coefficient matrix $A$ is symmetric, uniformly elliptic and $W^{1,q}$-regular $(q > n)$. We also discuss optimality of these assumptions on the data. The equation can be understood as a special coupling of two $A$-harmonic functions with an interface $Γ$. As such it plays an important role in several free boundary problems, as we shall discuss.

math.AP↗

arXiv2023

Polyharmonic equations involving surface measures

This article studies (optimal) $W^{2m-1,\infty}$-regularity for the polyharmonic equation $(-Δ)^m u = Q \; \mathcal{H}^{n-1} \llcorner Γ$, where $Γ$ is a (suitably regular) $(n-1)$-dimensional submanifold of $\mathbb{R}^n$, $\mathcal{H}^{n-1}$ is the Hausdorff measure, and $Q$ is some suitably regular density. We extend findings in [9], where the second-order equation $-\mathrm{div}(A(x)\nabla u) = Q \; \mathcal{H}^{n-1} \llcorner Γ$ is studied. As an application we derive (optimal) $W^{3,\infty}$-regularity for solutions of the biharmonic Alt-Caffarelli problem in two dimensions.

math.AP↗

arXiv2023

Short closed geodesics and the Willmore energy

We prove a lower bound on the length of closed geodesics for spheres with Willmore energy below $6\pi$. The energy threshold is optimal and the inequality cannot be extended to surfaces of higher genus. Moreover, we discuss consequences for the injectivity radius.

math.DG↗

arXiv2022

White-light generating molecular materials: correlation between the amorphous/crystalline structure and nonlinear optical properties

Amorphous materials are integral part of todays technology, they commonly are performant and versatile in integration. Consequently, future applications increasingly aim to harvest the potential of the amorphous state. Establishing its structure-property relationship, however, is inherently challenging using diffraction-based techniques yet is extremely desirable for developing advanced functionalities. In this article, we introduce a set of transmission electron microscopy-based techniques to locally quantify the structure of a material. This unique approach allows to clearly identify the spatial distribution of amorphous and crystalline regions and to quantify atomic arrangements of amorphous regions of a representative model system. We study an ensemble of well-defined, functionalized adamantane-type cluster molecules exhibiting exceptionally promising nonlinear optical properties of unclear origin. The nanoscopic structure for three model compounds ([(PhSn)4S6], [(NpSn)4S6], [(CpSn)4S6]) correlates with their characteristic optical responses. These results highlight the advantageous properties of amorphous molecular materials when understanding the microscopic origin.

cond-mat.mtrl-sci↗

arXiv2022

The elastic flow with obstacles: small obstacle results

We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.

math.AP↗

arXiv2021

The Poisson equation involving surface measures

We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-Δu = Q \; \mathcal{H}^{n-1} \llcorner Γ$ in a domain $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $Γ\subset \subset Ω$ is a compact (Lipschitz) manifold and $Q \in L^\infty(Γ)$. We also discuss optimality and necessity of the assumptions on $Q$ and $Γ$. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.

math.AP↗