# Marius MuellerUniversity of Freiburg | Albert-Ludwigs-Universität Freiburg · Analysis

Marius Mueller

Master of Science

## About

22

Publications

606

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34

Citations

Citations since 2017

Introduction

## Publications

Publications (22)

This erratum points out an error in “The Poisson equation involving surface measures” (Vol. 47 of Communications in Partial Differential Equations, (2022)) and provides a counterexample and discussion of the erroneous theorem.

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a $C^{1,\alpha}$-regular hypersurface, $Q\in C^{0,\alpha}$ is a density on $\Gamma$, and $A$ is symmetric, unifo...

This article studies (optimal) $W^{2m-1,\infty}$-regularity for the polyharmonic equation $(-\Delta)^m u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$, where $\Gamma$ 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 [...

We examine a variational free boundary problem of Alt–Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior $$C^2$$ C 2 -regularity of minimizers and that the free boundary consists of finitely many $$C^2$$ C 2 -hypersurfaces. With the aid of these results, we can prove that minimizers are i...

We consider the class $S^m_\perp(\Omega)$ of $m$-dimensional surfaces in $\overline{\Omega} \subset {\mathbb R}^n$ which intersect $S = \partial \Omega$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(\Omega)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$-integral of the second fundamental...

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 ma...

We prove the (optimal) W1,∞-regularity of weak solutions to the equation −Δu=Q Hn−1⌊Γ in a domain Ω⊂Rn with Dirichlet boundary conditions, where Γ⊂⊂Ω is a compact (Lipschitz) manifold and Q∈L∞(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...

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 dissipatio...

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

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational cha...

We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-\Delta u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $\Gamma \subset \subset \Omega$ is a compact (Lipschitz) manifold and $Q \in L^\infty(\Gamma)$. We also discuss optimality and ne...

By the classical Li-Yau inequality, an immersion of a closed surface in $\mathbb{R}^n$ with Willmore energy below $8\pi$ 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.

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 imporve the subconvergence to convergence. Qualitative aspects such as energy dissipatio...

We study long-time existence and asymptotic behavior for the L2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below $8\pi$ then the solution of the Willmore flow converges for $t \rightarrow \infty$ to the Clifford Torus, possibly resca...

We examine a steepest energy descent flow with obstacle constraint in higher order energy frameworks where the maximum principle is not available. We construct the flow under general assumptions using De Giorgi’s minimizing movement scheme. Our main application will be the elastic flow of graph curves with Navier boundary conditions for which we st...

We examine the L² -gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a const...

We examine a variational free boundary problem of Alt-Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior C2-regularity of minimizers and that the free boundary consists of finitely many C2-hypersurfaces. With the aid of these results, we can prove that minimizers are in general not unique...

We examine a variational free boundary problem of Alt-Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior C 2-regularity of minimizers and that the free boundary consists of finitely many C 2-hypersurfaces. With the aid of these results, we can prove that minimizers are in general not uniq...

We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization argument.

We examine a steepest energy descent flow with obstacle constraint in higher order energy frameworks where the maximum principle is not available. We construct the flow under general assumptions using De Giorgi's minimizing movement scheme. Our main application will be the elastic flow of graph curves with Navier boundary conditions for which we st...

We examine the L^2-gradient flow of Euler's elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a const...

We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization argument.

## Projects

Project (1)