Marius Mueller

Marius Mueller
University of Freiburg | Albert-Ludwigs-Universität Freiburg · Analysis

Master of Science

About

19
Publications
551
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31
Citations
Citations since 2016
19 Research Items
31 Citations
201620172018201920202021202202468101214
201620172018201920202021202202468101214
201620172018201920202021202202468101214
201620172018201920202021202202468101214

Publications

Publications (19)
Article
Full-text available
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...
Preprint
Full-text available
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...
Preprint
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...
Article
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...
Article
Full-text available
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...
Article
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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.
Preprint
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...
Preprint
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...
Article
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...
Article
Full-text available
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...
Preprint
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...
Preprint
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...
Article
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.
Preprint
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...
Preprint
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...
Preprint
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.

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