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Bernd Kawohl

Publications and source records attributed to Bernd Kawohl.

8 recordsLinked to original sources
arXiv2023

On a formula for all sets of constant width in 3d

In the recent paper "On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the parametrization of the boundary of bodies of constant width in 3 dimensions, depending on one function defined on S^2. Each such function gives a minimal value r_0 and for all r \ge r_0 one finds a body of constant width 2r. Moreover, we show that all bodies of constant width in 3d have such a parametrization. The last result needs a tool that we describe as 'shadow domain' and that is explained in an appendix. The construction is explicit and and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and É. Oudet, "Analytic parametrization of three-dimensional bodies of constant width" in Arch. Ration. Mech. Anal., 186 (2007), 225-249.

math.MG↗
arXiv2018

On the first eigenvalue of the normalized p-Laplacian

We prove that, if $Ω$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $Ω$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.

math.AP↗
arXiv2018

From Symmetry to Monotonicity

We offer an alternative and shorter proof to a result by Jan J.Ubøe about monotonicity properties of a one-dimensional function that appeared in the Mathematical Intelligencer in 2015. Our proof is based on reducing the problem to symmetry properties of a two-dimensional surface.

math.CA↗
arXiv2018

Overdetermined problems for the normalized $p$-Laplacian

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-Δ_p^Nu=1$ in $Ω$, $u=0$ and $\tfrac{\partial u}{\partialν}=-c\neq 0$ on $\partialΩ$ can only exist on a bounded domain $Ω$ if $Ω$ is a ball.

math.AP↗
arXiv2016

Generalized elastica problems under area constraint

It was recently proved that the elastic energy $E(γ)=\tfrac{1}{2}\int_γκ^2 ds$ of a closed curve $γ$ with curvature $κ$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area $A(γ)$, and that the minimum is attained only for circles. Here we show under which hypothesis the result can be extended to other functionals involving the curvature. As an example we show that the optimal shape remains a circle for the $p$-elastic energy $\int_γ|κ|^p ds$, whenever $p>1$.

math.OC↗
arXiv2016

On the geometry of the $p$-Laplacian operator

The $p$-Laplacian operator $Δ_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $Ω$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We report also on results concerning the normalized or game-theoretic $p$-Laplacian $$Δ_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}Δ_pu=\tfrac{1}{p}Δ_1^Nu+\tfrac{p-1}{p}Δ_\infty^Nu$$ and its parabolic counterpart $u_t-Δ_p^N u=0$. These equations are homogeneous of degree 1 and $Δ_p^N$ is uniformly elliptic for any $p\in (1,\infty)$. In this respect it is more benign than the $p$-Laplacian, but it is not of divergence type.

math.AP↗
arXiv2015

The elastica problem under area constraint

We show that the elastic energy $E(γ)$ of a closed curve $γ$ has a minimizer among all plane simple regular closed curves of given enclosed area $A(γ)$, and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of $γ$ in a finite number of steps to construct some related convex sets with smaller energy.

math.OC↗