Overdetermined boundary value problems for the ∞\infty-Laplacian

We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $\Omega$ of $\R^n$ and discuss what kind of implications on the geometry of $\Omega$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.Comment: 9 pages, 1 figur

On rotationally symmetric mean curvature flow

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Minimal resistance of curves under the single impact assumption

We consider the hollow on the half-plane $\{ (x,y) : y \le 0 \} \subset \mathbb{R}^2$ defined by a function $u : (-1,\, 1) \to \mathbb{R}$, $u(x) < 0$, and a vertical flow of point particles incident on the hollow. It is assumed that $u$ satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph$(u)$ and goes away without hitting the graph of $u$ anymore. We solve the problem: find the function $u$ minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the $y$-axis. Assuming that the resistance of $u \equiv 0$ equals 1, we show that the minimal resistance equals $\pi/2 - 2\arctan(1/2) \approx 0.6435$. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730--2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space $\{(x_1,\ldots,x_d, y) : y \le 0 \} \subset \mathbb{R}^{d+1}$ is defined by a radial function $U$ satisfying the SIC, $U(x) = u(|x|)$, with $x = (x_1,\ldots,x_d)$, $u(\xi) < 0$ for $0 \le \xi < 1$, and $u(\xi) = 0$ for $\xi \ge 1$, and the flow is parallel to the $y$-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as $d \to \infty$

Pflanzenvergiftungen — psychiatrische Aspekte

Zusammenfassung: Seit der Antike sind Pflanzenvergiftungen dokumentiert, trotzdem werden Intoxikationen mit pflanzlichen Giften im psychiatrischen Schrifttum wenig beachtet, und die Gefahr durch Giftpflanzen wird gemeinhin verkannt. In diesem Artikel wird diese Problematik erörtert. Es werden weiterhin entsprechende Empfehlungen zur Vermeidung von Pflanzenvergiftungen und zum allgemeinen Umgang mit deren Folgen gegebe

Examination of the effect of acute levodopa administration on the loudness dependence of auditory evoked potentials (LDAEP) in humans

Rationale: The loudness dependence of the auditory evoked potential (LDAEP) is considered a noninvasive in vivo marker of central serotonergic functioning in humans. Nevertheless, results of genetic association studies point towards a modulation of this biomarker by dopaminergic neurotransmission. Objective: We examined the effect of dopaminergic modulation on the LDAEP using L-3,4-dihydroxyphenylalanine (levodopa)/benserazide (Madopar®) as a challenge agent in healthy volunteers. Methods: A double-blind placebo-controlled challenge design was chosen. Forty-two healthy participants (21 females and 21 males) underwent two LDAEP measurements, following a baseline LDAEP measurement either placebo or levodopa (levodopa 200mg/benserazide 50mg) were given orally. Changes in the amplitude and dipole source activity of the N1/P2 intensities (60, 70, 80, 90, and 100dB) were analyzed. Results: The participants of neither the levodopa nor the placebo group showed any significant LDAEP alterations compared to the baseline measurement. The test-retest reliability (Cronbachs Alpha) between baseline and intervention was 0.966 in the verum group and 0.759 in the placebo group, respectively. Conclusions: The administration of levodopa showed no effect on the LDAEP. These findings are in line with other trials using dopamine receptor agonist

Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $(\lambda_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(\mu_{q,}u_{q})$ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $u_{q}$ to $e_{p}$ is in the $C^{1}$-norm and the rate of convergence of $\mu_{q}$ to $\lambda_{p}$ is at least $O(p-q)$. Numerical evidence is presented.Comment: Section 5 was rewritten. Jed Brown was added as autho

On a classical spectral optimization problem in linear elasticity

We consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the $N$-dimensional Euclidean space. We survey recent results concerning the analytic dependence of the elementary symmetric functions of the eigenvalues upon domain perturbation and the role of balls as critical points of such functions subject to volume constraint. Our discussion concerns Dirichlet and buckling-type problems for polyharmonic operators, the Neumann and the intermediate problems for the biharmonic operator, the Lam\'{e} and the Reissner-Mindlin systems.Comment: To appear in the proceedings of the workshop `New Trends in Shape Optimization', Friedrich-Alexander Universit\"{a}t Erlangen-Nuremberg, 23-27 September 201

Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions