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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The problem of minimal resistance for functions and domains

Here we solve the problem posed by Comte and Lachand-Robert in [SIAM J. Math. Anal., 34 (2002), pp. 101–120]. Take a bounded domain Ω ⊂ R2 and a piecewise smooth nonpositive function u : ¯Ω → R vanishing on ∂Ω. Consider a flow of point particles falling vertically down and reflected elastically from the graph of u. It is assumed that each particle is reflected no more than once (no multiple reflections are allowed); then the resistance of the graph to the flow is expressed as R(u; Ω) = 1 |Ω| Ω(1 + |∇u(x)|2)−1dx. We need to find infΩ,u R(u;Ω). One can easily see that |∇u(x)| 1/2. We prove that the infimum of R is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, and partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem [Amer. Math. Monthly, 70 (1963), pp. 697–706]

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 $d \to \infty$

Schwere Lithiumintoxikationen bei normalen Serumspiegeln

Anliegen Unser Ziel ist es, Faktoren zu identifizieren, die das Risiko einer Lithiumintoxikation trotz normaler Serumspiegel erhöhen. Methode Wir beschreiben zwei eigene Fälle und bewerten diese im Kontext der Literatur. Ergebnisse Alter, Begleiterkrankungen und psychopharmakologische Komedikation erhöhen das Risiko einer Lithiumintoxikation bei normalen Serumspiegeln. Diskussion Bei älteren, multimorbiden Patienten sollte eine engmaschige klinische Kontrolle inklusive Spiegelbestimmung und EEG erfolgen, bei klinischen Anzeichen der Intoxikation sollte auch bei unauffälligen Spiegeln ein Absetzen erwogen werden

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

Level Set Segmentation with Shape and Appearance Models Using Affine Moment Descriptors

We propose a level set based variational approach that incorporates shape priors into edge-based and region-based models. The evolution of the active contour depends on local and global information. It has been implemented using an efficient narrow band technique. For each boundary pixel we calculate its dynamic according to its gray level, the neighborhood and geometric properties established by training shapes. We also propose a criterion for shape aligning based on affine transformation using an image normalization procedure. Finally, we illustrate the benefits of the our approach on the liver segmentation from CT images

Distribution of Response Time, Cortical, and Cardiac Correlates during Emotional Interference in Persons with Subclinical Psychotic Symptoms

A psychosis phenotype can be observed below the threshold of clinical detection. The study aimed to investigate whether subclinical psychotic symptoms are associated with deficits in controlling emotional interference, and whether cortical brain and cardiac correlates of these deficits can be detected using functional near-infrared spectroscopy (fNIRS). A data set derived from a community sample was obtained from the Zurich Program for Sustainable Development of Mental Health Services. 174 subjects (mean age 29.67 ± 6.41, 91 females) were assigned to four groups ranging from low to high levels of subclinical psychotic symptoms (derived from the Symptom Checklist-90-R). Emotional interference was assessed using the emotional Stroop task comprising neutral, positive, and negative conditions. Statistical distributional methods based on delta plots [behavioral response time (RT) data] and quantile analysis (fNIRS data) were applied to evaluate the emotional interference effects. Results showed that both interference effects and disorder-specific (i.e., group-specific) effects could be detected, based on behavioral RTs, cortical hemodynamic signals (brain correlates), and heart rate variability (cardiac correlates). Subjects with high compared to low subclinical psychotic symptoms revealed significantly reduced amplitudes in dorsolateral prefrontal cortices (interference effect, p < 0.001) and middle temporal gyrus (disorder-specific group effect, p < 0.001), supported by behavioral and heart rate results. The present findings indicate that distributional analyses methods can support the detection of emotional interference effects in the emotional Stroop. The results suggested that subjects with high subclinical psychosis exhibit enhanced emotional interference effects. Based on these observations, subclinical psychosis may therefore prove to represent a valid extension of the clinical psychosis phenotype