A parabolic free boundary problem with Bernoulli type condition on the free boundary

Consider the parabolic free boundary problem $$\Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} .$$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon = \beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free boundary $\partial\{u>0\}$ can be decomposed into an {\em open} regular set (relative to $\partial\{u>0\}$) which is locally a surface with H\"older-continuous space normal, and a closed singular set. Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems

IN VIVO EVALUATION OF ANKLE LIGAMENT FORCES USING A FIBER OPTIC TRANSDUCER

INTRODUCTION: Successful injury prevention, treatment and rehabilitation require a clear understanding of ligament function and forces acting on these ligaments, especially for injuries to the lateral ankle ligaments, which are very common in many kinds of sports. Several authors (Bahr et al. 1998; Renström, et al., 1988; Sauer et al., 1978) investigated forces or tensile strength of the ligament talofibulare anterior (LTFA) in vitro. There is, however, a lack of information of investigations with direct measurement of forces in this structure. The aim of this study was to apply a fiber optic transducer in vivo in order to register forces in the LTFA during different natural movements of the ankle joint under varied load conditions

PREVENTIVE ANKLE TAPING -EVALUATION OF MECHANICAL, NEUROMUSCULAR AND THERMAL EFFECTS BEFORE AND AFTER EXERCISE

INTRODUCTION: . Athletes and coaches in different sport disciplines use preventive adhesive tape or bandages in order to avoid ankle sprains since Paul Beiersdorf has invented tape about 100 years ago. Many authors tried to evaluate the main effects of taping in respect to its mechanical support to the ankle joint and its preventive effect concerning injuries in high risk sports like basketball, football, athletics or gymnastics. Clinical studies have shown that ankle taping reduces the risk of injury although it has been demonstrated that the mechanical support does decrease even after a few minutes of practise. The aim of the study was to investigate the mechanical as well as the neuromuscular influence of different tape materials and techniques before and after exercise. Method: 12 subjects underwent different trials in a randomized order with two materials (B,C) and two taping techniques (D,E) and without tape (A). This sense of exercises were used to stress the ankle: 5 Drop jumps (36 cm Height) 10 minutes running (tread mill) 3 minutes jumping using a special “jump-device” with slope surfaces ( increased inversion/eversion and dorsal/plantar flexion during the landing phase). The course was performed two times. The main test was the simulated ankle inversion injury by using a special tilt device to apply randomized ankle movements (300 inversion + 150 Plantarflexion) while the subject is standing on the tilt platform. Magnitude and velocity of ankle joint motion was recorded from 2axial goniometers (Penny&Giles). EMG activity was measured from the mm. peroneus 1., tibialis a., gastrocnemius and vastus medialis. Skin temperature alterations beneath the adhesive tape were recorded after each exercise by thermocouples with an accuracy of 0.1 K. RESULTS: Amplitude and velocity (relative to value without tape) of simulated inversion injury was initially reduced by all tapes significantly: Inversion angle relativ to the trial without tapeJA): If no tape was used then EMG -activity during simulation was reduced up to 12% after 20 minutes of exercise. If the ankle was taped -the EMG reduction after 20 minutes was at a lower level ( 5%). Skin temperature under the tape increased significantly compared to untaped ankle. DISCUSSION: The preventive effect of adhesive taping is achieved at the intitial exercise phase by the mechanical stabilization properties of the material. In the later exercise phase (when the mechanical support decreases) the risc of injury could be reduced by the proprioceptive effect, which can be concluded by the increased EMG-activities. REFERENCES: Fumich, R. M., et al. The measured effect of taping on combined foot and ankle motion before and after exercise. AMJSM, 9(1981):165-170

Three charged particles in the continuum. Astrophysical examples

We suggest a new adiabatic approach for description of three charged particles in the continuum. This approach is based on the Coulomb-Fourier transformation (CFT) of three body Hamiltonian, which allows to develop a scheme, alternative to Born-Oppenheimer one. The approach appears as an expansion of the kernels of corresponding integral transformations in terms of small mass-ratio parameter. To be specific, the results are presented for the system $ppe$ in the continuum. The wave function of a such system is compared with that one which is used for estimation of the rate for triple reaction $p+p+e\to d+\nu,$ which take place as a step of $pp$-cycle in the center of the Sun. The problem of microscopic screening for this particular reaction is discussed

N-d scattering above the deuteron breakup threshold

The complex Kohn variational principle and the (correlated) Hyperspherical Harmonics technique are applied to study the N--d scattering above the deuteron breakup threshold. The configuration with three outgoing nucleons is explicitly taken into account by solving a set of differential equations with outgoing boundary conditions. A convenient procedure is used to obtain the correct boundary conditions at values of the hyperradius $\approx 100$ fm. The inclusion of the Coulomb potential is straightforward and does not give additional difficulties. Numerical results have been obtained for a simple s-wave central potential. They are in nice agreement with the benchmarks produced by different groups using the Faddeev technique. Comparisons are also done with experimental elastic N--d cross section at several energies.Comment: LaTeX, 13 pages, 3 figure

Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions

The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions

Sγ3 switch sequences function in place of endogenous Sγ1 to mediate antibody class switching

Immunoglobulin heavy chain (IgH) class switch recombination (CSR) replaces the initially expressed IgH Cμ exons with a set of downstream IgH constant region (CH) exons. Individual sets of CH exons are flanked upstream by long (1–10-kb) repetitive switch (S) regions, with CSR involving a deletional recombination event between the donor Sμ region and a downstream S region. Targeting CSR to specific S regions might be mediated by S region–specific factors. To test the role of endogenous S region sequences in targeting specific CSR events, we generated mutant B cells in which the endogenous 10-kb Sγ1 region was replaced with wild-type (WT) or synthetic 2-kb Sγ3 sequences or a synthetic 2-kb Sγ1 sequence. We found that both the inserted endogenous and synthetic Sγ3 sequences functioned similarly to a size-matched synthetic Sγ1 sequence to mediate substantial CSR to IgG1 in mutant B cells activated under conditions that stimulate IgG1 switching in WT B cells. We conclude that Sγ3 can function similarly to Sγ1 in mediating endogenous CSR to IgG1. The approach that we have developed will facilitate assays for IgH isotype–specific functions of other endogenous S regions

Variational Principles for Water Waves

We describe the Hamiltonian structures, including the Poisson brackets and Hamiltonians, for free boundary problems for incompressible fluid flows with vorticity. The Hamiltonian structure is used to obtain variational principles for stationary gravity waves both for irrotational flows as well as flows with vorticity.Comment: 20 page

An Experimental Investigation of the Scaling of Columnar Joints

Columnar jointing is a fracture pattern common in igneous rocks in which cracks self-organize into a roughly hexagonal arrangement, leaving behind an ordered colonnade. We report observations of columnar jointing in a laboratory analog system, desiccated corn starch slurries. Using measurements of moisture density, evaporation rates, and fracture advance rates as evidence, we suggest an advective-diffusive system is responsible for the rough scaling behavior of columnar joints. This theory explains the order of magnitude difference in scales between jointing in lavas and in starches. We investigated the scaling of average columnar cross-sectional areas due to the evaporation rate, the analog of the cooling rate of igneous columnar joints. We measured column areas in experiments where the evaporation rate depended on lamp height and time, in experiments where the evaporation rate was fixed using feedback methods, and in experiments where gelatin was added to vary the rheology of the starch. Our results suggest that the column area at a particular depth is related to both the current conditions, and hysteretically to the geometry of the pattern at previous depths. We argue that there exists a range of stable column scales allowed for any particular evaporation rate.Comment: 12 pages, 11 figures, for supporting online movies, go to http://www.physics.utoronto.ca/nonlinear/movies/starch_movies.htm