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[en] The question on the Hoelder continuity of solutions of the p-Laplace equation with measurable summability index p=p(x) bounded away from one and infinity is studied. In the case when the domain of definition D subset of R, n≥2, of the equation is partitioned by a hyperplane Σ into parts D(1) and D(2) such that p(x) has a logarithmic modulus of continuity at a point x0 element of D intersection Σ from either side it is proved that solutions of the equation are Hoelder-continuous at x0. The case when p(x) has a logarithmic modulus of continuity in D(1) and D(2) is considered separately. It is proved that smooth functions in D are dense in the class of solutions.
[en] A magnification device for the Laplace equation is proposed and verified in this work. Unlike designs based on transformation optics, which are hard to realize due to the complementary media needed and the anisotropy and inhomogeneity of the resultant materials, our design only requires isotropic and homogeneous materials with positive values. The method of separation of variables is utilized to realize the magnification device, and the first experiment demonstration for the device is given, where we utilize the resistor network to mimic the problem. The measurement results validate the exact magnifying property of our proposal. Our design is suitable for the time-varying fields under quasi-static conditions, it applies directly in thermodynamics, and other problems governed by the Laplace equation. (paper)
[en] Analogies in physics are unusual coincidences that can be very useful to solve problems and to clarify some theoretical concepts. Apart from their own curiosity, analogies are attractive tools because they reduce the abstraction of some complex phenomena in such a way that these can be understood by means of other phenomena closer to daily experience. Usually, two analogous systems share a common aspect, like the movement of particles or transport of matter. On account of this, the analogy presented is exceptional since the involved phenomena are a priori disjoined. The most important equation of capillarity, the Young-Laplace equation, has the same structure as the Gullstrand equation of geometrical optics, which relates the optic power of a thick lens to its geometry and the properties of the media
[en] A one-dimensional ideal gas with negative compressibility described by quasi-Chaplygin equations is discussed. Its reduction to a Laplace equation is shown, and an evolutionary principle for selecting spontaneous solutions is summarized. Three extremely simple spontaneous solutions are obtained along with multidimensional self-similar solutions. The Buneman instability in a plasma is considered as an example. 17 references
[en] Cauchy problem for the Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface is considered. Discretization was performed and proved to obtain a numerical solution. An economic algorithm is proposed.
[en] It has been reported that the inverse-square law of gravity is violated over a range of a few hundred meters. I present a different method for the analysis of the data from that experiment. In this method, the experimental error can be evaluated analytically and I confirm the previous analysis but show that it is a 2σ effect. The method can also be used to design new experiments that will yield minimum errors for a fixed number of data points
[en] The aim of this article is to study whether such a critical exponent exists on IHn, which is the most approachable rank one symmetric space G/K of the noncompact type. Notice that, though linear wave equations have ben well studied on symmetric spaces, few authors were interested in nonlinear wave equations on such spaces. We are only concerned here in low dimensions (n = 2 or 3) and our results may be summarized by Theorems A and B below: these are based on assumptions similar to Asakura's, the comparison scale being defined here by functions related to the areas of spheres, namely (sinh r)n-1 on IHn
[en] In this work, we introduce a framework for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. Exact solutions are developed by using the He's variational iteration method (VIM). The work confirms the power of the method in reducing the size of calculations