Biblioteca Digital

We consider a Dirichlet problem in divergence form with variable growth, modeled on the p(x)-Laplace equation. We obtain existence and uniqueness of an entropy solution for L1 data, extending the work of B´enilan et al. [5] to nonconstant exponents, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponent; CMUC/FCT and MCYT grants BMF2002- 04613-C03, MTM2005-07660-C02 (first author); CMUC/FCT and Project POCI/MAT/57546/2004 (second author)

O objetivo deste trabalho é investigar os limites assimptóticos das soluções do problema de Dirichlet homogéneo associado à equação de evolução duplamente não linear $u_{t} = Delta_{p}u^{m} + g$, quando os parâmetros $p$ e $m$ tendem para infinito. Esta equação combina a não linearidade da equação dos meios porosos, que corresponde ao caso $p=2$, com a não linearidade da equação de $p$-Laplace, que corresponde ao caso $m=1$. A contribuição principal deste trabalho é generalizar alguns dos resultados conhecidos para os limites assimptóticos das soluções de problemas de valor inicial associados à equação dos meios porosos, quando $m$ tende para infinito e à equação de $p$-Laplace, quando $p$ tende para infinito. A motivação para o estudo do comportamento no limite das soluções destas equações radica nas suas aplicações físicas, uma vez que constituem modelos matemáticos para problemas físicos em diferentes contextos, por exemplo no estudo dos fluidos não Newtonianos, do fluxo turbulento de um gás em meios porosos e em glaciologia. Adicionalmente, sob certas condições iniciais, encontramos no limite problemas com propriedades completamente diferentes, com aplicações físicas que são interessantes por si sós e que exigem uma abordagem analítica inovadora. Estudaremos os limites em $p$ e $m$ separadamente e em sequência...

Theoretical models have been proposed in this article (Parts I and II) to predict the vertical wicking behaviour of yarns and fabrics based on different fibre, yarn and fabric parameters. The first part of this article deals with the modelling of flow through yarn during vertical wicking, whereas the second part deals with the modelling of vertical wicking through the fabric. The yarn model has been developed based on the Laplace equation and the Hagen– Poiseuille’s equation on fluid flow; pore geometry has been determined as per the yarn structure. Factors such as fibre contact angle, number of filaments in a yarn, fibre denier, fibre cross-sectional shape, yarn denier and twist level in the yarn have been taken into account for development of the model. Lambertw, a mathematical function, has been incorporated, which helps to predict vertical wicking height at any given time, considering the gravitational effects. Experimental verification of the model has been carried out using polyester yarns. The model was found to predict the wicking height with time through the yarns with reasonable accuracy. Based on the proposed yarn model, a mathematical model has been developed to predict the vertical wicking through plain woven fabric in the second part of this article.

Phase equilibrium calculations for fluids confined inside capillary tubes or porous media are formulated using the isofugacity equations. In this situation, the phase pressures are not equal and it is assumed that they are related by the Laplace equation. With this formulation, existing procedures for phase equilibrium calculations can be readily modified to include capillary effects. In this paper, we review some of the main authors who have studied the behavior of fluids inside porous media and perform bubble- and dew-point calculations for pure components and mixtures, using the Peng-Robinson equation of state to model the coexisting phases and several planar surface tension models. Comparisons with results from the literature indicate that the proposed formulation is adequate for representing phase equilibrium inside capillary tubes.

En contraste con las ecuaciones diferenciales ordinarias, no existe una teoría unificada para el estudio de las ecuaciones en derivadas parciales. Algunas ecuaciones en derivadas parciales poseen sus propias teorías, mientras que otras aún no las poseen. La razón para esto es por la complejidad de su geometría. En el caso de una ecuación diferencial ordinaria un campo vectorial local es definido en una variedad. Para una ecuación diferencial parcial un subconjunto de la tangente al un espacio de dimensión mayor que 1 es definido en cada punto de la variedad. Como es sabido, inclusive para un campo bidimensional inmerso en uno tridimensional en general no es integrable. Una teoría consolidada para el estudio de la ecuación de Laplace es la considerada en este artículo, la teoría de las funciones armónicas. En este artículo se estudiaran algunas propiedades de las funciones armónicas para la solución de la ecuación de Laplace.

The usual method of separation of variables to find a basis of solutions of Laplace's equation in toroidal coordinates is particularly appropriate for axially symmetric applications; for example, to find the potential outside a charged conducting torus. A

The conditions for R-separation of variables for the conformally invariant Laplace equation on an n-dimensional Riemannian manifold are determined and compared with the conditions for the additive separation of the null geodesic Hamilton-Jacobi equation. The case of 3-dimensions is examined in detail and it is proven that on any conformally flat manifold the two equations separate in the same coordinates.; Comment: 13 pages, submitted to Journal of Geometry and Physics. Replaced due to a factor of 1/4 error found in some presented formulae; note this does not affect the results derived and discussed - only in the initial presentation

We prove a general theorem which allows the determination of Lie symmetries of Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of Laplace equation (i.e. the wave equation) in Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to General Relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein's equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.; Comment: Accepted for publication in Journal of Geometry and Physics (22 pages)

We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of-war game is identical to the original except near the boundary of the domain $\partial \Omega$, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit $\epsilon \to 0$, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation. We also obtain several new results for the normalized infinity Laplace equation $-\Delta_\infty u = f$. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous $f$...

The stability for the viscosity solutions of a differential equation with a perturbation term added to the Infinity-Laplace Operator is studied. This is the so-called Infinity-Laplace Equation with variable exponent infinity. An approximation of the identity is crucial for the proofs.

We take attention to the singular behavior of the Laplace operator in spherical coordinates, which was established in our earlier work. This singularity has many non-trivial consequences. In this article we consider only the simplest ones, which are connected to the solution of Laplace equation in Feynman classical books and Lectures. Feynman was upset looking in his derived solutions, which have a fictitious singular behavior at the origin. We show how these inconsistencies can be avoided.; Comment: 8 pages

In this paper, the iterative method developed by Daftardar-Gejji and Jafari (DJ method) is employed for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. The method is demonstrated by several physical models of Laplace equation. The obtained results show that the present approach is highly accurate and requires reduced amount of calculations compared with the existing iterative methods.; Comment: 8 Pages

Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.

We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the $p$-Laplace equation $-\diver(\abs{Du}^{p-2}Du)=0$ coincide. Our proof is more direct and transparent than the original one by Juutinen, Lindqvist and Manfredi \cite{jlm}, which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the $p$-Laplace equation.

The main purpose of this paper is to show that there exists a positive number $\lambda_{1}$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $\lambda=\lambda_{1}$ and that $\lambda_{1}$ is simple, i.e., with respect to \textit{the first eigenvalue} solutions, which are not equal to zero a. e., of the $p(x)$-Laplace equation forms an one dimensional subset. Furthermore, by developing Moser method we obtained some results concerning H\"{o}lder continuity and bounded properties of the solutions. Our works are done in the setting of the Generalized-Sobolev Space. There are many perfect results about $p$-Laplace equations, but about $p(x)$-Laplace equation there are few results. The main reason is that a lot of methods which are very useful in dealing with $p$-Laplace equations are no longer valid for $p(x)$-Laplace equations. In this paper, many results are obtained by imposing some conditions on $p(x)$. Stimulated by the development of the study of elastic mechanics, interest in variational problems and differential equations has grown in recent decades, while Laplace equations with nonstandard growth conditions share a part. The equation discussed in this paper is derived from the elastic mechanics.

These notes are written up after my lectures at the University of Pittsburgh in March 2014 and at Tsinghua University in May 2014. My objective is the $\infty$-Laplace Equation, a marvellous kin to the ordinary Laplace Equation. The $\infty$-Laplace Equation has delightful counterparts to the Dirichlet integral, the Mean Value Theorem, the Brownian Motion, Harnack's Inequality and so on. It has applications to image processing and to mass transfer problems and provides optimal Lipschitz extensions of boundary values. My treaty of this "fully non-linear" degenerate equation is far from complete and generalizations are deliberately avoided.

This paper studies the third boundary problem of the Laplace equation with azimuthal symmetry.Many solutions of the boundary value problems in spherical coordinates are available in the form of infinite series of Legendre polynomials but the evaluation of the summing series shows many computational difficulties. Integral transform is a challenge as it involves an inverse Legendre transform. Here, the closed-form solution of the Laplace equation with the Robin boundary conditions on a sphere is solved by the Legendre transform. This analytical solution is expressed with the Appell hypergeometric function F1. The Robin boundary conditions is a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. In many experimental approaches, this weight h, the Robin coefficient, is the main unknown parameter for example in transport phenomena where the Robin coefficient is the dimensionless Biot number. The usefulness of this formula is illustrated by some examples of inverse problems in mass and heat transfer, in optics, in corrosion detection and in physical geodesy.

By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identities for special functions. A convenient tool for deriving this type of identities is the so-called \emph{global relation}, which has appeared recently in a wide range of boundary value problems. As a concrete application, we analyze the Neumann boundary value problem for the Laplace equation in the exterior of the so-called Hankel contour, which is the contour that appears in the definition of both the gamma and the Riemann zeta functions. By utilizing the explicit solution of this problem, we derive a plethora of novel identities involving the hypergeometric function.

A two-dimensional Laplace equation is separable in elliptic coordinates and leads to a Chebyshev-like differential equation for both angular and radial variables. In the case of the angular variable η (-1 < η < 1), the solutions are the well known first class Chebyshev polynomials. However, in the case of the radial variable ξ (1 < ξ < ∞) it is necessary to construct another independent solution which, to our knowledge, has not been previously reported in the current literature nor in textbooks; this new solution can be constructed either by a Fröbenius series representation or by using the standard methods through the knowledge of the first solution (first-class Chebyshev polynomials). In any case, either must lead to the same result because of linear independence. Once we know these functions, the complete solution of a two-dimensional Laplace equation in this coordinate system can be constructed accordingly, and it could be used to study a variety of boundary-value electrostatic problems involving infinite dielectric or conducting cylinders and lines of charge of this shape, since with this information, the corresponding Green's function for the Laplace operator can also be readily obtained using the procedures outlined in standard textbooks on mathematical physics. These aspects are dealt with and discussed in the present work and some useful trends regarding applications of the results are also given in the case of an explicit example...