Classical Laplace Equation Research Articles

A modified walk‐on‐sphere method for high dimensional fractional Poisson equation

AbstractWe develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2–10 dimensions verify our theory and the efficiency of the modified walk‐on‐sphere method.

МАССОПЕРЕНОС ПРИ ГАЛЬВАНИЧЕСКОМ ОСАЖДЕНИИ ПОКРЫТИЙ С ВЫРАВНИВАЮЩЕЙ ДОБАВКОЙ

The ability to control the roughness of the formed deposits with the provision of the leveling ability of the workpieces is important for obtaining dimensional sealed galvanic coatings when restoring power hydraulic equipment. A model that describes the process of convective diffusion of composite particles into the galvanic matrix of the formed coating is considered. It is established that the presented problem can be described by the classical Laplace equation. However, the analytical solution in the existing formulation isn’t integrated in quadratures. The numerical method described by Watson and Edwards was used. It allows to get a connection between the electrical characteristics of the field due to the decomposition of the secondary electric field into the primary field and the polarization field. It is assumed that the diffusion of the leveling additive occurs in a field similar to the primary electric field, and the concentration of the additive directly at the electrode surface is constant and equal to zero. A qualitative agreement between the calculated leveling ability and the experimentally found values for the chromium electrolyte was established when determining the amount of the leveling additive from the potential-concentration curves at a constant current strength. The dependences obtained in the course of the study indicate the exponential nature of the change in the microprofile of the formed galvanic coating with the time of electrolysis, both in the case of positive and negative equalization. This is confirmed by a lot of experimental data concerning the development of roughness in the case of diffusion restrictions. The proposed algorithm is stated quite clearly, which makes it possible to automate it for the production of practical calculations on the numerical integration of a system of differential equations in partial derivatives using a computer.

Theoretical fractional formulation of a three-dimensional radio frequency ion trap (Paul-trap) for optimum mass separation

We investigate the dynamics of an ion confined in a Paul–trap supplied by a fractional periodic impulsional potential. The Cantor–type cylindrical coordinate method is a powerful tool to convert differential equations on Cantor sets from cantorian–coordinate systems to Cantor–type cylindrical coordinate systems. By applying this method to the classical Laplace equation, a fractional Laplace equation in the Cantor–type cylindrical coordinate is obtained. The fractional Laplace equation is solved in the Cantor–type cylindrical coordinate, then the ions is modelled and studied for confined ions inside a Paul–trap characterized by a fractional potential. In addition, the effect of the fractional parameter on the stability regions, ion trajectories, phase space, maximum trapping voltage, spacing between two signals and fractional resolution is investigated and discussed.

Further results on modified harmonic functions in three dimensions

The Weinstein equation , with , considered in , is a modification of the classical Laplace equation . Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative.The main result of this article is the statement that in case the negative integers are even, i.e., , we still have a Fischer‐type decomposition. For , the classical harmonic functions, this decomposition is well known. But also in case , a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case or and probably in all higher negative odd cases, the decomposition doesn't hold.In case , we give a complete description of the vector space of homogeneous k‐modified harmonic polynomials of degree n in . Such a result is also at hand in case . Finally, in case of the classical harmonic functions, we give a description of the vector space .

On a nonlocal boundary value problem with an oblique derivative

The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the Rn space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique.

Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator

In this paper, we study the multiplicity of weak solutions to the boundary value problem where Ω is a bounded domain with smooth boundary in is odd in ξ and is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Here we do not require that f satisfies the Ambrosetti-Rabinowitz (AR) condition. The conditions on f and g are relatively weak and our result is new even in the case , i.e. for the classical Laplace equation with the Dirichlet boundary condition.

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1),(-\Delta )^{s}$$ is the fractional Laplacian operator, $$\mu $$ is a positive real parameter, $$q\in [2, 2^*_s)$$ and $$2^*_s=2n/(n-2s)$$ is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of $$\Omega $$ . Precisely, we show that the problem has at least $$cat_{\Omega }(\Omega )$$ nontrivial solutions, provided that $$q=2$$ and $$n\geqslant 4s$$ or $$q\in (2, 2^*_s)$$ and $$n>2s(q+2)/q$$ , extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators

In this paper, we consider the following perturbed nonlocal elliptic equation \begin{document} $\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right. $ \end{document} where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.

Fractional Laplacian equations with critical Sobolev exponent

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$ $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $$s\in (0,1),\, \Omega $$ is an open bounded set of $${\mathbb {R}}^n,\, n>2s$$ , with continuous boundary, $$\lambda $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is a fractional critical Sobolev exponent and $$f$$ is a lower order perturbation of the critical power $$|u|^{2^*-2}u$$ , while $$\mathcal {L}_K$$ is the integrodifferential operator defined as $$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Under suitable growth condition on $$f$$ , we show that this problem admits non-trivial solutions for any positive parameter $$\lambda $$ . This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $$K(x)=|x|^{-(n+2s)}$$ (this gives rise to the fractional Laplace operator $$-(-\Delta )^s$$ ), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

A molecular dynamics simulation study of nanoscale liquid threads

The properties of liquid threads in angular direction are studied. On the basis of Gibbs theory, a thermodynamic model is constructed and a formula of surface of tension is derived. For the determination of surface tension, different methods are presented. We investigate seven different systems (the numbers of molecules N are 1600, 2240, 2880, 3360, 4000, 4800, and 5280) by molecular dynamics simulations. We analyze the surface tension obtained by various routes, then whether the classical Laplace equation is applicable in nanoscale can be determined. The results obtained by molecular dynamics simulations support that surface tension is dependent on the dividing surface curvature.

On the equilibrium conditions for a spherical-cap liquid drop on a solid surface——Also comments on “Thermodynamic mechanism for the condensation of liquid drops on the condense surface” etc.

For a liquid drop of first kind (with the size much bigger than the interface thickness), whether on or far away from a solid surface, it is proved that the classical Laplace equation is valid and the related arguments are clarified. The formula of additional pressure for the liquid drop of the first kind dependent on contact angle, given in Ref. 48 1823] is wrong, the comment of Ref. is correct.

Solutions of Navier equations and their representation structure

Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector O ( n , R ) -invariant generalization of the classical Laplace equation. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible O ( n , R ) -submodules and construct an explicit basis for each irreducible summand. Moreover, we explicitly solve the initial value problems for Navier equations and their wave-type extension—Lamé equations by Fourier expansion and Xu's method of solving flag partial differential equations. Our work might be counted as a continuation of Olver's important work on the algebraic study of elasticity in a certain sense.

Generating Singularities of Weak Solutions of $p$-Laplace Equations On Fractal Sets

We study $p$-Laplace equations $-\Delta_p u=F(x)$ possessing weak solutions in the Sobolev space $W_0^{; ; 1, p}; ; (\Omega)$, $\Omega\subset\mathbb{; ; R}; ; ^N$, that are singular on prescribed fractal sets having large Hausdorff dimension. With an appropriate choice of $F\in L^{; ; p'}; ; (\Omega)$ the Hausdorff dimension of singular set of the weak solution can be made arbitrarily close to $N-pp'$ if $pp'<N$. For $p=2$, that is, for the classical Laplace equation, the bound $N-4$ is optimal, provided $N\ge4$. Moreovoer, there exist maximally singular solutions, that is, such that the bound is achieved. The proof is obtained via an explicit lower a-priori bound of supersolutions corresponding to special choice of right hand-sides that are singular near a fractal set.

Equilibrium of Multi-Phase Systems in Gravitational Fields

Four necessary conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field were derived by Ward and Sasges (1998) in a unified setting, by using an entropy maximization approach, and under the assumption that the liquid-vapor surface tension does not depend on elevation. These are thermal equilibrium, the Laplace and Young equations, and a condition on the chemical potentials of the components present in the system. Gibbs (1876) had obtained the Young equation in a derivation separate from the derivation of the other three conditions and by using an energy minimization approach. However, Gibbs had derived a more general form of the Laplace equation than Ward and Sasges's. Gibbs's equation contained a term expressing the contribution of the variation of surface tension with elevation. This equation has since been neglected by most of the scientific community. In the present paper, the same approach as Ward and Sasges's is used to derive, in an unified setting, the conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field but under the assumption that the liquid-vapor surface tension may depend on elevation. The four well-known conditions for equilibrium are obtained, with Gibbs's generalized Laplace equation instead of the classical Laplace equation. The derivations in this paper were carried out for two different system geometries, namely, for a sessile drop and for a conical capillary tube, and similar conditions for equilibrium were obtained.

A general stability criterion for droplets on structured substrates

Wetting morphologies on solid substrates, which may be chemically or topographically structured, are studied theoretically by variation of the free energy which contains contributions from the substrate surface, the fluid– fluid interface and the three-phase contact line. The first variation of this free energy leads to two equations—the classical Laplace equation and a generalized contact line equation—which determine stationary wetting morphologies. From the second variation of the free energy we derive a general spectral stability criterion for stationary morphologies. In order to incorporate the constraint that the displaced contact line must lie within the substrate surface, we consider only normal interface displacements but introduce a variation of the domains of parametrization.

Pressure difference between inside and outside of a drop and its critical radius in dropwise condensation

It was proved using the thermodynamic method that the pressure difference between the inside and outside of a liquid drop sitting on a solid surface follows the classical Laplace equation, and this was further validated using the mechanics method. Such a pressure difference depends only on the surface tension of the liquid and the radius of the drop but is independent of the liquidsolid contact angle.Similarly, the critical radius of a drop in dropwise condensation is also independent of the contact angle and can be calculated using the classical Kalvin equation.