Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball

We find a polynomial solution of the Dirichlet problem for the inhomogeneous biharmonic equation with polynomial right-hand side and polynomial boundary data in the unit ball. To this end, we use the closed-form representation of harmonic functions in the Almansi formula.     

Radial solutions for a Dirichlet problem in a ball

The Ambrosetti-Prodi boundary value problem with an asymptotically linear nonlinearity is considered. Under general conditions on the nonlinearity it is shown that there exist positive and negative solutions. In the case when the domain is a ball in Rⁿ and the nonlinearity “crosses” the first n eigenvalues, corresponding to radial eigenfunctions, it is proved that there are at least n + 1 radial solution.     

On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball

Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous Neumann problem for a polyharmonic equation in the unit ball.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

Dirichlet problem in the ball for an equation with a delta-like coefficient

We study the Dirichlet problem in the ball for the Helmholtz equation with an additional term in the form of the product by the delta function. The additional term is approximated by the simplest integral expression, and the solution of the original equation is defined as the limit of solutions of the regularized equations. We obtain exact solutions depending on the chosen method of approximation.     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

Unique solvability of the Dirichlet problem for an ultrahyperbolic equation in a ball

We obtain a criterion in terms of zeros of Jacobi polynomials for the uniqueness of the solution of the first boundary value problem for an ultrahyperbolic equation in a ball. The nonuniqueness in the Dirichlet problem proves to occur if and only if the coefficient of the equation belongs to a countable dense subset of the real line.     

On a new method for constructing the Green function of the Dirichlet problem for the polyharmonic equation

By using a special decomposition of the fundamental solution, we construct a new representation of the Green function of the Dirichlet problem for the polyharmonic equation in a ball.     

Convergence of difference solutions to generalized solutions of the Dirichlet problem for a second-order elliptic equation

Exact difference scheme operators are applied to construct a difference scheme for the Dirichlet problem for a secondorder elliptic equation with variable coefficients in a rectangle. If the solution belongs to the class W ₂ ² (), the scheme is of first-order accuracy in the grid norm of W ₂ ¹ ().Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 21–26, 1985.     

A problem for the polyharmonic equation in the sphere

Tashkent. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 51–58, September–October, 1991.     

Regularity of solutions to the polyharmonic equation in general domains

The present paper establishes boundedness of $[m-\frac{n}{2}+\frac{1}{2} ]$ derivatives for the solutions to the polyharmonic equation of order 2m in arbitrary bounded open sets of ${\mathbb{R}}^{n}$ , 2≤n≤2m+1, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.     

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem

     

Asymptotic solutions of the Dirichlet problem for the heat equation with impulses

We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet problem for the heat equation with impulses. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 427–430, March, 2006.     

Application of the almansi formula for constructing polynomial solutions to the dirichlet problem for a second-order equation

We obtain the Almansi decomposition for the second-order partial differential operators with constant coefficients. This decomposition is used for constructing a polynomial solution to the Dirichlet problem.     

Positive solutions for a Dirichlet problem

1. IntroductionSince the work of Ambrosetti and Rabinowitz[l], the problems similar to{;t2:<::l">, (l.l)have been studied extensively But it is well known that, for applying the Moulltain PassTheOrem, we atway8 assum that g(x, 8) is suPerlineax in s at indnity; moeove) a strongercondition like (AR) (see later on) is required. If these conditions are not satisfied, can wealso get solutions for problem (1.1) by a Mountain Pass Theorem? So, ill this paPer, westudy the following Dirichlet pr…