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We consider the optimal recovery problem for the solution of the Dirichlet problem for the Laplace equation in the unit d-dimensional ball on a sphere of radius ρ from a finite collection of inaccurately specified Fourier coefficients of the solution on a sphere of radius r, 0 < r < ρ < 1. The methods are required to be exact on certain subspaces of spherical harmonics. [ABSTRACT FROM AUTHOR]

The wave equation on an infinite homogeneous tree is studied. For the Laplace operator, the Kirchhoff conditions are taken as the matching conditions at the vertices. A solution of the Cauchy problem is obtained and the behavior of the wave energy as time tends to infinity is described. It is shown that part of the energy does not go to infinity, but remains on the edges of the trees. The part of the energy remaining on the edges depends on the branching number. [ABSTRACT FROM AUTHOR]

In this paper, we prove the nonexistence of global solutions to the quasilinear backward parabolic inequality with homogeneous Dirichlet boundary condition and bounded integrable sign-changing initial function, where Ω is a bounded smooth domain in ℝ. The proof is based on the derivation of a priori estimates for the solutions and involves the algebraic analysis of the integral form of the inequality with an optimal choice of test functions. We establish conditions for the nonexistence of solutions based on the weak formulation of the problem with test functions of the form where u and u are the positive and negative parts of the solution u of the problem and φ is the standard cut-off function whose support depends on the parameter R. [ABSTRACT FROM AUTHOR]

An analog of the Oleinik-Hopf normal derivative lemma for the Laplace operator on a polyhedral set is considered. [ABSTRACT FROM AUTHOR]

On the basis of determining equations written out in terms of Laplace invariants, some classes of Bianchi equations of third order similar to well-known classes of hyperbolic equations with two independent variables are singled out. [ABSTRACT FROM AUTHOR]