Gellerstedt Equation Research Articles

The Bitsadze-Samarskii problem with the Frankl condition on the degeneration line for a mixed-type equation with singular coefficient

We consider the Bitsadze-Samarskii problem with the Frankl condition for the Gellerstedt equation with a singular coefficient and prove its well-posedness.

A combination of the Tricomi problem and a boundary-value problem with a shift for the Gellerstedt equation

We study a problem whose statement combines the Tricomi problem and the problem with a shift considered by V. I. Zhegalov and A. M. Nakhushev for the Gellerstedt equation with a singular coefficient. We prove its solvability by the method of integral equations, and we do the uniqueness of the solution with the help of the extremum principle.

Well-posedness of the Dirichlet and Poincaré problems for a multidimensional Gellerstedt equation in a cylindrical domain

We prove the unique solvability of the Dirichlet and Poincare problems for a multidimensional Gellerstedt equation in a cylindrical domain. We also obtain a criterion for the unique solvability of these problems.

Problem with Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics for the gellerstedt equation with a singular coefficient

We study the well-posedness of a problem for the Gellerstedt equation with a singular coefficient and with the Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics.

A problem with a Frankl condition on the boundary of the ellipticity domain for the Gellerstedt equation with a singular coefficient

We study a problem with the Frankl and Bitsadze-Samarskii conditions on the elliptic boundary and on the degeneration line for the Gellerstedt equation with a singular coefficient. We prove the correctness of the stated problem.

Mutually-adjoint boundary-value problems with deviation from the characteristic for a multidimensional Gellerstedt equation

In this paper, we study mutually-adjoint boundary-value problems with a deviation from the characteristic for multidimensional Gellerstedt equation.

A problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation

In this paper we study a problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation. We prove that the stated problem is well-posed.

The weighted [formula omitted] estimate on the solution of the Gellerstedt equation in the upper half space

In this note, we are concerned with the weighted W 2 , p estimate for the Gellerstedt equation, which is degenerate on the boundary ∂ R + 2 . Based on the estimates of the generalized Airy functions and Mihlin's multiplier theorem, we establish a weighted W 2 , p a priori estimate.

A nonlocal boundary-value problem for the gellerstedt equation

We consider the boundary-value problem for the Gellerstedt equation $$Signy\left| y \right|^m u_{xx} + u_{yy} = 0,$$ wherem=const > 0, in a mixed region; in contrast to the Tricomi problem, nonlocal conditions pointwise connecting the boundary valuesu(x, y) with the values on an inner curve and on the line of degeneracy are assumed on some arcs of the elliptic part of the boundary, and a condition with displacement is assumed on the characteristic parts of the boundary. Under certain constraints on the functions in the boundary conditions, we prove the unique solvability of the problem considered.