On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition

The Frankl problem without the spectral parameter was considered by Bitsadze and Smirnov. The present paper gives the eigenvalues and eigenfunctions of the Frankl problem with the odd parity condition. We prove the completeness of eigenfunctions. The Frankl problem with a nonlocal parity condition for the Lavrent’ev-Bitsadze equation is studied. The eigenvalues and eigenfunctions are found, and the basis property of the eigenfunctions in the elliptic part of the domain in the space L ₂ is proved.     

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition of the second kind

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.     

Basis property of eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equation. We show that these eigenfunctions form a Riesz basis in the elliptic part of the domain. In addition, we prove the Riesz basis property on [0, π/2] of the system of cosines occurring in the expressions for the eigenfunctions. Earlier, the Riesz basis property was proved for the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with continuous solution gradient.     

Basis property of the eigenfunctions of a generalized gasdynamic Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

We find the eigenfunctions of a generalized Frankl problem with the use of Bessel functions. We prove that these eigenfunctions form a Riesz basis in the space L ₂(D ₊), where D ₊ is the elliptic part of the domain. In addition, we prove the Riesz basis property of a trigonometric function system and the completeness of this system in the space L ₂(0, π/2).     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.     

On the completeness of eigenfunctions of some higher order operators

The completeness of diverse eigenfunction systems is of interest, e.g., in connection with their use for expansion purposes. Whereas second-order operators have been treated extensively, it is more difficult to find valid proofs even for comparatively simple fourth- and higher-order operators. A criterion is developed that allows known properties of second-order differential operators to be used as a basis for conclusions regarding the completeness of the eigenfunctions of related fourth-order operators. The completeness of the “flatclamped-plate modes” is proved as an example, and it is demonstrated that the detailed form of the “conditions of finiteness at the singular point” can be crucial for the definition of operators corresponding to differential expressions with singular points.     

On the completeness of the generalized eigenfunctions of elliptic operators on manifolds with conical singularities

We show the completeness of the system of generalized eigenfunctions of closed extensions of elliptic cone operators under suitable conditions on the symbols (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A problem with an analog of Frankl condition on the characteristics for gellerstedt equation with singular coefficient

We investigate the problem with an analog of Frankl condition on boundary characteristics for generalized Tricomi equation. We prove that the formulated problem is correct.     

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.     

Gellerstedt problem with a generalized Frankl matching condition on the type change line with data on external characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with nonclassical matching conditions for the gradient of the solution (in the sense of Frankl) on the type change line of the equation. We prove that the inhomogeneous Gellerstedt problem with data on the external characteristics of the equation is solvable either uniquely or modulo a nontrivial solution of the homogeneous problem. We obtain integral representations of the solution of the problem in both the elliptic and the hyperbolic parts of the domain. The solution proves to be regular.     

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.     

On a problem with a shift and with the Frankl condition on a segment of the degeneration line for a class of equations of mixed type

We consider the Frankl-Nakhushev problem. By using the maximum principle, we prove the uniqueness of the solution of the problem in the class of Hölder functions, and by using the method of integral equations, in particular, the recently developed method of Wiener-Hopf equations, we prove its existence.