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.     

Solvability of the Gellerstedt problem with data on parallel characteristics

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with boundary conditions on parallel characteristics in the hyperbolic domain of the equation. Three distinct types of conditions on the type change lines are considered, and existence and uniqueness theorems for the corresponding problems are proved.     

Problem with displacement conditions on internal characteristics and the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient

We study the well-posedness of a problem with displacement conditions on internal characteristics and an analog of the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient. The uniqueness of a solution is proved with the use of an extremum principle. The proof of the existence uses the method of integral equations.     

Initial-boundary value problem for a parabolic-hyperbolic equation with power-law degeneration on the type change line

We consider the first boundary value problem for equations of mixed type in a rectangular domain. A criterion for the solution uniqueness is proved by the spectral expansion method. The solution is constructed in the form of a series in the eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution with respect to the initial function is proved.     

Eigenfunctions of the Tricomi problem with an inclined type change line

We construct the eigenfunctions of the Tricomi problem for the case in which the type change line of the elliptic-hyperbolic equation is inclined and forms an arbitrary angle α with the x-axis. These eigenfunctions form a basis in the elliptic domain. In addition, we find an integral constraint on the inclined type change line.     

Gellerstedt problem for a differential-difference equation of mixed type with advanced-retarded multiple deviations of the argument

We consider the Gellerstedt problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part and with advanced-retarded multiple deviations of the argument in the derivatives and the function. We prove the uniqueness theorem for the problem without restrictions on the deviation value. The problem is uniquely solvable. We derive closed-form integral representations of the solutions.     

Problem with shift on parallel characteristics for Gellerstedt equation with singular coefficient

We study correctness for a problemwith an analog of Frankl condition on a degeneration line segment and dislocation conditions on parallel characteristics for Gellerstedt equation with singular coefficient. With the help of maximum principle we prove uniqueness of a solution to the problem and with the method of integral equations we prove the existence of a solution to the problem.     

Regularity theorem for a one-sided problem with convex constraints on the gradient of the solution

One considers a one-sided problem for a second-order linear elliptic operator, according to the conditions of which the gradient of the solution at each point x must belong to a given strictly convex set K(x). Under certain conditions ensuring the solvability of the problem in the class one proves that the first-order derivatives of the solution are locally Lipschitz continuous.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 166–171, 1984.     

Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu?f(u)l(|∇u|) and Δφu?f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality Δu?f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.     

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.     

Convergence conditions for restarted conjugate gradient methods with inaccurate line searches

Convergence properties of restarted conjugate gradient methods are investigated for the case where the usual requirement that an exact line search be performed at each iteration is relaxed.The objective function is assumed to have continuous second derivatives and the eigenvalues of the Hessian are assumed to be bounded above and below by positive constants. It is further assumed that a Lipschitz condition on the second derivatives is satisfied at the location of the minimum.A class of descent methods is described which exhibitn-step quadratic convergence when restarted even though errors are permitted in the line search. It is then shown that two conjugate gradient methods belong to this class.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Initial boundary value problem for the radiative transfer equation with diffusion matching conditions

Under study is the Cauchy problemfor the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface. The solvability of the initial-boundary value problem is proved. Some stabilization conditions for the nonstationary solution are obtained.     

Bitsadze-Samarskii problem with a lacking shift condition for the Gellerstedt equation with a singular coefficient

For the Gellerstedt equation with a singular coefficient, we study the well-posedness of the problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of the degeneration line and with a shift condition on parts of boundary characteristics. We use the maximum principle to prove the uniqueness of the solution of the problem in the class of Hölder functions and the method of integral equations to prove its existence.     

An interior estimate of the gradient of a solution to the Dirichlet problem for equations of curvature type

The Dirichlet problem for equations of curvature type is considered. It is proved that the gradient of the solution is a priori bounded at an interior point of the domain. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 188–195.     

Asymptotic solution of the Cauchy problem for equations with complex characteristics

Differential equations with a small parameter in the derivative are considered. A method is developed for constructing formal asymptotic solutions for the case of complex characteristics. For this a new class of manifolds is introduced which is a natural generalization of real Lagrangian manifolds to the complex case. The theory of the canonical Maslov operator is constructed in this class of manifolds. Asymptotic solutions are expressed in terms of the canonical Maslov operator.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 41–136, 1977.