Some Nonclassical Boundary Value Problems for Linear Mixed-Type Equations

Well-posedness of nonlocal boundary value problems is studied for some class of mixed-type equations which includes the Chaplygin equation and parabolic equations with varying time direction.     

On the Question of Existence of a Solution to the Tricomi Problem for One Class of Systems of Mixed-Type Equations

We establish extremal properties of regular and weak solutions to the Tricomi problem for a system of mixed-type equations. Using a Schwartz-type alternation method, we prove unique weak solvability of the Tricomi problem under some constraints on coefficients.     

On the Equations of Linear Shallow Shell Theory

The paper presents a formulation of the two-dimensional theory of shallow shells, including the effects of transverse shear deformation and of moments turning about the normal to the middle surface. The present formulation includes, as it must, Marguerre's theory. At the same time it is consistent with recent formulations of general linear shell theory, in particular in regard to the preservation of the static-geometric duality. Various reductions of the equation of the theory are considered. Of particular significance and effectiveness among these are reductions for the special cases of (1) shells without moments about the middle surface normal, (where an earlier result of Naghdi is extended) (2) the shell without transverse shear deformability, (the static-geometrical dual of case (1)). As an application of the general equations an explicit solution is obtained for the problem of stretching, twisting and bending of pretwisted rectangular plates.     

Spectral Properties of a Solution to the Gellerstedt Problem for Mixed-Type Equations and Their Applications

We find eigenvalues and the corresponding eigenfunctions of the Gellerstedt problem for mixed-type equations with the Lavrent'ev–Bitsadze operator. We study the spectral properties of the system of eigenfunctions and present their applications to construction of a solution to the Gellerstedt problem in the form of the sum of a series.     

A Boundary Value Problem for Mixed-Type Equations in Domains with Multiply Connected Hyperbolicity Subdomains

Unlike the familiar problems for mixed-type equations, we pose a boundary value problem for the Lavrent'ev–Bitsadze equation in domains with unbounded multiply connected hyperbolicity subdomains. We prove unique solvability in explicit form. We also study this problem for the general Lavrent'ev–Bitsadze equation.     

On a Problem for a Mixed-Type Equation With Fractional Derivative

For a mixed-type equation with the Riemann–Liouville partial fractional derivative we study a problem where the boundary condition contains a linear combination of generalized fractional operators with the Gauss hypergeometric function. We find a solution to the considered problem explicitly by solving an equation with fractional derivatives of various orders and prove the uniqueness of the solution for various values of parameters of the mentioned operators.     

Morse理论及其在一类N-Laplacian方程中的应用

研究了一类次临界增长的N-Laplacian方程.利用Trudinger-Moser不等式和Morse理论,证明了在不具有山路引理的几何性质、不满足Ambrosetti-Rabinowitz条件及整体符号条件下,该方程非平凡解的存在性.     

On a Frankl-Type Boundary-Value Problem for a Mixed-Type Degenerating Equation

Ukrainian Mathematical Journal - We study the existence and uniqueness of solutions for an analog of the Frankl-type boundary-value problem for a parabolic-hyperbolic-type equation. The uniqueness...     

低温等离子模型中的一个混合型方程的闭Dirichlet问题

描述理想的低温等离子体中电磁波传播的模型是一个椭圆双曲混合型方程.证明了该方程闭Dirichlet问题弱解的存在唯一性.该结果关于区域的几何结构要求较少.由于这里所讨论的方程的奇异性与Keldysh方程的奇异性有相似性质,而后者的奇异性比Tricomi方程更强,因此关于其正则性的研究是很有意义的.作者给出了一个内正则性结果.     

低温等离子模型中的一个混合型方程的闭Dirichlet问题

描述理想的低温等离子体中电磁波传播的模型是一个椭圆双曲混合型方程.证明了该方程闭Dirichlet问题弱解的存在唯一性.该结果关于区域的几何结构要求较少.由于这里所讨论的方程的奇异性与Keldysh方程的奇异性有相似性质,而后者的奇异性比Tricomi方程更强,因此关于其正则性的研究是很有意义的.作者给出了一个内正则性结果.