含伪微分算子边界条件下的伪微分方程组两点边值问题

O. Makarov, I. Nikolenko
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引用次数: 0

摘要

研究了含伪微分算子边界条件下的二阶伪微分方程和伪微分方程的两点边值问题。考虑伪微分算子的需要有两个原因,一是在应用问题中越来越多地出现这样的方程,二是通过考虑这样的方程,可以在Schwartz空间S及其对偶空间中实现边值问题的适定性。首先,我们考虑一个标量伪微分方程,其符号属于空间$C_{-\infty}^{\infty}$,由多项式生长的无穷可微函数组成。对于该方程,得到了某一特定类型的边值问题在空间S中适定的边界条件,并给出了该边值问题在空间S中适定的微分-差分方程的一个例子和带有卷积型伪微分算子的特定边界条件。然后我们考虑一个由两个伪微分方程组成的系统,其符号来自空间$C _ {-\infty} ^ {\infty}$。对于这一系统,我们证明了空间s中一个适定边值问题的存在性。在证明中使用了傅里叶变换和系统的三角化。在这种情况下,我们还给出了在空间s中该边值问题正确的一个系统和特定的边界条件的例子,从而证明了对于任何伪微分方程,以及对于两个伪微分方程的系统,在$S$空间中总是存在一个正确的边值问题,而边界条件包含伪微分算子。给出了构造正确边界条件的算法。它们是伪微分算子,其符号依赖于伪微分方程的符号。
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Two-point boundary value problem for systems of pseudo-differential equations under boundary conditions containing pseudo-differential operators
This paper deals with a two-point boundary value problem for pseudodifferential equations and for systems of second order pseudodifferential equations under boundary conditions containing pseudodifferential operators. The need to consider pseudodifferential operators is caused by two reasons, first, such equations appear more and more often in applied problems, and second, by considering such equations, it is possible to achieve the well-posedness of the boundary value problem in the Schwartz space S and in its dual space.First, we consider a scalar pseudodifferential equation with a symbol belonging to the space $C_{-\infty}^{\infty}$, consists of infinitely differentiable functions of polinomial growth. For this equation it is found of the boundary condition under which a specific type the boundary value problem is well-posed in the space S. In addition, an example of a differential-difference equation and a specific boundary condition with a convolution-type pseudo-differential operator under which this boundary value problem is well-posed in the space S are given.Then we consider a system of two pseudodifferential equations with symbols from the space $C _ {-\infty} ^ {\infty}$. For this system, we prove the existence of a well posed boundary value problem in the space S. The Fourier transform and the reduction of the system to a triangular form are used in the proof. In this case, we also give an example of a system and a specific boundary condition under which this boundary value problem is correct in the space S.Thus, the work proves that for any pseudo-differential equation, as well as for a system of two pseudo-differential equations, there is always a correct boundary value problem in the $S$ space, while the boundary conditions contain pseudo-differential operators. The algorithm for constructing correct boundary conditions is also indicated. They are pseudo-differential operators whose symbols depend on the symbols of pseudo-differential equations.
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