复微分算子方程指数型dirichlet-taylor级数空间中的非局部边值问题

V. Il'kiv, N. Strap, I. Volyanska
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引用次数: 0

摘要

偏微分方程的非局部条件问题是当今微分方程理论的一个重要组成部分。这类问题主要在Hadamard意义上存在,它们的可解性与小分母问题有关。本文研究了一类具有广义微分算子$B=zd/dz$的偏微分方程的非局部边值问题,该算子作用于标量复变量$z$的函数。在狄利克雷-泰勒级数的函数空间中,建立了这些问题的唯一可解判据及其解存在的充分条件。证明了这些空间中问题解的统一定理和存在性定理。在许多广义微分算子的情况下所考虑的问题在Hadamard意义上是不正确的,它的可解性取决于在解的构造中出现的小分母。本文证明了在单变量的情况下,相应的分母并不小,并由一些常数从下面估计。给出了问题的哈达玛后的正确性。它区别于具有许多空间变量的非条件后哈达玛问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
NONLOCAL BOUNDARY VALUE PROBLEM IN SPACES OF EXPONENTIAL TYPE OF DIRICHLET-TAYLOR SERIES FOR THE EQUATION WITH COMPLEX DIFFERENTIATION OPERATOR
Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.
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