Abstract

Boundary-value problems for loaded equations in the plane in the case of loaded parts consist of traces of an unknown solution or its first normal derivatives, are well studied. The multidimensional loaded differential equations are relatively less investigated. Moveover, when the loaded part consists of not only the traces of the solution or its first normal derivatives, but also second derivatives of the solutions, the classical methods are not effective. Therefore, in this paper, we propose a method which overcomes these difficulties. Under some conditions on coefficients of the loaded multidimensional Chaplygin’s equation, we prove existence and uniqueness of a solution of a nonlocal boundary-value problem in the Sobolev space \(W_{2}^{3}(Q)\).

中文翻译：

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Sobolev空间中一维多维Chaplygin方程非局部边值问题的唯一可解性

摘要

很好地研究了在加载零件包含未知解或其一阶导数的痕迹的情况下，平面中加载方程的边值问题。相对较少研究多维加载的微分方程。移动时，当加载的零件不仅包含溶液的迹线或其一阶正导数，还包含溶液的二阶导数时，传统方法无效。因此，在本文中，我们提出了一种克服这些困难的方法。在加载的多维Chaplygin方程系数的某些条件下，我们证明了Sobolev空间\（W_ {2} ^ {3}（Q）\）中非局部边值问题解的存在性和唯一性。