AN ANALOGUE OF SAINT-VENANT'S PRINCIPLE AND THE UNIQUENESS OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR PARABOLIC EQUATIONS IN UNBOUNDED DOMAINS
AN ANALOGUE OF SAINT-VENANT'S PRINCIPLE AND THE UNIQUENESS OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR PARABOLIC EQUATIONS IN UNBOUNDED DOMAINS
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模拟了圣维南原理和无界域抛物方程边值问题解的唯一性
DOI：
10.1070/rm1976v031n06abeh001583
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发表时间：
1976
影响因子：
0.9
通讯作者：
G. Iosif'yan
中科院分区：
文献类型：
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作者：
O. Oleinik;G. Iosif'yan
吉洪诺夫的论文“A unique theorem for the equation of heat conduction”[1]，发表于1935年，对偏微分方程理论的发展产生了重大影响。在本文中，他证明了唯一性定理的解决方案柯西问题的方程热传导在某些类的职能指数增长，并建造的例子解决方案，以显示非唯一性更广泛的类的职能。许多研究一直致力于吉洪诺夫的论文所产生的问题，并随后推广和发展他的结果（见[2]-[10]，和其他地方）;这项研究形成了显着贡献的理论偏微分方程。在这里，我们研究的问题的唯一性的解决方案的柯西问题，边值问题，并没有初始条件的问题。我们还研究了二阶抛物型方程解的渐近性质，通过使用一种基于推导解的先验估计的方法，类似于弹性理论中的圣维南原理[11]。文[8]-[10]给出了另一种新的方法，它使我们能够研究具有一般边界条件的一般抛物型方程组的这些问题，并得到类似于Tikhonov定理的结果。它使用解析的解决方案，某些辅助抛物型系统相对于一个额外的独立变量。
Tikhonov's paper "A uniqueness theorem for the equation of heat conduction" [1], published in 1935, has had a great influence on the development of the theory of partial differential equations. In this paper he proved a uniqueness theorem for the solution of the Cauchy problem for the equation of heat conduction in certain classes of functions of exponential growth, and constructed examples of solutions to show non-uniqueness in wider classes of functions. Much research has been devoted to problems arising from Tikhonov's paper, and to the subsequent generalization and development of his results (see [2]-[10], and elsewhere); this research forms a significant contribution to the theory of partial differential equations. Here we study the question of the uniqueness of the solution of the Cauchy problem, of boundary value problems, and of a problem without initial conditions. We also study the asymptotic properties of solutions of second order parabolic equations, by using a method based on the derivation of a priori estimates for the solutions that are similar to Saint-Venant's principle in the theory of elasticity [11]. Another new approach, which allows us to investigate these questions for general parabolic systems with general boundary conditions, and to obtain an analogue of Tikhonov's theorem, is given in [8]-[10]. It uses the analyticity of solutions of certain auxiliary parabolic systems with respect to an additional independent variable.