{"title":"多维Lavrentiev - Bitsadze方程主要混合问题的适定性","authors":"S. Aldashev","doi":"10.18287/2541-7525-2021-27-3-7-13","DOIUrl":null,"url":null,"abstract":"It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"231 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation\",\"authors\":\"S. Aldashev\",\"doi\":\"10.18287/2541-7525-2021-27-3-7-13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.\",\"PeriodicalId\":427884,\"journal\":{\"name\":\"Vestnik of Samara University. Natural Science Series\",\"volume\":\"231 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Vestnik of Samara University. Natural Science Series\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.18287/2541-7525-2021-27-3-7-13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik of Samara University. Natural Science Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18287/2541-7525-2021-27-3-7-13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
已知弹性膜在空间中的振荡是用偏微分方程来模拟的。如果膜的挠度被认为是u(x;t);X = (x1;:::;xm); m 2;然后,根据汉密尔顿原理,我们得到一个多维波动方程。假设膜在弯曲位置处于平衡状态,我们也从哈密顿原理得到了多维拉普拉斯方程。因此,弹性膜在空间中的振荡可以用多维Lavrentiev Bitsadze方程来建模。研究了广义函数空间中多维双曲型方程柱面上的主要混合问题。在作者的著作中,证明了该问题对于多维双曲型和椭圆型方程的适定性,并得到了经典解的显式形式。据我们所知,这些问题对于多维双曲-椭圆方程还没有研究过。具有边值条件的多维Lavrentiev - Bitsazde方程混合问题是不适定的。本文证明了具有边界和初始条件的多维Lavrentiev Bitsadze方程的主要混合问题的唯一可解性,并得到了经典解的显式形式。
Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation
It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.