| 刘思童,梁波.四阶椭圆型方程弱解的存在性[J].井冈山大学自然版,2024,45(1):1-7 |
| 四阶椭圆型方程弱解的存在性 |
| EXISTENCE OF WEAK SOLUTIONS FOR A CLASS OF FOURTH-ORDER ELLIPTIC EQUATIONS |
| 投稿时间：2023-09-12 修订日期：2023-11-10 |
| DOI：10.3969/j.issn.1674-8085.2024.01.001 |
| 中文关键词: 四阶椭圆型方程 Lax-Milgram定理 变分法 存在性 唯一性 |
| 英文关键词: fourth-order elliptic equation Lax-Milgram theorem variational method existence uniqueness |
| 基金项目:辽宁省教育厅高校科研项目(LJKMZ20220832) |
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| 中文摘要: |
| 针对四阶椭圆方程的不同形式，分别应用Lax-Milgram定理及变分法对两类四阶椭圆型方程进行研究。本文第一部分运用Lax-Milgram验证在Hilbert空间H2上恒存在唯一的解u，使得H2上的有界强制双线性型与H2上任一有界线性泛函相等。进而证明出存在唯一弱解满足第一类含有一阶项的四阶椭圆型方程。第二部分运用变分方法解决另一类含有p次二阶项四阶椭圆型方程。在方法上，首先定义方程弱解，其次找出与方程相对应的泛函，进而将问题转化为求相应泛函的极值元，证明泛函极值元的存在性，最后证明弱解的唯一性。 |
| 英文摘要: |
| For the different forms of fourth order elliptic equations, and applies the Lax Milgram theorem and variational method are respectively applied to study two types of fourth order elliptic equations. In the first part of this article, Lax Milgram is used to verify the existence and uniqueness of a solution u on a Hilbert spaceH2, so that the bounded mandatory bilinear form is equal to any bounded linear functional on H2. Furthermore, it is proven that there exists a unique weak solution for a class of fourth order elliptic equations with a first order term. In the second part of this article, the variational method is used to solve another class of fourth order elliptic equations with second order terms of degree p. In the method, firstly, the weak solution of the equation is defined, then the corresponding functional is found out, and then the problem is transformed into the extremal element of the corresponding functional, the existence of the extremal element of the functional is proved, and finally the uniqueness of the weak solution is proved. |
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