Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems
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- 作者：Hideo Kozono (1)
Taku Yanagisawa (2) - 关键词：35Q40 ; 35F99
- 刊名：manuscripta mathematica
- 出版年：2013
- 出版时间：4 - July 2013
- 年：2013
- 卷：141
- 期：3
- 页码：637-662
- 全文大小：306KB
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Taku Yanagisawa (2)
1. Department of Mathematics, Waseda University, Tokyo, 169-8555, Japan
2. Department of Mathematics, Nara Women鈥檚 University, Nara, 630-8506, Japan - ISSN：1432-1785
文摘
We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N X and N Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.