, Volume 141, Issue 3-4, pp 637-662
Date: 03 Nov 2012

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

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Abstract

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.