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

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that a(·, ·) is a continuous bilinear form on the product {X × Y} of Banach spaces X and Y, where Y is reflexive. If null spaces N _X and N _Y associated with {a(·, ·)} have complements in X and in Y, respectively, and if {a(·, ·)} satisfies certain variational inequalities both in X and in Y, then for every F ∈ N⊥/Y, i.e., F ∈ Y* with F(φ) = 0} for all φ ∈ N_Y, there exists at least one u ∈ X such that a(u, φ) = F(φ) holds for all φ ∈ Y with {double pipe}u{double pipe}X ≤ C{double pipe}F{double pipe}Y*. 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.