TY - JOUR

T1 - Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

AU - Kozono, Hideo

AU - Yanagisawa, Taku

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/7

Y1 - 2013/7

N2 - 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 ∈ NY, 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.

AB - 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 ∈ NY, 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.

UR - http://www.scopus.com/inward/record.url?scp=84878760790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878760790&partnerID=8YFLogxK

U2 - 10.1007/s00229-012-0586-6

DO - 10.1007/s00229-012-0586-6

M3 - Article

AN - SCOPUS:84878760790

VL - 141

SP - 637

EP - 662

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3-4

ER -