### Abstract

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.

Original language | English |
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Pages (from-to) | 637-662 |

Number of pages | 26 |

Journal | Manuscripta Mathematica |

Volume | 141 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2013 Jul |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Manuscripta Mathematica*,

*141*(3-4), 637-662. https://doi.org/10.1007/s00229-012-0586-6