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.
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References
Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Comp., New York (1965)
Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964)
Borchers W., Miyakawa T.: Algebraic L 2 decay for Navier-Stokes flows in exterior domains. Acta Math. 165, 189–227 (1990)
Borchers W., Sohr H.: On the equations v = g and dive u = f with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)
Bourguignon J.P., Brezis H.: Remarks on the Euler equations. J. Funct. Anal. 15, 341–363 (1974)
Cattabriga L.: Su un problema al contorno relativo al sistema di equazinoi di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)
Chang I.D., Finn R.: On the solutions of a class of equations occurring in continuum mechanics with application to the Stokes paradox. Arch. Rational Mech. Anal 7, 388–401 (1961)
Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976)
Faedo S.: Su un principo di esistenza nell’analisi lineare. Ann. Scuola Norm. Sup Pisa Cl. Sci. 11, 1–8 (1957)
Fichera, G.: Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari. Convegno Inter. Eq. Lin. Der. Parz. Trieste, 174–227 (1954)
Finn R.: Mathematical questions relating to viscous fluid flow in an exterior domain. Rocky Mountain J. Math. 3, 107–140 (1973)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I. Springer-Verlag, New York (1994)
Galdi G., Simader C.G: Existence, uniqueness and L q-estimates for the Stokes problem in an exterior domain. Arch. Rational Mech. Anal. 112, 291–318 (1990)
Georgescu V.: Some boundary value problems for differential forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. 122, 159–198 (1979)
Giga Y., Sohr H.: On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sect. IA Math 36, 103–130 (1989)
Kozono H.: L 1-solutions of the Navier-Stokes equations in exterior domains. Math. Ann. 312, 319–340 (1998)
Kozono H., Sohr H: On a new class of generalized solutions for the Stokes equations in exterior domains. Ann. Scuola Norm. Sup Pisa Cl. Sci. 19, 155–181 (1992)
Kozono H., Sohr H.: Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains. J. Math. Soc. Jpn. 44, 307–330 (1992)
Kozono H., Yanagisawa T.: L r-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains. Indiana Univ. Math. J. 58, 1853–1920 (2009)
Kozono, H., Yanagisawa, T.: Global compensated compactness theorem for general differential operators of first order. Arch. Rational Mech. Anal. (To appear)
Kozono, H., Yanagisawa, T.: New approach to the L r-de Rham-Hodge-Kodaira decomposition on manifolds with the boundary (in preparation)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Grundlerhen 181. Springer-Verlag, Berlin (1972)
Miranda, C.: Partial Differential Equations of Elliptic type, 2nd edn. Springer-Verlag, Berlin (1966)
Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Grundlerhen 130. Springer-Verlag, Berlin (1966)
Peetre J.: Another approach to elliptic boundary value problems. Comm. Pure Appl. Math. 14, 711–731 (1961)
Ramaswamy, J.: The Lax-Milgram theorem for Banach spaces I. Proc. Jpn. Acad. Ser. A 56, 462–464 (1980)
Ramaswamy, J.: The Lax-Milgram theorem for Banach spaces II. Proc. Jpn. Acad. Ser. A 57, 29–33 (1981)
Schwarz, G.: Hodge Decomposition—a Method for Solving Boundary Value Problems. Lecture Note in Mathematics 1607. Springer-Verlag, Berlin (1995)
Simader, C.G.: On Dirichlet’s Boundary Value Problem. Lecture Notes in Mathematics 268. Springer-Verlag, Berlin (1972)
Simader, C.G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in L q-spaces for bounded and exterior domains. In: Galdi, G.P. (ed.) Mathematical Problems Relating to the Navier-Stokes Equations. Series on Advanced in Mathematics for Applied Sciences, pp. 1–35. World Scientific, Singapore (1992)
Simader, C.G., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series 360. Logman, Harlow (1996)
von Wahl, W.: Vorlesung über das Aussenraumproblem für die instationären Gleichungen von Navier-Stokes. Vorlesungsreihe des SFB 256 “Nichtlineare Patielle Differentialgleichungen” Bonn 11 (1990)
von Wahl W.: Estimating \({\nabla u}\) by dive u and curl u. Math. Method Appl. Sci. 15, 123–143 (1992)
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Kozono, H., Yanagisawa, T. Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems. manuscripta math. 141, 637–662 (2013). https://doi.org/10.1007/s00229-012-0586-6
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DOI: https://doi.org/10.1007/s00229-012-0586-6