Lagrange multipliers for functions derivable along directions in a linear subspace
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- by Le Hai An, Pham Xuan Du, Duong Minh Duc and Phan Van Tuoc PDF
- Proc. Amer. Math. Soc. 133 (2005), 595-604 Request permission
Abstract:
We prove a Lagrange multipliers theorem for a class of functions that are derivable along directions in a linear subspace of a Banach space where they are defined. Our result is available for topological linear vector spaces and is stronger than the classical one even for two-dimensional spaces, because we only require the differentiablity of functions at critical points. Applying these results we generalize the Lax-Milgram theorem. Some applications in variational inequalities and quasilinear elliptic equations are given.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- David Arcoya and Lucio Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal. 134 (1996), no. 3, 249–274. MR 1412429, DOI 10.1007/BF00379536
- David Arcoya and Lucio Boccardo, Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 1, 79–100. MR 1674782, DOI 10.1007/s000300050066
- H. Brezis. Analyse Fonctionelle (Masson,Paris) 1987.
- Pham Xuan Du and Duong Minh Duc, A non-homogeneous $p$-Laplace equation in border case, Acta Math. Vietnam. 27 (2002), no. 1, 69–75. MR 1914409
- Duong Minh Duc, Nonlinear singular elliptic equations, J. London Math. Soc. (2) 40 (1989), no. 3, 420–440. MR 1053612, DOI 10.1112/jlms/s2-40.3.420
- S. Lang. Analysis I. Addison-Wesley, Reading, 1969.
- Khoi Le Vy and Klaus Schmitt, Minimization problems for noncoercive functionals subject to constraints, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4485–4513. MR 1316854, DOI 10.1090/S0002-9947-1995-1316854-3
- Benedetta Pellacci, Critical points for some functionals of the calculus of variations, Topol. Methods Nonlinear Anal. 17 (2001), no. 2, 285–305. MR 1868902, DOI 10.12775/TMNA.2001.017
- S. Ramaswamy, The Lax-Milgram theorem for Banach spaces. I, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 10, 462–464. MR 605763, DOI 10.3792/pjaa.56.462
- S. Ramaswamy, The Lax-Milgram theorem for Banach spaces. I, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 10, 462–464. MR 605763, DOI 10.3792/pjaa.56.462
- Michael Struwe, Quasilinear elliptic eigenvalue problems, Comment. Math. Helv. 58 (1983), no. 3, 509–527. MR 727715, DOI 10.1007/BF02564649
- M.Struwe. Variational Methods, Springer-Verlag, New York, 1996.
- E.Zeidler. Nonlinear Functional Analysis and its Applications,Vol.3 , Springer-Verlag, New York, 1983.
Additional Information
- Le Hai An
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Email: anle@math.utah.edu
- Pham Xuan Du
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Email: dxpham@indiana.edu
- Duong Minh Duc
- Affiliation: Department of Mathematics, Informatics, National University of Hochiminh City, Vietnam
- Email: dmduc@hcmc.netnam.vn
- Phan Van Tuoc
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 736255
- Email: phan@math.umn.edu
- Received by editor(s): February 20, 2003
- Published electronically: September 20, 2004
- Additional Notes: This work was partially supported by CONACyT (Mexico), grant G36357-E
- Communicated by: Jonathan M. Borwein
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 595-604
- MSC (2000): Primary 58E05, 49J40, 35J25, 35J60
- DOI: https://doi.org/10.1090/S0002-9939-04-07711-1
- MathSciNet review: 2093084