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Lagrange multipliers for functions derivable along directions in a linear subspace
Author(s):
Le
Hai
An;
Pham
Xuan
Du;
Duong
Minh
Duc;
Phan Van
Tuoc
Journal:
Proc. Amer. Math. Soc.
133
(2005),
595-604.
MSC (2000):
Primary 58E05, 49J40, 35J25, 35J60
Posted:
September 20, 2004
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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:
-
- 1.
- R.A. Adams. Sobolev Spaces Academic Press, New York, 1975. MR 0450957 (56:9247)
- 2.
- D. Arcoya and L. Boccardo. Critical points for multiple integrals of calculus of variations Arch. Rat. Mech. Anal., 134(1996), 249-274. MR 1412429 (97h:49003)
- 3.
- D. Arcoya and L. Boccardo. Some remarks on critical point theory for nondifferentiable functionals Nonlinear Differ. Equ. Appl., 6(1999), 79-100. MR 1674782 (2000c:58018)
- 4.
- H. Brezis. Analyse Fonctionelle (Masson,Paris) 1987.
- 5.
- P.X. Du and D.M.Duc. A non-homogenuous p-Laplace equation in border case. Acta Mathematica Vietnamica, 27(2002) 69-75. MR 1914409 (2003b:35082)
- 6.
- D.M.Duc. Nonlinear Singular Elliptic Equations J.London.Math.Soc. 40 (1989) 420-440. MR 1053612 (91g:35107)
- 7.
- S. Lang. Analysis I. Addison-Wesley, Reading, 1969.
- 8.
- V.K.Le and K.Schmitt. Minimization problems for noncoercive functionals subject to constraints Transactions Amer. Math. Soc., 347(1995), 4485-4513. MR 1316854 (96d:49007)
- 9.
- B. Pellacci. Critical points for some functionals of the calulus of variations. Topol. Methods Nonlinear Anal. 17(2001), 285-305.. MR 1868902 (2002j:49025)
- 10.
- S. Ramaswamy. The Lax-Milgram theorem for Banach spaces. I Proc. Japan Acad. 56 (1980) 462-464. MR 0605763 (82e:47015a)
- 11.
- S. Ramaswamy. The Lax-Milgram theorem for Banach spaces. II Proc. Japan Acad. 57 (1981) 29-33. MR 0608386 (82e:47015b)
- 12.
- M.Struwe. Quasilinear elliptic eigenvalue problems Comment. Math. Helvetici. 58 (1983) 509-527. MR 0727715 (85h:58036)
- 13.
- M.Struwe. Variational Methods, Springer-Verlag, New York, 1996.
- 14.
- E.Zeidler. Nonlinear Functional Analysis and its Applications,Vol.3 , Springer-Verlag, New York, 1983.
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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
Email:
phan@math.umn.edu
DOI:
10.1090/S0002-9939-04-07711-1
PII:
S 0002-9939(04)07711-1
Keywords:
Lagrange multipliers theorem,
Lax-Milgram theorem,
variational inequalities,
quasilinear elliptic eigenvalue problems
Received by editor(s):
February 20, 2003
Posted:
September 20, 2004
Additional Notes:
This work was partially supported by CONACyT (Mexico), grant G36357-E
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2004,
American Mathematical Society
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