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The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction
An extension of the Fitzpatrick theory
1. | Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia |
Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.
References:
[1] |
G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58 (1977), 110. |
[2] |
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley and Sons, New York, 1984. |
[3] |
G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117. |
[4] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, Wiley and Sons, Chichester, 1984. |
[5] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. |
[6] |
H. H. Bauschke and X. Wang, The kernel average for two convex functions and its applications to the extension and representation of monotone operators, Trans. Amer. Math. Soc., 361 (2009), 5947-5965.
doi: 10.1090/S0002-9947-09-04698-4. |
[7] |
H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. |
[9] |
H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps et II. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and ibid. 1197-1198. |
[10] |
F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math., XVIII Part II. A.M.S., Providence, 1976. |
[11] |
F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. |
[12] |
M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104. |
[13] |
R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316.
doi: 10.1023/A:1020639314056. |
[14] |
R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383.
doi: 10.1090/S0002-9939-03-07053-9. |
[15] |
I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974. |
[16] |
K. Fan, A generalization of Tychonoff's theorem, Math. Ann., 142 (1961), 305-310. |
[17] |
K. Fan, A minimax inequality and applications. Inequalities, III, In: Proc. Third Sympos., Univ. California, Los Angeles 1969, pp. 103-113. Academic Press, New York, 1972. |
[18] |
W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ., 1953. |
[19] |
S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. |
[20] |
N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008. |
[21] |
N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135.
doi: 10.1007/s11784-008-0083-4. |
[22] |
N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330, (2004) 519-549.
doi: 10.1007/s00208-004-0558-6. |
[23] |
J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[24] |
Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer, Dordrecht, 1997.
doi: 10.1016/0362-546X(85)90097-5. |
[25] |
B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-138. |
[26] |
E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399. |
[27] |
J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. |
[28] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. |
[29] |
J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247. |
[30] |
J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46.
doi: 10.1007/s11228-004-4170-4. |
[31] |
J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product, Proc. Amer. Math. Soc., 136 (2008), 873-878.
doi: 10.1090/S0002-9939-07-09176-9. |
[32] |
G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. |
[33] |
B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038. |
[34] |
J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. R. Math. Acad. Sci. Paris, Ser. I, 338 (2004), 853-858.
doi: 10.1016/j.crma.2004.03.017. |
[35] |
J.-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871.
doi: 10.1016/j.na.2004.05.018. |
[36] |
T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., in press, 2014.
doi: 10.1137/130909391. |
[37] |
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969. |
[38] |
T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65. |
[39] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Birkhäuser, Basel, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[40] |
B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859.
doi: 10.1090/S0002-9939-03-07083-7. |
[41] |
A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650. |
[42] |
A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.
doi: 10.1007/s00526-012-0519-y. |
[43] |
A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270. |
[44] |
A. Visintin, Weak structural stability of pseudo-monotone equations, in press. |
[45] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators, Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[46] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4612-4566-7. |
show all references
References:
[1] |
G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58 (1977), 110. |
[2] |
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley and Sons, New York, 1984. |
[3] |
G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117. |
[4] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, Wiley and Sons, Chichester, 1984. |
[5] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. |
[6] |
H. H. Bauschke and X. Wang, The kernel average for two convex functions and its applications to the extension and representation of monotone operators, Trans. Amer. Math. Soc., 361 (2009), 5947-5965.
doi: 10.1090/S0002-9947-09-04698-4. |
[7] |
H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. |
[9] |
H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps et II. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and ibid. 1197-1198. |
[10] |
F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math., XVIII Part II. A.M.S., Providence, 1976. |
[11] |
F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. |
[12] |
M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104. |
[13] |
R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316.
doi: 10.1023/A:1020639314056. |
[14] |
R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383.
doi: 10.1090/S0002-9939-03-07053-9. |
[15] |
I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974. |
[16] |
K. Fan, A generalization of Tychonoff's theorem, Math. Ann., 142 (1961), 305-310. |
[17] |
K. Fan, A minimax inequality and applications. Inequalities, III, In: Proc. Third Sympos., Univ. California, Los Angeles 1969, pp. 103-113. Academic Press, New York, 1972. |
[18] |
W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ., 1953. |
[19] |
S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. |
[20] |
N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008. |
[21] |
N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135.
doi: 10.1007/s11784-008-0083-4. |
[22] |
N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330, (2004) 519-549.
doi: 10.1007/s00208-004-0558-6. |
[23] |
J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[24] |
Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer, Dordrecht, 1997.
doi: 10.1016/0362-546X(85)90097-5. |
[25] |
B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-138. |
[26] |
E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399. |
[27] |
J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. |
[28] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. |
[29] |
J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247. |
[30] |
J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46.
doi: 10.1007/s11228-004-4170-4. |
[31] |
J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product, Proc. Amer. Math. Soc., 136 (2008), 873-878.
doi: 10.1090/S0002-9939-07-09176-9. |
[32] |
G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. |
[33] |
B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038. |
[34] |
J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. R. Math. Acad. Sci. Paris, Ser. I, 338 (2004), 853-858.
doi: 10.1016/j.crma.2004.03.017. |
[35] |
J.-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871.
doi: 10.1016/j.na.2004.05.018. |
[36] |
T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., in press, 2014.
doi: 10.1137/130909391. |
[37] |
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969. |
[38] |
T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65. |
[39] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Birkhäuser, Basel, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[40] |
B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859.
doi: 10.1090/S0002-9939-03-07083-7. |
[41] |
A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650. |
[42] |
A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.
doi: 10.1007/s00526-012-0519-y. |
[43] |
A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270. |
[44] |
A. Visintin, Weak structural stability of pseudo-monotone equations, in press. |
[45] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators, Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[46] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4612-4566-7. |
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