June  2015, 35(6): 2763-2796. doi: 10.3934/dcds.2015.35.2763

Weak structural stability of pseudo-monotone equations

1. 

Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia

Received  January 2014 Revised  June 2014 Published  December 2014

The inclusion $\beta(u)\ni h$ in $V'$ is studied, assuming that $V$ is a reflexive Banach space, and that $\beta: V \to {\cal P}(V')$ is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242]. A notion of strict generalized pseudo-monotonicity is also introduced. The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional $V \!\times V'\to \mathbf{R} \cup \{+\infty\}$.
    A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi's notion of $\Gamma$-convergence. The compactness and the convergence of the family of operators $\beta$ provide the (weak) structural stability of the inclusion $\beta(u)\ni h$ with respect to variations of $\beta$ and $h$, under the only assumptions that the $\beta$s are equi-coercive and the $h$s are equi-bounded.
    These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form $D_t\partial\varphi(u) + \alpha(u) \ni h$, $\partial\varphi$ being the subdifferential of a convex lower semicontinuous mapping $\varphi$, and $\alpha$ a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.
Citation: Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763
References:
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H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.

[2]

A. Ambrosetti and C. Sbordone, $\Gamma$-convergenza e G-convergenza per problemi non lineari di tipo ellittico, Boll. Un. Mat. Ital. (5), 13 (1976), 352-362.

[3]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, Berlin, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[5]

G. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

[6]

L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problèmes quasi-linéaires, Portugal. Math., 41 (1982), 535-562.

[7]

A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[8]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998.

[9]

H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280.

[10]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[12]

F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Amer. Math. Soc., Providence, R. I., (1976), 1-308.

[13]

F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. doi: 10.1016/0022-1236(72)90070-5.

[14]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Vol. 580. Springer-Verlag, Berlin-New York, 1977.

[15]

V. Chiadò Piat, G. Dal Maso and A. Defranceschi, G-convergence of monotone operators, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 7 (1990), 123-160.

[16]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Univ. Press, New York, 1999.

[17]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.

[18]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850.

[19]

N. Dunford and J. Schwartz, Linear Operators, Vol. I. Interscience, New York, 1958.

[20]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974.

[21]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 20 (1988), 59-65.

[22]

G. Francfort, F. Murat and L. Tartar, Homogenization of monotone operators in divergence form with x-dependent multivalued graphs, Ann. Mat. Pura Appl. (4), 188 (2009), 631-652. doi: 10.1007/s10231-009-0094-9.

[23]

N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl., 146 (1987), 1-13. doi: 10.1007/BF01762357.

[24]

N. Fusco and G. Moscariello, Further results on the homogenization of quasilinear operators, Ricerche Mat., 35 (1986), 231-246.

[25]

P. Hartman and G. Stampacchia, On some non linear elliptic differential functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.

[26]

P. Hess, Variational inequalities for strongly nonlinear elliptic operators, J. Math. Pures Appl., 52 (1973), 285-297.

[27]

Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer, Dordrecht, 1979.

[28]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.

[29]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[30]

N. Kenmochi, Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Math. J., 4 (1974), 229-263.

[31]

Le and V. Khoi, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4.

[32]

J. Leray and J. L. Lions, Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.

[33]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[34]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer, Berlin, 1972. (French edition: Dunod, Paris 1968)

[35]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152. doi: 10.1007/BF02417888.

[36]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.

[37]

A. Pankov, G-Convergence and Homogenization of Nonlinear Partial Differential Equations, Kluwer, Dordrecht, 1997.

[38]

U.E. Raĭtum, On G-convergence of quasilinear elliptic operators with unbounded coefficients, (Russian) Dokl. Akad. Nauk SSSR, 261 (1981), 30-34.

[39]

T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., 52 (2014), 1071-1107. doi: 10.1137/130909391.

[40]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[41]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571-597; errata, ibid. (3), 22 (1968), p673.

[42]

L. Tartar, Course Peccot, Collège de France, Paris 1977. [Unpublished, partially written in Topics in the Mathematical Modelling of Composite Materials. (A. Cherkaev, R. Kohn, eds.) Birkhäuser, Boston, (1997), 21-43.]

[43]

L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Springer, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1.

[44]

R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam-New York, 1979.

[45]

A. Visintin, Strong convergence results related to strict convexity, Communications in P.D.E.s, 9 (1984), 439-466. doi: 10.1080/03605308408820337.

[46]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.

[47]

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.

[48]

A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270.

[49]

A. Visintin, An extension of the Fitzpatrick theory, Commun. Pure Appl. Anal., 13 (2014), 2039-2058. doi: 10.3934/cpaa.2014.13.2039.

[50]

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.

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.

[2]

A. Ambrosetti and C. Sbordone, $\Gamma$-convergenza e G-convergenza per problemi non lineari di tipo ellittico, Boll. Un. Mat. Ital. (5), 13 (1976), 352-362.

[3]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, Berlin, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[5]

G. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

[6]

L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problèmes quasi-linéaires, Portugal. Math., 41 (1982), 535-562.

[7]

A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[8]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998.

[9]

H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280.

[10]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[12]

F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Amer. Math. Soc., Providence, R. I., (1976), 1-308.

[13]

F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294. doi: 10.1016/0022-1236(72)90070-5.

[14]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Vol. 580. Springer-Verlag, Berlin-New York, 1977.

[15]

V. Chiadò Piat, G. Dal Maso and A. Defranceschi, G-convergence of monotone operators, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 7 (1990), 123-160.

[16]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Univ. Press, New York, 1999.

[17]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.

[18]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850.

[19]

N. Dunford and J. Schwartz, Linear Operators, Vol. I. Interscience, New York, 1958.

[20]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974.

[21]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 20 (1988), 59-65.

[22]

G. Francfort, F. Murat and L. Tartar, Homogenization of monotone operators in divergence form with x-dependent multivalued graphs, Ann. Mat. Pura Appl. (4), 188 (2009), 631-652. doi: 10.1007/s10231-009-0094-9.

[23]

N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl., 146 (1987), 1-13. doi: 10.1007/BF01762357.

[24]

N. Fusco and G. Moscariello, Further results on the homogenization of quasilinear operators, Ricerche Mat., 35 (1986), 231-246.

[25]

P. Hartman and G. Stampacchia, On some non linear elliptic differential functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.

[26]

P. Hess, Variational inequalities for strongly nonlinear elliptic operators, J. Math. Pures Appl., 52 (1973), 285-297.

[27]

Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer, Dordrecht, 1979.

[28]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.

[29]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[30]

N. Kenmochi, Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Math. J., 4 (1974), 229-263.

[31]

Le and V. Khoi, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658. doi: 10.1090/S0002-9939-2010-10594-4.

[32]

J. Leray and J. L. Lions, Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.

[33]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[34]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer, Berlin, 1972. (French edition: Dunod, Paris 1968)

[35]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152. doi: 10.1007/BF02417888.

[36]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.

[37]

A. Pankov, G-Convergence and Homogenization of Nonlinear Partial Differential Equations, Kluwer, Dordrecht, 1997.

[38]

U.E. Raĭtum, On G-convergence of quasilinear elliptic operators with unbounded coefficients, (Russian) Dokl. Akad. Nauk SSSR, 261 (1981), 30-34.

[39]

T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., 52 (2014), 1071-1107. doi: 10.1137/130909391.

[40]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[41]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571-597; errata, ibid. (3), 22 (1968), p673.

[42]

L. Tartar, Course Peccot, Collège de France, Paris 1977. [Unpublished, partially written in Topics in the Mathematical Modelling of Composite Materials. (A. Cherkaev, R. Kohn, eds.) Birkhäuser, Boston, (1997), 21-43.]

[43]

L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Springer, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1.

[44]

R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam-New York, 1979.

[45]

A. Visintin, Strong convergence results related to strict convexity, Communications in P.D.E.s, 9 (1984), 439-466. doi: 10.1080/03605308408820337.

[46]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.

[47]

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.

[48]

A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270.

[49]

A. Visintin, An extension of the Fitzpatrick theory, Commun. Pure Appl. Anal., 13 (2014), 2039-2058. doi: 10.3934/cpaa.2014.13.2039.

[50]

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.

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