Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 4, p. 669-691

We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations t (r h u)- div (a h ·Du) with r h (x,t)0, r h L (Ω×(0,T)). The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators (r h 0), G-convergence for parabolic operators (r h 1), singular perturbations of an elliptic operator (a h a and r h r, possibly r0).

DOI : https://doi.org/10.1051/cocv:2007029
Classification:  35J15,  35K10,  35M10,  45J45
Keywords: G-convergence, PDE of mixed type, linear elliptic and parabolic equations
@article{COCV_2007__13_4_669_0,
     author = {Paronetto, Fabio},
     title = {Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {4},
     year = {2007},
     pages = {669-691},
     doi = {10.1051/cocv:2007029},
     zbl = {pre05212108},
     mrnumber = {2351397},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2007__13_4_669_0}
}
Paronetto, Fabio. Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 4, pp. 669-691. doi : 10.1051/cocv:2007029. http://www.numdam.org/item/COCV_2007__13_4_669_0/

[1] R.W. Carroll and R.E. Showalter, Singular and Degenerate Cauchy Problems. Academic Press, New York (1976). | MR 460842

[2] 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. | Numdam | Zbl 0731.35033

[3] F. Colombini and S. Spagnolo, Sur la convergence de solutions d'équations paraboliques. J. Math. Pur. Appl. 56 (1977) 263-306. | Zbl 0354.35009

[4] G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993). | MR 1201152 | Zbl 0816.49001

[5] E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 8 (1973) 391-411. | Zbl 0274.35002

[6] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, USA (1992). | MR 1158660 | Zbl 0804.28001

[7] A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators. Kluwer Academic Publishers, Dordrecht (1997). | MR 1482803 | Zbl 0883.35001

[8] F. Paronetto, Existence results for a class of evolution equations of mixed type. J. Funct. Anal. 212 (2004) 324-356. | Zbl 1066.35064

[9] F. Paronetto, Homogenization of degenerate elliptic-parabolic equations. Asymptotic Anal. 37 (2004) 21-56. | Zbl 1052.35025

[10] R.E. Showalter, Degenerate parabolic initial-boundary value problems. J. Diff. Eq. 31 (1979) 296-312. | Zbl 0416.35038

[11] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society (1997). | MR 1422252 | Zbl 0870.35004

[12] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl 0629.46031

[13] S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21 (1967) 657-699. | Numdam | Zbl 0153.42103

[14] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968) 571-597. | Numdam | Zbl 0174.42101

[15] S. Spagnolo, Convergence of parabolic equations. Boll. Un. Mat. Ital. 14-B (1977) 547-568. | Zbl 0356.35042

[16] L. Tartar, Convergence d'operateurs defferentiels, Proceedings of the Meeting “Analisi convessa e Applicazioni”. Roma (1974).

[17] L. Tartar, Cours Peccot, Collège de France, 1977. Partially written in: F. Murat, H-convergence - Séminaire d'Analyse Fonctionnelle et Numérique, Université d'Alger, 1977-78. English translation: F. Murat and L. Tartar: H-Convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev, R. Kohn, Editors, Birkhäuser, Boston (1997) 21-43. | Zbl 0920.35019