Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach

Paronetto, Fabio

doi:10.1051/cocv:2007029

Paronetto, Fabio

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 669-691

Résumé

We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations $\partial_{t} (r_{h} u) - div (a_{h} \cdot D u)$ with $r_{h} (x, t) \geq 0$ , $r_{h} \in L^{\infty} (Ω \times (0, T))$ . The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain $G$ -convergence for elliptic operators $(r_{h} \equiv 0)$ , $G$ -convergence for parabolic operators $(r_{h} \equiv 1)$ , singular perturbations of an elliptic operator $(a_{h} \equiv a$ and $r_{h} \to r$ , possibly $r \equiv 0)$ .

MR

DOI : 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},
     pages = {669--691},
     year = {2007},
     publisher = {EDP Sciences},
     volume = {13},
     number = {4},
     doi = {10.1051/cocv:2007029},
     mrnumber = {2351397},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2007029/}
}

TY  - JOUR
AU  - Paronetto, Fabio
TI  - Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2007
SP  - 669
EP  - 691
VL  - 13
IS  - 4
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2007029/
DO  - 10.1051/cocv:2007029
LA  - en
ID  - COCV_2007__13_4_669_0
ER  -

%0 Journal Article
%A Paronetto, Fabio
%T Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 669-691
%V 13
%N 4
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2007029/
%R 10.1051/cocv:2007029
%G en
%F 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, Tome 13 (2007) no. 4, pp. 669-691. doi: 10.1051/cocv:2007029

Bibliographie
Cité par

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

[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. | Zbl | Numdam

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

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

[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

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

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

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

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

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

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

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

[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. | Zbl | Numdam

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

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

[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

Cité par Sources :