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

Fabio Paronetto

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 4, page 669-691
  • ISSN: 1292-8119

Abstract

top
We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations t ( r h u ) - div ( a h · D u ) 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 r 0 ) .

How to cite

top

Paronetto, Fabio. "Asymptotic behaviour of a class of degenerate elliptic-parabolic operators: a unitary approach." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 669-691. <http://eudml.org/doc/250006>.

@article{Paronetto2007,
abstract = { We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations $\partial_t (r_h u) - \{\rm div\}(a_h \cdot Du)$ with $r_h(x,t) \geq0$, $r_h \in L^\{\infty\}(\Omega\times (0,T))$. 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 $r\equiv 0)$. },
author = {Paronetto, Fabio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {G-convergence; PDE of mixed type; linear elliptic and parabolic equations; -convergence; singular perturbations; lack of compactness},
language = {eng},
month = {7},
number = {4},
pages = {669-691},
publisher = {EDP Sciences},
title = {Asymptotic behaviour of a class of degenerate elliptic-parabolic operators: a unitary approach},
url = {http://eudml.org/doc/250006},
volume = {13},
year = {2007},
}

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
DA - 2007/7//
PB - EDP Sciences
VL - 13
IS - 4
SP - 669
EP - 691
AB - We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations $\partial_t (r_h u) - {\rm div}(a_h \cdot Du)$ with $r_h(x,t) \geq0$, $r_h \in L^{\infty}(\Omega\times (0,T))$. 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 $r\equiv 0)$.
LA - eng
KW - G-convergence; PDE of mixed type; linear elliptic and parabolic equations; -convergence; singular perturbations; lack of compactness
UR - http://eudml.org/doc/250006
ER -

References

top
  1. R.W. Carroll and R.E. Showalter, Singular and Degenerate Cauchy Problems. Academic Press, New York (1976).  
  2. V. Chiadò Piat, G. Dal Maso and A. Defranceschi, G-convergence of monotone operators. Ann. Inst. H. Poincaré, Anal. Non Linéaire7 (1990) 123–160.  Zbl0731.35033
  3. F. Colombini and S. Spagnolo, Sur la convergence de solutions d'équations paraboliques. J. Math. Pur. Appl. 56 (1977) 263–306.  Zbl0354.35009
  4. G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993).  
  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.  Zbl0274.35002
  6. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, USA (1992).  Zbl0804.28001
  7. A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators. Kluwer Academic Publishers, Dordrecht (1997).  Zbl0883.35001
  8. F. Paronetto, Existence results for a class of evolution equations of mixed type. J. Funct. Anal. 212 (2004) 324–356.  Zbl1066.35064
  9. F. Paronetto, Homogenization of degenerate elliptic-parabolic equations. Asymptotic Anal. 37 (2004) 21–56.  Zbl1052.35025
  10. R.E. Showalter, Degenerate parabolic initial-boundary value problems. J. Diff. Eq. 31 (1979) 296–312.  Zbl0416.35038
  11. R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society (1997).  Zbl0870.35004
  12. J. Simon, Compact sets in the space L p ( 0 , T ; B ) . Ann. Mat. Pura Appl.146 (1987) 65–96.  Zbl0629.46031
  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.  Zbl0153.42103
  14. S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968) 571–597.  
  15. S. Spagnolo, Convergence of parabolic equations. Boll. Un. Mat. Ital. 14-B (1977) 547–568.  Zbl0356.35042
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.