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

ESAIM: COCV, Vol. 13, N°4, pp. 669-691
DOI: 10.1051/cocv:2007029

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

Fabio Paronetto

Dipartimento di Matematica "Ennio De Giorgi", Università del Salento, via per Arnesano, 73100 Lecce, Italy; fabio.paronetto@unile.it

(Received May 26, 2005. Revised January 5, 2006 and February 17, 2006. Published online July 20, 2007.)

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 $r\equiv 0)$ .

Mathematics Subject Classification. 35J15, 35K10, 35M10, 45J45

Key words: G-convergence, PDE of mixed type, linear elliptic and parabolic equations

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