\(G\)-convergence of parabolic operators. (English) Zbl 0933.35020

Svanstedt, Nils

Nonlinear Anal., Theory Methods Appl. 36, No. 7, A, 807-842 (1999).

The paper deals with initial boundary value problems of the form \[ u'-\text{div}\bigl(a(x,t,Du_h)\bigr)=f\qquad\text{ in }\Omega\times (0,T), \qquad\qquad u\in L^p\bigl(0,T;W^{1,p}_0(\Omega)\bigr), \] where \(p\geq 2\), and the coefficient \(a\) is assumed to be monotone and to satisfy some boundedness and coercivity assumptions. The dependence of \(u\) on \(a\) is investigated by extending to this general framework the Spagnolo \(G\)-convergence. In particular, \(G\)-convergence of the coefficient \(a\) implies weak convergence of the function \(u\) and of the moment \(a(x,t,Du)\). Also in this general setting the \(G\)-convergence has a local character and is sequentially compact, in any class of coefficients with uniform bounds. Moreover, if the coefficients are equi-continuous with respect to \(t\), the \(G\)-convergence is equivalent to the elliptic \(G\)-convergence with \(t\in [0,T]\) fixed. The paper contains also an application to a parabolic homogenization theorem.

Reviewer: L.Ambrosio (Pisa)

MSC:

35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

Keywords:

\(G\)-compactness; parabolic homogenization theorem; monotone operators