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