Abstract
In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$. We derive the sufficient conditions, which guarantee the coincidence of $H$-limit with the weak-* limit of such sequences in $L^{\infty} (D; \mathbb{R}^{n \times n})$.
Highlights
The aim of this paper із to discuss some additional properties of the Hconvergence, which plays the key role in the homogenization theory of boundary value problems
L°(Q,R"), by..,Banach-Alaoglu Theorem it follows that there exists a matrix А" € L?(Q:R”TM") such that 4, -—“~> 4* weakly-* in L7(Q;RTM")
Following [2], [4], we conclude this section by some results concerning the estimates for the solution of Dirichlet problem (5.6)
Summary
The aim of this paper із to discuss some additional properties of the Hconvergence, which plays the key role in the homogenization theory of boundary value problems. Making use of (16), it follows that м is a the solution ofthe limit boundary value а о Чо = fin (4.8). Оу «0, This problem has a unique solution, so the whole sequence {u, }0 converges to u weakly in Wy"2 (Q). By well-known existence results for nonlinear elliptic equations with strictly monotone demi-continucus cocrcive operators (see [5], 1124), one can- see that for every open 5сї © and every / € 7(Q) the nonlinear Dirichlet boundary value problem. Following [2], [4], we conclude this section by some results concerning the estimates for the solution of Dirichlet problem (5.6). We say that a matrix A= la, | is admissible to the nonlinear Dirichlet problem (5.6) if AE Uso. To begin with, we prove the followingresult: PROPOSITION 6.2.
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