Estimates of homogenization for a parabolic equation with periodic coefficients
The authors consider in the half-space \({\mathbb R}^{n+1}_{+}=\{(x,t): x\in {\mathbb R}^n\), \(t\geqslant 0\}\) the family of the Cauchy problems \[ \partial_tu^\varepsilon - \text{ div}\,a(\varepsilon^{-1}x)\nabla u^\varepsilon=0 \quad \forall t>0, \qquad u^\varepsilon(\cdot,0)=f(x)\in L^2({\mathbb R}^n), \] where \(\varepsilon \in (0,1]\) and \(a(y)\) is a measurable symmetric periodic matrix having a periodicity cell \(Y=[-1/2,1/2)^n\) and satisfying for every \(\xi\in {\mathbb R}^n\) the inequality \(\lambda\xi^2\leqslant a(y)\xi\cdot\xi\leqslant \lambda^{-1}\xi^2\) with a positive constant \(\lambda\). The main results of the article concern estimates of \(L^2({\mathbb R}^n)\)-norms of the differences \(u^\varepsilon(\cdot,t)-u^0(\cdot,t)\) and \(\nabla u^\varepsilon(\cdot,t)-\nabla v^\varepsilon(\cdot,t)\) for every \(t>0\), where \(u^0\) is the solution of the corresponding homogenized problem and \(v^\varepsilon\) is the so-called first approximation.
scientific article; zbMATH DE number 5136624