Homogenization of higher-order parabolic systems in a bounded domain

ABSTRACT

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{2p}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we consider matrix elliptic differential operators $A_{D,\mathit{\epsilon }}$ and $A_{N,\mathit{\epsilon }}$ of order 2p ($p⩾2$) with the Dirichlet or Neumann boundary conditions, respectively. The coefficients of $A_{D,\mathit{\epsilon }}$ and $A_{N,\mathit{\epsilon }}$ are periodic and depend on $\mathbf{x}/\mathit{\epsilon }$, $\mathit{\epsilon }>0$. The behavior of the operator $e^{-A_{†,\mathit{\epsilon }}t}$, $†=D,N$, for small $\mathit{\epsilon }$ is studied. It is shown that, for fixed $t>0$, the operator $e^{-A_{†,\mathit{\epsilon }}t}$ converges in the $L_2$-operator norm to $e^{-A_{†}^{0}t}$, as $\mathit{\epsilon }\to 0$. Here $A_{†}^0$ is the effective operator with constant coefficients. We obtain a sharp order estimate $\Vert e^{-A_{†,\mathit{\epsilon }}t}-e^{-A_{†}^{0}t}{\Vert }_{L_2\to L_2}⩽C\mathit{\epsilon }$. Also, we find approximation for $e^{-A_{†,\mathit{\epsilon }}t}$ in the $(L_2\to H^p)$-norm with error estimate of order $O({\mathit{\epsilon }}^{1/2})$. The results are applied to homogenization of the solutions of initial boundary value problems for parabolic systems.

KEYWORDS:

• Periodic differential operators
• parabolic systems of higher order
• homogenization
• operator error estimates

Notes

No potential conflict of interest was reported by the author.

Dedicated to the memory of V. V. Zhikov.

Additional information

Funding

This work was supported by Russian Science Foundation [project number 17-11-01069].