A Common Approach to Singular Perturbation and Homogenization II: Semilinear Elliptic Systems

TL;DR

The paper develops an abstract implicit-function-theorem framework for singular perturbation and periodic homogenization of 2D semilinear elliptic systems with $\mathbb{Z}^2$-periodic coefficients, proving existence and local uniqueness of $\varepsilon$-dependent weak solutions $u_\varepsilon$ near a nondegenerate homogenized solution $u_0$, with $\|u_\varepsilon-u_0\|_\infty\to0$ as $\varepsilon\to0$ and $\|u_\varepsilon-u_0\|_\infty=\mathcal{O}(\varepsilon^{1/p})$ under extra regularity. The core technique is a generalized implicit-function theorem for equations $F_\varepsilon(u)=0$ with approximate solutions $\bar{u}_\varepsilon$, where invertibility of the linearized operator is ensured via Gröger's maximal-regularity results, and homogenization cell problems supply the necessary correctors to build $\bar{u}_\varepsilon$. Two constructions of approximate solutions are developed: one using mollified derivatives and oscillatory correctors, another with direct first-order correctors, both yielding small residuals in $W^{-1,p}$ and sup-norm control. The framework is extended to nonlinear mixed Dirichlet-Robin boundary conditions, preserving existence, local uniqueness, and convergence results, thereby offering a unified approach to singular perturbation and homogenization with nonsmooth data and cross-diffusion in two dimensions.

Abstract

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{αβ}(x/\varepsilon)\partial_{x_j}u(x)+b_i^α(x,u(x))\right)=b^α(x,u(x)) \mbox{ for } x \in Ω. $$ For small $\varepsilon>0$ we prove existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of $\|u_\varepsilon-u_0\|_\infty$ for $\varepsilon \to 0$. Our assumptions are, roughly speaking, as follows: The functions $a_{ij}^{αβ}$ are bounded, measurable and $\mathbb{Z}^2$-periodic, the functions $b_i^α(\cdot,u)$ and $b^α(\cdot,u)$ are bounded and measurable, the functions $b_i^α(x,\cdot)$ and $b^α(x,\cdot)$ are $C^1$-smooth, and $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^2$. Neither global solution uniqueness is supposed nor growth restrictions of $b_i^α(x,\cdot)$ or $b^α(x,\cdot)$ nor higher regularity of $u_0$, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and and for homogenization problems.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• Remark 2.2
• Corollary 2.3