Existence and uniqueness of periodic cell correctors

On $V_{\mathrm{\#}}$ with gradient inner product, define $b(u,v)={\int }_{Y}a\mathrm{\nabla }u\cdot \mathrm{\nabla }v$ and $F_p(v)=-{\int }_{Y}ap\cdot \mathrm{\nabla }v$. By [F5] and Cauchy--Schwarz, $\mathrm{\mid }b(u,v)\mathrm{\mid }\le \mathrm{\Lambda }\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$, $b(v,v)\ge \lambda \mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2$, and $\mathrm{\mid }{F}_p(v)\mathrm{\mid }\le \mathrm{\Lambda }\mathrm{\mid }p\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$. Thus [F2]–[F4] apply and give a unique ${\chi }_p$ with $b({\chi }_p,v)=F_p(v)$ for every $v$. Rearranging is exactly [F1].

Test the equation with $v={\chi }_p$. Then $\lambda \mathrm{\parallel }\mathrm{\nabla }{\chi }_p{\mathrm{\parallel }}_2^2\le {\int }_{Y}a\mathrm{\mid }\mathrm{\nabla }{\chi }_p{\mathrm{\mid }}^2=-{\int }_{Y}ap\cdot \mathrm{\nabla }{\chi }_p\le \mathrm{\Lambda }\mathrm{\mid }p\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }{\chi }_p{\mathrm{\parallel }}_2$. If the gradient norm is zero the estimate is immediate; otherwise divide by it. The $L^2$ bound follows from [F2].

For $p,q\in {\mathbb{R}}^d$ and scalars $s,t$, the function $s{\chi }_p+t{\chi }_q$ satisfies the cell equation with datum $sp+tq$, by linearity of the integral in both $p$ and the corrector. Uniqueness in step 1.1 therefore gives ${\chi }_{sp+tq}=s{\chi }_p+t{\chi }_q$.