Periodic scalar Dirichlet homogenization on the unit cube

Equip $H_0^1(Q)$ with gradient norm, equivalent to its Sobolev norm by [F4]. The form $b_{ϵ}(u,v)={\int }_{Q}a(x\mathrm{/}ϵ)\mathrm{\nabla }u\cdot \mathrm{\nabla }v$ is bounded by $\mathrm{\Lambda }\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$ and coercive with constant $\lambda$; $F(v)={\int }_{Q}fv$ is bounded by $\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$. Thus [F5] gives a unique $u_{ϵ}$. The same argument using [F3] gives a unique $u_0$. Testing with $u_{ϵ}$ gives $\mathrm{\parallel }\mathrm{\nabla }{u}_{ϵ}{\mathrm{\parallel }}_2\le {\lambda }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_2$ (with the zero-gradient case immediate).

For smooth periodic mean-zero $w$, let $\widehat{w}(k)$ be its Fourier coefficients and $P_{N}w={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}\widehat{w}(k)e^{2\pi ik\cdot y}$. Periodic integration by parts gives $\widehat{{\mathrm{\partial }}_{j}w}(k)=2\pi i{k}_j\widehat{w}(k)$. Parseval [F10], applied to $w$ and all its first derivatives, yields $\mathrm{\parallel }w-P_{N}w{\mathrm{\parallel }}_{H^1(Y)}^2={\sum }_{\mathrm{\mid }k\mathrm{\mid }>N}(1+4{\pi }^2\mathrm{\mid }k{\mathrm{\mid }}^2)\mathrm{\mid }\widehat{w}(k){\mathrm{\mid }}^2⟶0.$ The zero coefficient vanishes, and the symmetric truncation preserves reality. For each finite truncation, approximate its finitely many coefficients by rational real and imaginary parts while imposing $c_{-k}=\overline{c_k}$; this gives a countable family of rational-coefficient, real, mean-zero trigonometric polynomials dense in the smooth periodic mean-zero functions in $H^1(Y)$. If $v\in V_{\mathrm{\#}}$, choose a smooth periodic $w$ arbitrarily close to $v$ in $H^1$ by [F10], then replace $w$ by $w-{\int }_{Y}w$; since ${\int }_{Y}v=0$, this subtraction still approximates $v$. The rational polynomials are therefore dense in all of $V_{\mathrm{\#}}$.

Fix any sequence ${ϵ}_n\to 0$. Apply [F6] to the bounded solutions. Along a subsequence, $u_{{ϵ}_n}\rightharpoonup u$ in $H_0^1$, strongly in $L^2$, and $\mathrm{\nabla }{u}_{{ϵ}_n}$ two-scale converges to $\mathrm{\nabla }u+{\mathrm{\nabla }}_y{u}_1$ for $u_1\in L^2(Q;V_{\mathrm{\#}})$. It remains to identify $u$.

Fix $\varphi ,\eta \in C_c^{\mathrm{\infty }}(Q)$ and a smooth periodic $w(y)$. The oscillating test $v_n(x)=\varphi (x)+{ϵ}_n\eta (x)w(x\mathrm{/}{ϵ}_n)$ lies in $H_0^1(Q)$ and has gradient $\mathrm{\nabla }\varphi +\eta {\mathrm{\nabla }}_{y}w(x\mathrm{/}{ϵ}_n)+{ϵ}_{n}w(x\mathrm{/}{ϵ}_n)\mathrm{\nabla }\eta$. The last term contributes a quantity tending to zero by boundedness of $\mathrm{\nabla }{u}_{{ϵ}_n}$ and the uniform rescaling bound [F7]. On the other terms, two-scale convergence gives ${\int }_{Q\times Y}a(y)(\mathrm{\nabla }u+{\mathrm{\nabla }}_y{u}_1)\cdot (\mathrm{\nabla }\varphi +\eta {\mathrm{\nabla }}_{y}w)={\int }_{Q}f\varphi ,$ where the test may contain the measurable factor $a(y)$: approximate each $a(y)$ times a smooth periodic factor in $L^2(Y)$ by smooth Fourier sums and use the uniform rescaling bound [F7] and boundedness of $\mathrm{\nabla }{u}_{{ϵ}_n}$. The right side converges because ${ϵ}_n\eta w(x\mathrm{/}{ϵ}_n)\to 0$ in $L^2$.

Set $\varphi =0$ in step 3.1 and let $\eta \in C_c^{\mathrm{\infty }}(Q)$ be arbitrary. For each smooth periodic $w$, this gives ${\int }_Q\eta (x)F_w(x)dx=0$, where $F_w(x):={\int }_{Y}a(y)(\mathrm{\nabla }u(x)+{\mathrm{\nabla }}_y{u}_1(x,y))\cdot {\mathrm{\nabla }}_{y}w(y)dy\mathrm{.}$ For almost every $x$, Cauchy--Schwarz in $y$ gives $\mathrm{\mid }{F}_w(x)\mathrm{\mid }\le \mathrm{\Lambda }\mathrm{\parallel }{\mathrm{\nabla }}_{y}w{\mathrm{\parallel }}_{L^2(Y)}(\mathrm{\mid }\mathrm{\nabla }u(x)\mathrm{\mid }+\mathrm{\parallel }{\mathrm{\nabla }}_y{u}_1(x,\cdot ){\mathrm{\parallel }}_{L^2(Y)})\mathrm{.}$ The right side belongs to $L^2(Q)$ because $u\in H_0^1(Q)$ and $u_1\in L^2(Q;V_{\mathrm{\#}})$. The absolute value of the integrand is integrable on $Q\times Y$, so Tonelli applied to its positive and negative parts gives measurability of the section integral [F12]. Thus $F_w\in L^2(Q)$. Since its pairing with every $\eta \in C_c^{\mathrm{\infty }}(Q)$ vanishes, density [F11] implies $F_w=0$ almost everywhere.

Use [A1] to take one common full-measure set for the countable dense family from step 1.2, and intersect it with the full-measure set where $u_1(x,\cdot )\in V_{\mathrm{\#}}$ and $\mathrm{\nabla }u(x)$ is defined. For each such $x$, the cell functional is continuous in the $H^1$ norm, so the equation extends from that dense family to every $w\in V_{\mathrm{\#}}$. Uniqueness in [F2] yields $u_1(x,\cdot )={\chi }_{\mathrm{\nabla }u(x)}$ in $V_{\mathrm{\#}}$.

Now use step 3.1 with $w=0$. Step 5.1 and the flux definition [F2] identify its averaged flux as ${\int }_{Y}a(\mathrm{\nabla }u+{\mathrm{\nabla }}_y{u}_1)=A^{\ast }\mathrm{\nabla }u$. Therefore ${\int }_Q{A}^{\ast }\mathrm{\nabla }u\cdot \mathrm{\nabla }\varphi ={\int }_{Q}f\varphi$ for every $\varphi \in C_c^{\mathrm{\infty }}(Q)$. Density [F8] and boundedness of the form extend this to all test functions in $H_0^1(Q)$, so uniqueness in [F5] gives $u=u_0$.

Every sequence ${ϵ}_n\to 0$ has a subsequence weakly converging to $u_0$ by steps 2.1–6.1. If the family failed to converge weakly to $u_0$, [F9] would give a bounded linear functional and a sequence ${ϵ}_n\to 0$ whose scalar values stay a fixed positive distance from its value at $u_0$; the subsequence just obtained contradicts that. Thus the full weak convergence holds.