Lemma 16.15.17E43893840E09

Proof. For the quantitative assertion, $C$ denotes constants depending only on the quantities listed there; $O'\Subset O\Subset O_1\Subset U$ denote open sets, and the constants may also depend on them. The qualitative assertion follows from the quantitative one. Indeed, if $V\Subset U'\Subset U$ with $U'$ open, then on $U'$ the matrix $G$ is bounded above and below and uniformly continuous, and $u$ has finite $C^1$ norm and bounded second derivatives. The quantitative assertion, applied on $U'$, therefore gives $u\in W^{\ell ,p}(V)$ for all $\ell $ and $p$, and Morrey's inequality (lemma 2.9.16), applied to the derivatives of $u$, shows that all derivatives of $u$ are continuous on $V$.

We prove the estimate by bootstrapping: an interior $L^ p$ estimate (Step 1), applied to difference quotients, raises the order of differentiability by one at a time (Steps 2 and 3), on smaller domains and with smaller exponents, and Step 4 chains these steps.

Step 1: an interior estimate. Let $(a^{i\bar j})$ be a function on $O$ with $\overline{a^{i\bar j}}=a^{j\bar i}$, $\Lambda ^{-1}|\zeta |^2\leq a^{i\bar j}\zeta _ i\bar\zeta _ j\leq \lambda ^{-1}|\zeta |^2$ and $|a(x)-a(y)|\leq \varpi '(|x-y|)$, where $\varpi '(r)\to 0$ as $r\to 0$. We claim that every real $w\in W^{2,p}_{\mathrm{loc}}(O)$ with $w\in W^{1,p}(O)$ satisfies

with $C$ depending also on $\varpi '$. Indeed, by (16.14.8), $a^{i\bar j}\partial _ i\partial _{\bar j}w=A:D^2w$ with a real symmetric $A\geq (4\Lambda )^{-1}I$, $|A|\leq C\lambda ^{-1}$ and $|A(x)-A(y)|\leq \varpi '(|x-y|)$. Let $\delta _0$ and $C_1$ be the constants of lemma 16.14.7 for $d=2n$, $p$ and $(4\Lambda )^{-1}$, and choose $r>0$ with $\varpi '(\varrho )\leq \delta _0$ for $0<\varrho <r$ and $r<\operatorname {dist}(O',\mathbb R^{2n}\setminus O)$. Cover $\overline{O'}$ by finitely many balls $B(x_\nu ,r/2)$ with $x_\nu \in \overline{O'}$, and let $\chi _\nu =\chi (\cdot -x_\nu )$, where $\chi \in C_ c^\infty (B(0,r))$ equals $1$ on $B(0,r/2)$. By lemma 2.9.14(1) and lemma 16.14.7,

and $A:D^2(\chi _\nu w)=\chi _\nu \, A:D^2w+2\sum _{\mu ,\sigma }A^{\mu \sigma }\partial _\mu \chi _\nu \, \partial _\sigma w+w\, A:D^2\chi _\nu $. Summing over $\nu $ proves the claim.

Step 2: $u\in W^{3,p}_{\mathrm{loc}}$. Let $e$ be a real coordinate unit vector, and let $0<|s|<\operatorname {dist}(O,\mathbb R^{2n}\setminus O_1)$, with the notation $\tau _ s$, $\delta _ s$ of lemma 16.15.16. For $x\in O$ and $0\leq t\leq 1$ the matrix $G_ t(x)=(1-t)G(x)+tG(x+se)$ satisfies $\lambda I\leq G_ t\leq \Lambda I$; let $(g_ t^{i\bar j})$ be its inverse and $a_ s^{i\bar j}=\int _0^1g_ t^{i\bar j}\, \mathrm dt$. By Jacobi's formula, $\frac{\mathrm d}{\mathrm dt}\log \det G_ t=g_ t^{i\bar j}(G_{i\bar j}(x+se)-G_{i\bar j}(x))$, so integration over $0\leq t\leq 1$ gives $\delta _ s\log \det G=a_ s^{i\bar j}\delta _ sG_{i\bar j}$. Since $\delta _ sG_{i\bar j}=\delta _ sk_{i\bar j}+\partial _ i\partial _{\bar j}\delta _ su$, equation (16.15.18) gives

The right side is bounded by $\sup _ U|\nabla \phi _0|+|\beta |\sup _ U|\nabla u|+C\lambda ^{-1}\sup _ U|\nabla k|$. The matrices $a_ s$ satisfy the hypotheses of Step 1 with $\varpi '=2\lambda ^{-2}\varpi $, because they are averages of inverses of matrices between $\lambda I$ and $\Lambda I$, and $|M^{-1}-N^{-1}|=|M^{-1}(N-M)N^{-1}|\leq \lambda ^{-2}|M-N|$ for such matrices. By lemma 16.15.16(1), $\| \delta _ su\| _{W^{1,p}(O)}\leq \| u\| _{W^{2,p}(O_1)}$. Step 1 therefore gives $\| D^2\delta _ su\| _{L^ p(O')}\leq C$, uniformly in $s$. Since $\delta _ s\partial _\mu \partial _\sigma u=\partial _\mu \partial _\sigma \delta _ su$, lemma 16.15.16(2), applied to $Z=\partial _\mu \partial _\sigma u$, shows that $\partial _ e\partial _\mu \partial _\sigma u\in L^ p(O')$, with norm at most $C$. Thus $\| u\| _{W^{3,p}(O')}\leq C$, where $C$ depends on $M$ through $\| u\| _{C^1(U)}$ and $\| D^2u\| _{L^ p(O_1)}$.

Step 3: from $W^{\ell }$ to $W^{\ell +1}$. Let $\ell \geq 3$ and suppose that $u\in W^{\ell ,q'}(O_1)$ for some $q'\geq 2p+2n$. By Morrey's inequality (lemma 2.9.16), applied to the derivatives of $u$ of order at most $\ell -2$, which lie in $W^{2,q'}(O_1)$, the function $u$ is of class $C^{\ell -1}$ on $O_1$, and on a slightly smaller set, which we again call $O_1$, its derivatives of order at most $\ell -1$ are bounded by $C\| u\| _{W^{\ell ,q'}}$. Thus $G\in C^{\ell -3}(O_1)$, its derivatives of order at most $\ell -3$ lie in $W^{1,2p}(O_1)$, and those of order $\ell -2$ lie in $L^{2p}(O_1)$.

We first record a formula for the derivatives of $\log \det G$. For a smooth function $G$ with values in the positive definite Hermitian matrices and a multi-index $\gamma $ with $|\gamma |\geq 1$,

where $P_\gamma (G)$ is a linear combination, with coefficients depending only on $\gamma $, of functions $\operatorname {tr}(G^{-1}\partial ^{\beta _1}G\, G^{-1}\partial ^{\beta _2}G\cdots G^{-1}\partial ^{\beta _ r}G)$ with $r\geq 2$, $|\beta _ i|\geq 1$ and $\beta _1+\cdots +\beta _ r=\gamma $; in particular $|\beta _ i|\leq |\gamma |-1$. For $|\gamma |=1$ this is Jacobi's formula, with $P_\gamma =0$. If it holds for $\gamma $, differentiate once more: by the Leibniz rule and $\partial _\mu (G^{-1})=-G^{-1}(\partial _\mu G)G^{-1}$, the derivative of each term $\operatorname {tr}(G^{-1}\partial ^{\beta _1}G\cdots )$ is a sum of terms of the same form, with $r$ or $r+1$ factors and total multi-index $\gamma +e_\mu $, and the derivative of $\operatorname {tr}(G^{-1}\partial ^\gamma G)$ is $\operatorname {tr}(G^{-1}\partial ^{\gamma +e_\mu }G)-\operatorname {tr}(G^{-1}\partial _\mu G\, G^{-1}\partial ^\gamma G)$. Note that $\operatorname {tr}(G^{-1}\partial ^\gamma G)=g^{i\bar j}\partial ^\gamma G_{i\bar j}$.

Now fix $\gamma $ with $|\gamma |=\ell -2\geq 1$. We claim that, almost everywhere on $O_1$,

where $\partial ^\gamma G$ is formed with weak derivatives. Let $K\subset O_1$ be compact, and let $u_\varepsilon =u*\rho _\varepsilon $ and $G_\varepsilon =k+(\partial _ i\partial _{\bar j}u_\varepsilon )$. By proposition 2.5.3(1) and lemma 2.5.1(1),(2), as $\varepsilon \to 0$ the derivatives of $u_\varepsilon $ of order at most $\ell -1$ converge uniformly on $K$ to those of $u$, and those of order $\ell $ converge in $L^{2p}(K)$. Hence $G_\varepsilon $ is positive definite on $K$ for small $\varepsilon $, its derivatives of order at most $\ell -3$ converge uniformly on $K$ to those of $G$, and $\partial ^\gamma G_\varepsilon \to \partial ^\gamma G$ in $L^{2p}(K)$. The formula above holds for the smooth functions $G_\varepsilon $ on the interior of $K$. Its left side converges to $\partial ^\gamma \log \det G=\partial ^\gamma \phi _0+\beta \partial ^\gamma u$ in the sense of distributions, since $\log \det G_\varepsilon \to \log \det G$ uniformly on $K$ and $\log \det G=\phi _0+\beta u$, and its right side converges in $L^1(K)$, since every factor of $P_\gamma (G_\varepsilon )$ involves derivatives of $G_\varepsilon $ of order at most $\ell -3$. Hence the identity holds in the sense of distributions on the interior of $K$, and therefore almost everywhere (proposition 2.5.3(4)). This proves the claim.

Put $v=\partial ^\gamma u\in W^{2,2p}(O_1)$. Since $\operatorname {tr}(G^{-1}\partial ^\gamma G)=D_ Gv+g^{i\bar j}\partial ^\gamma k_{i\bar j}$, the claim says that

For $x\in O$ and small $s$, as in Step 2, the discrete product rule $\delta _ s(FF')=(\tau _ sF)\delta _ sF'+(\delta _ sF)F'$ gives

We bound the right side in $L^ p(O)$ uniformly in $s$, using lemma 16.15.16(1) on $O\Subset O_1$. First, $\delta _ s(G^{-1})=-(\tau _ sG)^{-1}(\delta _ sG)G^{-1}$, so $\| \delta _ sG^{-1}\| _{L^{2p}(O)}\leq \lambda ^{-2}\| \partial _ eG\| _{L^{2p}(O_1)}\leq C$, as $\partial _ eG$ involves derivatives of $u$ of order three. Hence the last term is bounded by $\| \delta _ sG^{-1}\| _{L^{2p}}\| \partial \bar\partial v\| _{L^{2p}}\leq C$. Next, $\| \delta _ s\partial ^\gamma \phi _0\| _\infty \leq C$ and $\| \delta _ sv\| _{L^ p(O)}\leq \| \partial _ ev\| _{L^ p(O_1)}\leq C$. Finally, $g^{i\bar j}\partial ^\gamma k_{i\bar j}$ and $P_\gamma (G)$ are sums of products of bounded continuous factors, namely entries of $G^{-1}$, derivatives of $k$, and derivatives of $G$ of order at most $\ell -3$, each of which has difference quotients bounded in $L^ p(O)$; by the discrete product rule their difference quotients are bounded in $L^ p(O)$. Thus the right side is bounded in $L^ p(O)$, and $\| \delta _ sv\| _{W^{1,p}(O)}\leq \| \partial _ ev\| _{W^{1,p}(O_1)}\leq C$. The matrices $\tau _ sG^{-1}$ satisfy the hypotheses of Step 1 with $\varpi '=\lambda ^{-2}\varpi $. By Step 1, $\| D^2\delta _ sv\| _{L^ p(O')}\leq C$ uniformly in $s$, and lemma 16.15.16(2), applied to the second derivatives of $v$, shows that the third derivatives of $v$ lie in $L^ p(O')$, with norm at most $C$. Every multi-index of length $\ell +1$ is the sum of a multi-index of length $\ell -2$ and of three coordinate unit vectors, so $\| u\| _{W^{\ell +1,p}(O')}\leq C$, where $C$ depends on $u$ only through $\| u\| _{W^{\ell ,q'}(O_1)}$.

Step 4: conclusion. Choose open sets $V=O_{3\ell }\Subset O_{3\ell -1}\Subset \cdots \Subset O_0=U$, depending only on $V$, $U$ and $\ell $, and put $p_ m=2^{\ell -m}(p+2n)-2n$ for $3\leq m\leq \ell $. Then $p_\ell =p\geq 2$, $p_ m=2p_{m+1}+2n$, and $p_3\leq q$. Step 2, applied with the exponent $p_3$ in place of $p$ and with the sets $O_1$, $O_2$, $O_3$ in the roles of $O_1$, $O$, $O'$, bounds $\| u\| _{W^{3,p_3}(O_3)}$, since $\| D^2u\| _{L^{p_3}(O_1)}\leq |O_1|^{1/p_3-1/q}\| D^2u\| _{L^ q(U)}$. For $m=3,\ldots ,\ell -1$ we then apply Step 3 with $\ell $ replaced by $m$, with the exponent $p_{m+1}$ in place of $p$ and $q'=p_ m$, and with the sets $O_{3m}$ (shrunk to $O_{3m+1}$), $O_{3m+2}$, $O_{3m+3}$ in the roles of $O_1$, $O$, $O'$; this bounds $\| u\| _{W^{m+1,p_{m+1}}(O_{3m+3})}$ in terms of $\| u\| _{W^{m,p_ m}(O_{3m})}$. For $m=\ell -1$ we obtain the bound for $\| u\| _{W^{\ell ,p}(V)}$. $\square$