Week 10: Higher interior regularity (continued), boundary regularity

Higher interior regularity

We now prove a higher regularity theorem which states that if the coefficients of $L$ are sufficiently smooth and $f$ is $k$-times weakly differentiable, then a solution $u\in W^{1,2}(\Omega)$ to $Lu=f$ on $\Omega$ is $(k+2)$-times weakly differentiable.

Theorem 10.1 (higher regularity). Suppose $u\in W^{1,2}(\Omega)$ solves

with $a_{ij},b^{i},c\in C^{k+1}(\Omega)$ and $f\in H^{k,2}(\Omega)$ for some $k\in\mathbb{N}\cup\{0\}$. Then $u\in \wloc^{k+2,2}(\Omega)$ and for each $U\Subset\Omega$,

where $C_{k}>0$ depends only on $k$, $U$, $\Omega$ and $L$.

Proof. We have already proved the case $k=0$. Suppose the claim is true for $k\in\mathbb{N}$ fixed. If $u\in W^{1,2}(\Omega)$ solves $Lu=f$ with $a_{ij},b^{i},c\in C^{k+2}(\Omega)$ and $f\in W^{k+1,2}(\Omega)$, then the inductive hypothesis implies that $u\in \wloc^{k+2,2}(\Omega)$. Fix $r\in\{1,\dots,n\}$, $V\Subset\Omega$ and $\varphi\in C_{0}^{\infty}(V)$. Plugging $v=-\partial_{r}\varphi$ into (*) and using the fact that $u\in W^{2,2}(V)$, we see that the left-hand side may be written as

$\int a_{ij}\partial_{i}u\cdot\partial_{j}(-\partial_{r}\varphi)= \int a_{ij}\partial_{i}(\partial_{r}u)\cdot \partial_{j}\varphi - \int\left(\partial_{j}\partial_{r}a_{ij}\cdot \partial_{i}u+\partial_{r}a_{ij}\cdot \partial_{i}\partial_{j}u \right)\cdot \varphi.$

On the other hand, the right-hand side may be written as

$\int(f-b^{i}\partial_{i}u-cu)\cdot (-\partial_{r}\varphi) =\int(\partial_{r}f - \partial_{r}b^{i}\cdot \partial_{i}u - b^{i}\cdot \partial_{i}\partial_{r}u - \partial_{r}c \cdot u - c\partial_{r}u)\varphi.$

Rearranging, we deduce that $L\partial_{r}u=f_{1}$ weakly, where

$f_{1}=\partial_{r}f -\partial_{r}b^{i}\partial_{i}u-\partial_{r}c\cdot u+\partial_{j}\partial_{r}a_{ij}\cdot \partial_{i}u + \partial_{r}a_{ij}\cdot \partial_{i}\partial_{j}u\in H^{k}(V).$

Therefore, by the inductive hypothesis, $\partial_{r}u\in \wloc^{k+2,2}(V)$ so that $\partial_{r}u\in\wloc^{k+2,2}(\Omega)$ and ultimately $u\in \wloc^{k+3,2}(\Omega)$. The inductive hypothesis also yields the estimate for $U\Subset V\Subset\Omega$

$||\partial_{k}u||_{k+2,2,U}\leq C_{k}'\left(||f_{1}||_{k,2,V} + ||\partial_{k}u||_{2,V}\right).$

However, it follows from Hölder's inequality and the Leibniz rule, as well as the fact that the coefficients of $L$ and their derivatives up to order $k+2$ are bounded on $V$ that

$||f_{1}||_{k,2,V}\leq C_{k}''\left(||f||_{k+1,2} + ||u||_{k+2,2,V}\right),$

where $C_{k}''$ depends only on $k$, $V$ and the coefficients of $L$. Finally, combining these two estimates and the inductive step's estimate yields the desired estimate.

Corollary 10.2 (smoothness). Suppose $u\in W^{1,2}(\Omega)$ solves $Lu=f$ on $\Omega$ with $a_{ij},b^{i},c,f\in C^{\infty}(\Omega)$. Then $u\in C^{\infty}(\Omega)$.

Proof. By the preceding theorem, for each $k\in \mathbb{N}$, we see that $u\in \wloc^{k,2}(\Omega)$. By the Sobolev embedding theorem, for each $m\in\mathbb{N}$, we may choose $k$ sufficiently large so as to deduce that $u$ is equal to a $C^{m}(\Omega)$ function almost everywhere. Since $m$ is arbitrary, we have established the claim.

Boundary regularity

We now turn to the question of whether weak solutions to $Lu=f$ lie in $W^{k,2}(\Omega)$ for $k>1$ and under appropriate additional assumptions on $L$ and $\Omega$, a phenomenon we refer to as boundary regularity. Our approach will be similar to the case of interior regularity. We first note the following lemma, which is an analogue of Lemma 6.3 and Corollary 6.6.

Lemma 10.3. Let $0<r_{1}<r_{2}<1$ and $p\in\left]1,\infty\right[$.

• If $u\in H^{k,p}(B^{+}(0,r_{2}))$, then whenever $0<|h|< r_{2}-r_{1}$ and $i\in\{1,\dots,n-1\}$, the inequality

holds.

• If $u\in H^{k-1,p}(B^{+}(0,r_{2}))$ and there exists a constant $C$ such that for all $0<r_{1}<r_{2}$, $0<|h|<r_{2}-r_{1}$ and $i\in\{1,\dots,n-1\}$

then $\partial_{1}u,\dots,\partial_{n-1}u\in H^{k-1,p}(B^{+}(0,r_{2}))$ and $||\partial_{i}u||_{k-1,p,B^{+}(0,r_{2})}\leq C$.

We first consider the case $\Omega=B^{+}(0,r)$ for appropriate $r>0$. The following lemma gets us regularity up to (part of) the boundary lying in the hyperplane $\{x^{n}=0\}$.

Lemma 10.4. Let $r\in\left]\frac{1}{2},1\right[$ and suppose $L$ is an elliptic operator over $B^{+}(0,r)$ satisfying the additional condition $a_{ij}\in C^{1}(\overline{B^{+}(0,r)})$. If $u\in H_{0}^{1,2}(B^{+}(0,r))$ is a weak solution to the equation

on $B^{+}(0,r)$, then $u\in W^{2,2}(B^{+}(0,\frac{1}{2}))$ and

Proof. Let $\eta\in \cs(\mathbb{R}^{n})$ be such that $\left.\eta\right|_{B(0,\frac{1}{2})}\equiv 1$, $0\leq \eta\leq 1$ and $\supp\eta\subset B(0,r)$. As in the proof of Theorem 9.7, we may write the condition '$Lu=f$ weakly' as the equation

$\int a_{ij}\cdot\partial_{i}u\cdot\partial_{j}v=\int\widetilde{f}\cdot v,$

which should hold for all $v\in H_{0}^{1,2}(B^{+}(0,r))$, where $\widetilde{f}=f-b^{i}\partial_{i}u-cu$. We now let $v=-\delta_{k}^{-h}(\eta^{2}\delta_{k}^{h}u)$ for small $h\in\mathbb{R}$ and $k\in\{1,\dots,n-1\}$. Note that $v$ is clearly an element of $W^{1,2}(B^{+}(0,r))$. To see that it lies in $H_{0}^{1,2}(B^{+}(0,r))$, we write it out more explicitly:

$v(x)=\frac{1}{h^{2}}\left(\eta^{2}(x-he_{k})\cdot\left(u(x) - u(x-he_{k}) + \eta^{2}(x)\cdot\left(u(x) - u(x+he_{k}) \right) \right) \right)$

Taking traces of both sides and using the properties of trace operator (and the fact that $h$ is small), we see that $v$ has trace zero so that it lies in $H_{0}^{1,2}(B^{+}(0,r))$. Now, proceeding analogously to Theorem 9.7, we deduce that

$\int a_{ij}\partial_{i}u\cdot\partial_{j}v\geq \frac{\lambda_{0}}{2}\int\eta^{2}|\D \delta_{k}^{h}u|^{2} - c_{1}\int_{B^{+}(0,r)}|\D u|^{2}$

and

$\int\widetilde{f}v\leq \frac{\lambda_{0}}{4}\int\eta^{2}|\D\delta_{k}^{h}u|^{2} + c_{2}\int_{B^{+}(0,r)}|f|^{2} + |u|^{2} + |\D u|^{2}$

for $c_{1},c_{2}>0$ depending only on $L$ and $r$ so that

$\int_{B^{+}(0,\frac{1}{2})}|\D\delta_{k}^{h}u|^{2}\leq \int \eta^{2}|\D\delta_{k}^{h}u|^{2}\leq c_{3}\int_{B^{+}(0,r)}|f|^{2} + |u|^{2} + |\D u|^{2}.$

Using the preceding lemma and its proof, we deduce that for all $k\in \{1,\dots,n-1\}$, $\partial_{k}u\in W^{1,2}(B^{+}(0,\frac{1}{2}))$ and

$\int_{B^{+}(0,\frac{1}{2})}|\D \partial_{k}u|^{2}\leq c_{3}\int_{B^{+}(0,r)}|f|^{2} + |u|^{2} + |\D u|^{2}.$

As for $\partial_{n}u$, we first note by interior regularity (Theorem 9.7) that $u\in \wloc^{2,2}(B^{+}(0,r))$ so that the equation $Lu=f$ actually holds almost everywhere. Therefore,

$-\partial_{i}(a_{ij}\partial_{j}u) = \widetilde{f}$

almost everywhere. In particular, the left-hand side is equal to

$-a_{nn}\partial_{n}^{2}u - \sum_{i+j<2n}a_{ij}\partial_{i}\partial_{j}u - \sum_{i,j=1}^{n}\partial_{i}a_{ij}\cdot \partial_{j}u.$

Since for all $v\in\mathbb{R}^{n}$ we have the condition $a_{ij}v^{i}v^{j}\geq \lambda_{0}|v|^{2}$, taking $v^{i}=\delta^{i}_{n}$ yields the inequality $a_{nn}\geq \lambda_{0}$. In particular, we deduce that

$|\partial_{n}^{2}u|^{2}\leq C(L)\left( \sum_{i+j<2n}|\partial_{i}\partial_{j}u|^{2} + |\D u|^{2} + |\widetilde{f}|^{2} \right).$

Since each term on the right-hand side is summable by the preceding computation, we deduce that $\partial_{n}^{2}u\in L^{2}(B^{+}(0,\frac{1}{2})$, and estimating $|\widetilde{f}|^{2}$ from above as usual, we deduce that

$||u||_{2,2,B^{+}(0,\frac{1}{2})}\leq c_{4}\left(||f||_{2} + ||u||_{1,2} \right).$

Finally, taking $v=u$ in the equation `$Lu=f$ weakly,' we obtain after carrying out a similar computation to the above that $||u||_{1,2,B^{+}(0,r)}\leq c_{5}\left(||f||_{2} + ||u||_{2} \right)$. Combining the above two inequalities establishes the claim.

Theorem 10.5 (boundary regularity). Suppose $\Omega$ is a bounded domain of class $C^{2}$ and $u\in H_{0}^{1,2}(\Omega)$ is a weak solution to the equation

where $L$ is the usual (divergence-form) elliptic operator satisfying the additional condition $a_{ij}\in C^{1}(\overline{\Omega})$. Then $u\in W^{2,2}(\Omega)$ and the estimate

where $C$ depends only on $\Omega$ and $L$.

Proof. Recall that by definition, for each $x\in \partial\Omega$, we may find a neighbourhood $N_{x}\ni x$ and a $C^{2}$ diffeomorphism $\Phi_{x}:N_{x}\rightarrow B(0,1)$ such that $\Phi_{x}(x)=0$, $\Phi_{x}(N_{x}\cap \Omega)=B^{+}(0,1)$ and $\Phi_{x}(N_{x}\cap \partial\Omega)=B(0,1)\cap \{x^{n}=0\}$. It suffices to show that for each $x\in\partial\Omega$,

$||u||_{2,2,\Phi_{x}^{-1}(B^{+}(0,\frac{1}{2}))}\leq C\left(||f||_{2} + ||u||_{1,2} \right),$

since we may pick a partition of unity subordinate to a (finite) covering $\{\Phi_{x_{i}}^{-1}(B(0,\frac{1}{2}))\}_{i=1}^{N}\cup \{\Omega_{\delta}\}$ of $\overline{\Omega}$ to deduce an estimate of the form

$||u||_{2,2}\leq C\left(||f||_{2} + ||u||_{1,2} \right);\tag{\#}$

Choosing $v=u$ in the defining equation of a weak solution to $Lu=f$ and applying the usual tricks then yields a bound of the form $||u||_{1,2}\leq C'\cdot (||u||_{2} + ||f||_{2})$.

To streamline the proof, we define the following functions on $B^{+}(0,3/4)$:

$\begin{aligned} \underline{u}&:=u\circ \Phi_{x}^{-1}\\ \underline{c}&:=c\circ \Phi_{x}^{-1}\\ \underline{f}&:=f\circ \Phi_{x}^{-1}\\ \underline{b}^{i}&:=(b^{k}\cdot \partial_{k}\Phi_{x}^{i})\circ \Phi_{x}^{-1}\\ \underline{a}_{ij}&:=(a_{kl}\cdot \partial_{k}\Phi_{x}^{i}\cdot\partial_{l}\Phi_{x}^{j})\circ \Phi_{x}^{-1} \end{aligned}$

Note that $\underline{u}\in W^{1,2}(B^{+}(0,3/4))$, $\underline{c},\underline{b}^{i}\in L^{\infty}(B^{+}(0,3/4))$, $\underline{a}_{ij}=\underline{a}_{ji}\in C^{1}(\overline{B^{+}(0,3/4)})$ and $\underline{f}\in L^{2}(B^{+}(0,3/4))$. Moreover, $\underline{a}_{ij}$ is positive definite, and for any $v\in \mathbb{R}^{n}$,

$(\underline{a}_{ij}\circ \Phi_{x})v^{i}v^{j}=a_{kl}\left(\partial_{k}\Phi_{x}^{i} v^{i} \right) \left(\partial_{l}\Phi_{x}^{j}v^{j} \right)\geq \lambda_{0}|\D\Phi_{x}^{T}\cdot v |^{2}\geq\lambda_{0}\lambda_{1}|v|^{2},$

where $\lambda_{1}:=\inf_{|v|=1,y\in \Phi_{x}^{-1}(B(0,\frac{3}{4}) )}|\D\Phi_{x}^{T}(y)\cdot v|^{2}>0$ since $\Phi_{x}^{-1}(B(0,\frac{3}{4}))\Subset N_{x}$.

Now, fixing $\varphi\in C_{0}^{1}(\Phi_{x}^{-1}(B^{+}(0,3/4)))$, the equation $B_{L}(u,\varphi)=\int f\varphi$ implies after a brief computation that

$\int\left( \underline{a}_{kl}\partial_{k}\underline{u}\partial_{l}(\varphi\circ \Phi_{x}^{-1}) + \underline{b}^{k}\partial_{k}\underline{u}\cdot (\varphi\circ \Phi_{x}^{-1}) + \underline{c}\underline{u}\cdot (\varphi\circ \Phi_{x}^{-1})\right)\circ \Phi_{x}(t)dt= \int (\underline{f}\cdot(\varphi\circ \Phi_{x}^{-1}))\circ \Phi_{x}(t) dt.$

Making the change of variables $t=\Phi_{x}^{-1}(s)$ and letting $\varphi:=(\psi/|\det \D\Phi_{x}^{-1}|)\circ \Phi_{x}$ for $\psi\in C_{0}^{\infty}(B^{+}(0,3/4))$, we deduce that

$\partial_{i}(\underline{a}_{ij}\partial_{j}\underline{u})+\underline{b}^{i}\partial_{i}\underline{u} + \underline{c}\cdot \underline{u}=:\underline{L}\underline{u}=\underline{f}$

in the weak sense so that by Lemma 10.4, $\underline{u}\in W^{2,2}(B^{+}(0,1/2))$ and we have an estimate of the form

$||\underline{u}||_{2,2,B^{+}(0,1/2)}\leq C\left(||\underline{f}||_{2,B^{+}(0,3/4)} + ||\underline{u}||_{2,B^{+}(0,3/4)} \right)$

where $C$ depends only on $L$ and $\Omega$ (through $\Phi_{x}$). Writing $u$ and $f$ in terms of $\underline{u}$ and $\underline{f}$, we may then use this inequality to establish (#).

As with interior regularity, we may argue by induction to establish higher-order boundary regularity subject to additional conditions on $\Omega$ and $L$. In this case, we again first consider the case of a half ball. We state the relevant result without proof.

Theorem 10.6 (higher boundary regularity). Suppose $\Omega$ is a bounded domain of class $C^{k+2}$ and $u\in H_{0}^{1,2}(\Omega)$ is a weak solution to the equation

where $L$ is the usual (divergence-form) elliptic operator satisfying the additional conditions $a_{ij},b^{i},c\in C^{k+1}(\overline{\Omega})$ and $f\in H^{k}(\Omega)$. Then $u\in W^{k+2,2}(\Omega)$ and the estimate

where $C_{k}$ depends only on $k$, $\Omega$ and $L$. Moreover, if $\Omega$ is a bounded domain of class $C^{\infty}$ and $a_{ij},b^{i},c,f\in C^{\infty}(\overline\Omega)$, then $u\in C^{\infty}(\overline\Omega)$.

Remark 10.7. If a solution $u\in H_{0}^{1,2}(\Omega)$ to $Lu=f$ is known to be unique, then we may combine Theorem 9.6 and Theorem 10.6 to obtain an estimate of the form $||u||_{k+2,2}\leq C_{k}'||f||_{k,2}$.