Elliptic equations

Variable coefficient elliptic operators

We now consider second order linear equations with variable coefficients. Let $U \subseteq \R^d$ be a domain, and define $$\label{e:L}\tag{L} Lu \defeq \sum_{i,j=1}^d a^{ij}(x)\partial_{ij}u + \sum_{i=1}^d b^i(x)\partial_i u + c(x)u\,. $$ We may assume $A=(a^{ij})$ is symmetric, since the Hessian is symmetric.

Definition 1. We say $L$ is uniformly elliptic in $U$ if there exist $0 < \lambda\leq\Lambda$ such that $$ \lambda\abs{\xi}^2 \leq \xi \cdot A(x)\xi \leq \Lambda\abs{\xi}^2 $$ for every $x\in U$ and $\xi\in\R^d$.

Maximum principles

Suppose now $U$ is bounded, $a^{ij},b^i,c$ are bounded and continuous.

Theorem 2 (Weak maximum principle). Suppose $u\in C^2(U)\cap C(\bar U)$ and $Lu\geq0$ in $U$.

  1. If $c \leq 0$, then a nonnegative maximum of $u$ is attained on the boundary. That is, $$ \max_{\bar U}u\leq \max_{\partial U}u_{+}\,. $$
  2. If $c=0$, then a maximum of $u$ is attained on the boundary. That is, $$ \max_{\bar U}u=\max_{\partial U}u \,. $$

Lemma 3. If $-Lu < 0$ in $U$, then $u$ does not attain a positive maximum in $U$.

Proof sketch. If $u$ attains a positive maximum at $x_0 \in U$, then $\grad u=0$ and $D^2u\leq0$, which implies $-Lu\geq0$, which is a contradiction.

Proof sketch of Theorem 2.

  1. For $c \leq 0$, let $u_\epsilon=u+\epsilon e^{k x_1}$. For sufficiently large $k$, $L(e^{k x_1}) > 0$ in $U$, and hence $$ \max_{\bar U} u \leq \max_{\bar U} u_\epsilon \leq \max_{\partial U} u_{\epsilon, +} \leq \max_{\partial U} u_{\epsilon} + \epsilon C(U, k) \,, $$ and send $\epsilon \to 0$.
  2. When $c=0$, use the same argument after adding a constant to $u$.

Corollary 4. If $-Lu=f$ in $U$ and $u=g$ on $\partial U$, there is at most one solution in $C^2(U)\cap C(\bar U)$.

Lemma 5 (Hopf lemma). Let $U$ be a bounded domain satisfying the interior sphere condition. Suppose $u\in C^2(U)\cap C(\bar U)$ satisfies $-Lu\leq0$. Let $x_0\in\partial U$ admit an interior tangent ball $B\subseteq U$. If $u$ attains a strict local maximum at $x_0 \in \partial U$, and $u(x_0) \geq 0$, $$ \liminf_{t\to 0}\frac{u(x_0)-u(x_0-t\hat n)}{t} > 0\,, $$ where $\hat n$ is the outward normal to the tangent ball at $x_0$. In particular, if $u$ is $C^1$ at $x_0$, then $\hat n\cdot\grad u(x_0) > 0$.

Proof sketch. Find a ball $B=B(y,R) \subseteq U$ be such that $x_0 \in \partial B$ and $u(x_0) > u(x)$ for all $x \in B$. Consider the annulus $R/2 < \abs{x-y} < R$, and construct a barrier function of the form $$ v(x)=u(x_0) + \epsilon \paren[\big]{ e^{-\alpha R^2} - e^{-\alpha\abs{x-y}^2} } \,. $$

Theorem 6 (Strong maximum principle). Suppose $U$ is connected, $u\in C^2(U)\cap C(\bar U)$ and $Lu\geq0$ in $U$.

  1. If $c \leq 0$, and $u$ attains a nonnegative interior maximum in $U$, then $u$ is constant.
  2. If $c=0$ and $u$ attains an interior maximum in $U$, then $u$ is constant.

Proof sketch. Use the Hopf lemma on a ball that touches an interior maximum.

Problem 7. Show that the weak maximum principle can fail when $c > 0$.

Problem 8. Let $A$ be a constant symmetric positive-definite matrix. Show that there exists a linear change of variables transforms the equation $\sum_{i,j} a^{ij}\partial_{ij}u=0$ into Laplace’s equation. (Consequently interior smoothness and the other properties we proved for Laplace’s equation will still hold.)

Interior regularity: energy methods

Given the tools we have so far we can’t prove everything here rigorously, but will do enough to hopefully convince you that this is believable.

For regularity it is more convenient to work with equations in divergence form: $$\label{e:D}\tag{D} -\dv(A(x)\grad u)=f\quad\text{in }U\,. $$ Assume $A \in C^\infty(\bar U)$ is uniformly elliptic. Notice $$ -\dv(A\grad u)=-a^{ij}\partial_{ij}u-(\partial_i a^{ij})\partial_j u\,. $$

Definition 9. Let $C \subseteq \R^d$ be a bounded $C^1$ domain, and $k \in \N$. Define the $H^k$ norm by $$ \norm{u}_{H^k}^2 = \sum_{\abs{\alpha} \leq k} \norm{D^\alpha u}_{L^2}^2 \,, $$ and the Sobolev space of index $k$ (notation $H^k(U)$) to be the completion of $C^\infty(\bar U)$ under the $H^k$ norm.

Definition 10. We say $u\in H^1(U)$ is a weak solution of \eqref{e:D} if $$ \int_U A\grad u\cdot\grad\phi\,dx=\int_U f\phi\,dx $$ for every $\phi\in C_c^\infty(U)$.

Lemma 11 (Caccioppoli inequality). There exists a constant $C = C(\lambda, \Lambda)$ such that for every $u\in H^1(U)$ which is a weak solution of \eqref{e:D} with $f\in L^2(U)$, and every $\eta\in C_c^\infty(U)$ we have $$ \int_U\eta^2\abs{\grad u}^2\,dx \leq C\int_U\paren[\big]{\abs{\grad\eta}^2u^2+\eta^2 f^2+\eta^2u^2}\,dx\,. $$

Proof sketch. Use $\phi=\eta^2u$ as a test function, expand $\grad(\eta^2u)$, and apply ellipticity, Cauchy–Schwarz, and Young’s inequality. (Making this rigorous is done via an approximating sequence $u_\epsilon \to u$ with $u_\epsilon \in C^\infty(\bar U)$, which is routine once the appropriate tools have been developed.)

Theorem 12 (Interior $H^2$ regularity). Suppose $A\in C^1(U)$ is bounded and uniformly elliptic, $f\in L^2(U)$, and $u\in H^1(U)$ is a weak solution of \eqref{e:D}. For every $C^1$ domain $V\Subset U$, we have $u\in H^2(V)$ and $$\label{e:H2}\tag{H2} \norm{u}_{H^2(V)}\leq C\paren[\big]{\norm{f}_{L^2(U)}+\norm{u}_{H^1(U)}}\,. $$ Here $C$ depends on $U$, $V$, $A$, and $\lambda$, but not on $u$ or $f$.

Lemma 13. Theorem 12 holds if $u \in C^\infty(\bar U)$.

Proof sketch. Differentiate \eqref{e:D} in the $x_k$ direction, and write $v=\partial_k u$: $$ -\dv(A\grad v)=\partial_k f+\dv((\partial_k A)\grad u)\,. $$ Multiply by $\eta^2v$, integrate by parts, use ellipticity and Young’s inequality to obtain $$ \int_U\eta^2\abs{\grad v}^2\,dx \leq C\paren[\Big]{\int_U\abs{f}^2+\abs{\grad u}^2\,dx}\,. $$ Choose $\eta=1$ on $V$ and sum over $k$.

Proof sketch of Theorem 12. One needs to replace $\partial_k u$ in the previous lemma by the difference quotient $$ D_k^h u(x)=\frac{u(x+he_k)-u(x)}{h} \,, $$ apply the same energy argument to the difference-quotient equation, and pass to a limit as $h \to 0$. (Doing this requires choosing convergent subsequences etc, and developing this machinery fully is beyond the scope of this course. )

Remark 14. If the coefficients and $f$ have more derivatives, the same argument can be iterated to obtain higher Sobolev regularity.