Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part II.

This is the second of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post.

Some consequences of Harnack’s inequality the Mean value property

The mean value property is characteristic of harmonic functions, but the fact that harmonic functions control their pointwise values by their local average is a general fact that is characteristic of elliptic equations (as we will see later, less sharp but more general theorems for nonlinear elliptic equations still have this flavor and are at the very heart of the regularity theory of fully nonlinear elliptic PDEs). Let me mention a few of its consequences, I already talked last time about Harnack’s inequality, as it follows from the mean value theorem, the mean value theorem (at least for harmonic functions) is more fundamental.

First, and perhaps the most important consequence, is the pointwise a priori estimate for the derivatives of a harmonic function in terms of its supremum:

Theorem 1 (A priori gradient estimate for harmonic functions)

Let {u} be a harmonic function in a ball {B_r(y)}, then

\displaystyle |\nabla u(y)| \leq \frac{C_n}{r} \sup \limits_{B_r(y)}u(x)

Proof: Let {u} be a harmonic function in {B_r(y)}, by the mean value property, {u(y+h)= (w_nr^n)^{-1}\int_{B_{r/2}(y+h)}u(x)dx} as long as {|h|<\frac{r}{2}}, therefore

\displaystyle |u(y+h)-u(y)| \leq C_nr^{-n} \int_{B_{r/2}(y+h) \Delta B_{r/2}(y)}|u(x)|dx

(Recall {A \Delta B = A \setminus B \cup B \setminus A} is called the symmetric difference of sets), now the set {B_{r/2}(y+h) \Delta B_{r/2}(y)} lies in the union of two annuli with radii {r} and {r+h}, thus its volume is not larger than {C_nr^{n-1}|h|} for a dimensional constant {C_n}. We then have

\displaystyle  \int_{B_{r/2}(y+h) \Delta B_{r/2}(y)}|u(x)|dx \leq C_nr^{n-1}|h| \sup \limits_{B_r(y)}u(x)

\displaystyle |u(y+h)-u(x)|\leq \frac{C_n}{r}|h| \sup \limits_{B_r(y)}u(x)

since the direction of {h} is arbitrary and {u} is {C^1}, dividing both sides by {|h|} and taking {h \rightarrow 0} we obtain the a priori estimate. \Box

One may iterate this to estimate higher derivatives (thanks to the fact that the derivatives of a harmonic function are themselves harmonic). To obtain the estimate

\displaystyle |D^{(k)}u(y)| \leq \frac{C_{n,k}}{r^k} \sup \limits_{B_r(y)}u(x)

I emphasize that these are a priori estimates, one needs to know {u} is already smooth to prove them, what they say is that the derivatives of all orders {u} are all controlled by the supremum of {u}!. In particular, a family of uniformly bounded harmonic functions is compact in every {C^k}. Usually, the first time you learn about this phenomenon is when studying Montel’s theorem in a complex analysis.

The a priori estimates and Harnack’s inequality also give a quick proof (which I will omit) of another classical result, but in potential theory, which I mentioned because it was due to Harnack himself:

Theorem 2 (Harnack’s convergence theorem) Let {u_n:\Omega \rightarrow \mathbb{R}} be a decreasing sequence of functions which are continuous in {\bar{\Omega}} and harmonic in the interior. Then they converge uniformly in compact sets of {\Omega} to a smooth function {u:\Omega \rightarrow \mathbb{R}} which is harmonic.

Since I just mentioned a priori estimates, I should recall the (one of many ) proofs of the fact that being harmonic even in some weak sense forces a function to be smooth and harmonic in the classical sense. Let’s say for instance, harmonic in the sense of distributions (we will revisit this theorem for other weak notions of harmonicity):

Theorem 3 (Weak harmonic implies harmonic) Let {u} be a bounded measurable function in {\Omega} such that

\displaystyle \int_\Omega u(x) \Delta \phi (x) dx = 0

for any smooth function {\phi} with compact support in {\Omega}. Then {u} (after modifying it in at most a set of measure zero) is smooth in the interior of {\Omega} and harmonic.

Proof: Let {\psi_\epsilon} be an approximation to the identity given by a {C^\infty_c} kernel {\psi_1} which is radially symmetric and supported in {B_1(0)}. Let {u_\epsilon = u \star \psi_\epsilon}, then for each {\epsilon>0} and any compact {K \subset \Omega} the functions {u_\epsilon} ({\epsilon} small enough depending on the distance between {K} and {\partial \Omega}) are smooth with bounded derivatives of any orders, moreover, they are all uniformly bounded in {L^\infty} by the boundedness assumption on {u}. Now, using the symmetry of the kernel and Fubini’s theorem, one can see that for any {\phi \in C^\infty_c(\Omega)}

\displaystyle \int_\Omega u_\epsilon(x) \Delta \phi (x) dx = \int_\Omega u(x) \Delta \phi_\epsilon(x)dx=0

and since each {u_\epsilon} is smooth we have {\Delta u_\epsilon = 0}. Furthermore, by the a priori estimates the functions {u_\epsilon} are also uniformly bounded {C^k(K)}, for any {k \geq 0}. Then we know (by Arzela-Ascoli) that a suitable subsequence of {u_\epsilon} with {\epsilon} converges uniformly in {K} to a function which lies in {C^k} (for any {k}), since they also must converge to {u(x)} a.e.(by Lebesgue’s differentiation theorem) we conclude that {u} agrees a.e. with a smooth harmonic function, as we wanted to prove.

\Box

I think that I will stop here for now. Tomorrow: I will review the Poisson kernel to give the potential theoretic proof of the Mean value property, Harnack’s inequality and the (a priori) gradient estimates for harmonic functions, and after that, it will be the Calderón-Zygmund estimates.

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