Harmonic functions

Our aim is now to study Harmonic functions.

Definition 1. We say a function $u$ is Harmonic in the domain $U$ if $$ -\lap u = 0 \quad\text{in } U \,. $$

Newton potentials

Since $\lap$ is radially symmetric, we look for solutions to $$\label{e:H}\tag{H} -\lap u = 0 \quad\text{in } \R^d\,, $$ which are of the form $u(x) = f(\abs{x})$.

Lemma 2. If $u(x) = f(\abs{x})$ solves \eqref{e:H} on $\R^d - \set{0}$, then $$ \frac{1}{r^{d-1}} \partial_r \paren[\big]{ r^{d-1} \partial_r f } = 0 \,. $$ Consequently, there exists $a, b \in \R$ such that $u = a \Gamma + b $, where $\Gamma$ is the Newton kernel, defined by $$ \Gamma(x) = \begin{cases} \displaystyle -\frac{\ln \abs{x}}{2\pi} & \quad d = 2 \,,\\ \displaystyle \frac{1}{d(d-2) V_d \abs{x}^{d - 2}} & \quad d \geq 3 \,. \end{cases} $$ Here $V_d$ is the volume of the unit ball $B(0, 1) \subseteq \R^d$.

Remark 3. The constants above are chosen so that $$ \int_{\partial B(0, 1)} \grad \Gamma \cdot \hat n \, dS = -1 \,, $$ where $\hat n(x) = x / \abs{x}$ is the outward pointing unit normal on $\partial B(0, 1)$.

Remark 4. It may be tempting to think $-\lap \Gamma = 0$ on all of $\R^d$. But this isn’t right (as we will see shortly). It turns out $-\lap \Gamma = \delta_0$, where $\delta_0$ is the Dirac $\delta$-distribution at $0$. That is, $\Gamma$ is a stationary solution to the heat equation with a point source of heat at the origin.

Approximating a function $f$ as a sum of point sources suggests the solution of the Poisson equation $$ \label{e:P}\tag{P} -\lap u = f \quad \text{in } \R^d $$ should be given by $$\label{e:N}\tag{N} u(x) = \int_{\R^d} \Gamma(x - y) f(y) \, dy \,. $$ This is called the Newton potential of $f$.

Proposition 5. If $f \in C^2_c(\R^d)$, and $u$ is defined by \eqref{e:N} then $u \in C^2(\R^d)$ and solves \eqref{e:P}.

Lemma 6 (Green’s identities). Let $U \subset \R^d$ be $C^1$ and bounded, and $u, v \in C^2(U) \cap C^1(\bar U)$. Then \begin{gather} \label{e:G1}\tag{G1} \int_U \left( u \lap v + \grad u \cdot \grad v \right) \, dx = \int_{\partial U} u \frac{\partial v}{\partial \hat n} \, dS, \\ \label{e:G2}\tag{G2} \int_U \left( u \lap v - v \lap u \right) \, dx = \int_{\partial U} \bigl( u \frac{\partial v}{\partial \hat n} - v \frac{\partial u}{\partial \hat n} \bigr) \, dS \,. \end{gather}

Proof sketch of Proposition 5. Checking $u \in C^2$ follows from the fact that $\Gamma$ is integrable near $0$. Checking $-\lap u = f$ follows by removing an $\epsilon$-ball near the origin, and using Green’s identities.

Mean value property

Proposition 7 (Mean value property). Suppose $u$ is harmonic in $U$, $x \in U$, $B(x, r) \subseteq U$, then $$\label{e:MVP}\tag{MVP} u(x) = \frac{1}{\abs{\partial B(x, r)}} \int_{\partial B(x, r)} u(y) \, dS(y) \,. $$

Proof sketch 1. Write $$ 0 = \int_{B(x, r) - B(x, \epsilon)} (u \lap \Gamma - \Gamma \lap u) \, dx $$ and send $\epsilon \to 0$.

Proof sketch 2. Define $$ \phi(r) = \frac{1}{\abs{B(x, r)}} \int_{B(x, r)} u(y, r) \, dS(y) = \frac{1}{d V_d} \int_{B(0, 1)} u(x + r y, r) \, dS(y) \,, $$ and show $\phi’ = 0$.

Proposition 8. If $u \in C^2(U)$ satisfies the mean value property \eqref{e:MVP}, then $u$ is harmonic in $U$.

Remark 9. We will shortly see that if $u \in C(U)$ satisfies the mean value property, then $u \in C^\infty$, and $u$ is harmonic.

Problem 10. Suppose $d \geq 3$ and $u$ solves \begin{align} -\lap u &= f &&\text{in } B(0, r) \,,\\ u &= g &&\text{on } \partial B(0, r) \,. \end{align} Show that $$ u(0) = \frac{1}{d V_d r^{d-1}} \int_{\partial B(0, r)} g \, dS + \frac{1}{d (d - 2) V_d} \int_{B(0, r)} \paren[\Big]{ \frac{1}{\abs{x}^{d - 2}} - \frac{1}{r^{d - 2}} } \, f \, dx \,. $$

Maximum principle

Theorem 11 (Strong maximum principle). Suppose $U \subseteq \R^d$ is open and bounded, $u \in C(\bar U) \cap C^2(U)$ is harmonic in $U$.

The maximum of $u$ in $\bar U$ is attained on $\partial U$.
If $U$ is connected, and $u$ attains an interior maximum at $x_0 \in U$, then $u$ is constant on the connected component containing $x_0$.

Corollary 12. Let $g \in C(\partial U)$, $f \in C(U)$. There exists at most one solution $u \in C^2(U) \cap C(\bar U)$ of the Dirichlet problem \begin{align} \label{e:D1}\tag{D1} -\lap u &= f &&\text{in } U\,,\\ \label{e:D2}\tag{D2} u &= g &&\text{on } \partial U\,. \end{align}

Problem 13. We say $u$ is sub-harmonic if $u$ is $C^2$ and $- \lap u \leq 0$.

If $v$ is sub-harmonic in $U$, show that $v$ satisfies the sub-mean value property: $$ v(x) \leq \frac{1}{\abs{B_r}} \int_{B(x, r)} v \, dy\,, $$ for every $x, r$ such that $B(x, r) \subseteq U$.
If $v$ is sub-harmonic show that $v$ satisfies the strong maximum principle.
Does $v$ satisfy a minimum principle? That is, is the minimum of $v$ in $\bar U$ attained on $\partial U$? Prove it, or find a counter example.

Lemma 14 (Hopf). Let $U \subseteq \R^d$ be a domain that satisfies the interior sphere condition: That is, for every $x_0 \in \partial U$, there exists a ball $B \subseteq U$ such that $x_0 \in \partial B$. (Recall a domain is an open connected set. Every $C^2$ domain satisfies the interior sphere condition; but $C^1$ domains need not satisfy it.)

Suppose $u \in C^2(U) \cap C^1(\bar U)$ is sub-harmonic in $U$ and attains a maximum at $x_0 \in \partial U$. Then $\hat n \cdot \grad u(x_0) > 0$ unless $u$ is identically constant. (Here $\hat n$ is an outward pointing unit normal.)

Proof sketch.

In an annulus $A(r, R) = B(0, R) - \bar B(0, r)$, find all radial harmonic functions.
Construct a function $v$ so that $\grad v \cdot \hat n(x_0) > 0$, and $u \leq v$ near $x_0$.

Problem 15. Show that the Hopf lemma is false if $U$ does not satisfy the interior sphere condition. That is, suppose $u \in C^2(U) \cap C^1(\bar U)$ is a non-constant sub-harmonic function that attains a maximum at $x_0 \in \partial U$. Show that we could have $\grad u(x_0) = 0$.

Proof sketch. It’s easy to find a piecewise $C^1$ domain that provides a counter-example. (It’s possible, but less easy, to find a $C^1$ domain that doesn’t satisfy the interior sphere condition that provides a counter-example.)

Corollary 16. Let $U \subseteq \R^d$ be a domain that satisfies the interior sphere condition. Let $g \in C(\partial U)$, $f \in C(U)$. Then $C^2(U) \cap C^1(\bar U)$ solutions of the Neumann problem \begin{align} -\lap u &= f &&\text{in } U\,,\\ \hat n \cdot \grad u &= g &&\text{on } \partial U\,. \end{align} are unique, up to an additive constant.

Problem 17. Suppose $U \subseteq \R^d$ is a bounded domain, and $u$ satisfies the Dirichlet problem \eqref{e:D1}–\eqref{e:D2} with $g = 0$. Show that $$ \max_{U} u \leq \frac{\diam(U)^2}{8d} \max_{U} f $$

Problem 18. Let $U \subseteq \R^d$ be a bounded domain. Is there a harmonic function $u\colon U \to \R$ such that for every $y \in \partial U$, $\lim_{x \to y} u(x) = \infty$? Prove or disprove.

Regularity

Theorem 19. If $u \in C(U)$ satisfies the mean value property \eqref{e:MVP} for every $x, r$ such that $B(x, r) \subseteq U$, then $u \in C^\infty(U)$.

Proof sketch. Let $\phi$ be any radial mollifier supported in $B(0,1)$, and set $u_\epsilon = u * \phi_\epsilon$. Show that $u = u_\epsilon$ at all points that are at least $\epsilon$ away from $\partial U$.

Proposition 20. Say $u$ is harmonic on $U$. For every $x_0, r$ such that $B(x_0, r) \subseteq U$ we have $$ \abs{D^\alpha u(x_0)} \leq \frac{C_k}{r^{d + k}} \norm{u}_{L^1(B(x_0, r))} $$ where $\alpha$ is a multi-index of order $k$, and $$ C_k \defeq \begin{cases} \displaystyle \frac{1}{V_d} & k = 0 \,,\\ \displaystyle \frac{ (2^{d+1} d k)^k }{V_d} & k \geq 1 \,. \end{cases} $$

Proposition 21. If $u$ is harmonic in $U$, then $u$ is real analytic in $U$.

Problem 22.

If $\phi \colon \R \to \R$ is $C^2$ and convex, and $u \in C^2(U)$ is harmonic, then show that $\phi(u)$ is sub-harmonic.
If $u$ is harmonic, show that $\abs{\grad u}^2$ is sub-harmonic.

Liouville’s theorem

Theorem 23 (Liouville’s theorem). If $u \colon \R^d \to \R$ is harmonic and bounded, then $u$ is constant.

Corollary 24. If $f \in C^2_c(\R^d)$, and $d \geq 3$, then any bounded solution of the Poisson problem \eqref{e:P} is given by $$ u = \Gamma * f + C $$ for some constant $C \in \R$.

Remark 25. When $d = 2$, note $\Gamma * f$ need not be bounded.

Harnack inequality

Theorem 26 (Harnack inequality). Let $V$ be a connected, bounded, open set with $\bar V \subseteq U$. There exists a constant $C = C(V)$ such that for every function $u \geq 0$ which is harmonic in $U$, we have $$ \sup_V u \leq C \inf_V u \,. $$

Proof sketch. Let $r = d(V, \partial U) / 4$ and show that if $x, y \in V$ are such that $\abs{x - y} < r$ then $$ u(x) \geq \frac{u(y)}{2^d} \,. $$

Theorem 27 (Harnack convergence theorem). Let $U \subseteq \R^d$ be a domain. Suppose $\set{u_n}_{n \in \N}$ is an increasing sequence of harmonic functions in $U$. Then either $u_n \to \infty$ as $n \to \infty$ locally uniformly in $U$, or there exists a harmonic function $u \colon U \to \R$ such that $u_n \to u$ as $n \to \infty$ locally uniformly in $U$.

Energy methods

Proposition 28. The uniqueness in Corollary 12 and Corollary 16 hold.

Proposition 29 (Dirichlet’s principle). Let $U$ be a bounded, $C^1$ domain and $g \in C(\partial U)$. Define $$ \mathcal A = \set{ w \in C^2(\bar U) \st w = g \text{ on } \partial U} \,. $$ Then $u \in \mathcal A$ is solves the Dirichlet problem \eqref{e:D1}–\eqref{e:D2} if and only if $$ u = \argmin_{w \in \mathcal A} \mathcal E(w) \,, \qquad\text{where}\quad \mathcal E(w) \defeq \int_U \paren[\Big]{ \frac{1}{2} \abs{\grad w}^2 - w f} \, dx $$