Harmonic function

A harmonic function satisfies Laplace's equation $\Delta u=0$.

For $r<1$, average the real part of the convergent power series for $\log (1+r{e}^{i\theta })$: every nonconstant Fourier mode vanishes. For $r>1$, factor out $r$ and apply the same argument to $r^{-1}$. The boundary singularity at $r=1$ is logarithmically integrable. This identity computes the expected logarithmic distance after an isotropic Gaussian displacement.

For a smooth harmonic function on a manifold without boundary, assume its nonzero superlevel sets are compact. On a regular positive superlevel set $\{U>\epsilon \}$, integration by parts gives $\int |\nabla U{|}^2=-\epsilon {\int }_{\partial \{U>\epsilon \}}|\nabla U|\le 0$. Applying the same argument to $-U$ proves $U=0$. This avoids assuming an unstated decay rate for the boundary flux at infinity.

An element $h\in H_{\mathrm{loc}}^1(U)$ is weakly harmonic on an open set $U$ when ${\int }_U\nabla h\cdot \nabla \varphi =0$ for every test function $\varphi \in C_c^{\infty }(U)$. This is the statement $\Delta h=0$ in distributions. It requires no regularity of the domain boundary. The Weyl lemma, a form of elliptic regularity, identifies such an element with a smooth harmonic function.

A real function $h$ on the vertices of a graph of an unweighted locally finite graph is a discrete harmonic function at $u$ when $\mathrm{deg}(u)h(u)={\sum }_{v\sim u}h(v)$, equivalently $Lh(u)=0$ for the Graph Laplacian. For positive degree this says its value is the average over its graph neighbours; at an isolated vertex of a graph the Laplacian condition is vacuous. On an electrical network the corresponding equation uses conductance weights and the Weighted graph Laplacian. Voltages away from sources and sinks, and hitting probabilities away from their prescribed boundaries, are discrete harmonic functions.

Every bounded harmonic function on a graph is constant if the connected locally finite graph is recurrent. Along its random walk on a graph, the harmonic values form a bounded martingale and converge almost surely. Recurrence visits each vertex infinitely often, making every vertex value a subsequential limit of the same convergent sequence. Thus every value agrees.

A real function $h$ on a finite connected graph satisfying $Lh=0$ everywhere is constant. At a maximum graph vertex, the identity $\mathrm{deg}(u)h(u)={\sum }_{v\sim u}h(v)$ forces every graph neighbour to have the same maximum. Connectedness propagates it to every graph vertex. This proves uniqueness of voltage up to an additive constant, and identifies sums of oppositely directed occupation voltages.

Every bounded discrete harmonic function on ${\mathbb{Z}}^d$ is constant. After rescaling its range to $[0,1]$, place it in the compact convex set of all such functions; translation invariance and the mean-value identity force every extreme point to be constant, and the Krein-Milman theorem finishes the proof.

A harmonic function on a punctured ball that is bounded near the missing point extends uniquely to a harmonic function on the whole ball. The extension can be defined by the mean value property for harmonic functions; interior estimates then give smoothness across the point.

The harmonic replacement of $u$ in a domain $D$ is the harmonic function $w$ having the same boundary trace as $u$. Thus $u-w\in H_0^1(D)$, and subtracting the weak equations permits local energy estimates for the error $u-w$.

For every closed ball contained in the domain of a harmonic function, the value at its centre equals both its average over the ball and its average over the boundary sphere.

If $u\ge 0$ is a harmonic function on a neighbourhood of $\overline{B_{4r}(y)}$, thenFor $x,z\in B_r(y)$, use $B_r(x)\subset B_{3r}(z)\subset B_{4r}(y)$ and the mean value property for harmonic functions. Nonnegativity lets us enlarge the integral, and the ratio of volumes is $3^n$. This compares values without requiring derivative bounds.

If a continuous function has the spherical mean value property at every point for some sequence of radii decreasing to zero, then it is harmonic. Comparison on each relatively compact ball with the harmonic function having the same boundary data proves the result by propagating any hypothetical interior extremum along one of the admissible spheres.

If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.

The Weyl lemma says that every locally integrable distributional solution of $\Delta u=0$ agrees almost everywhere with a smooth harmonic function. Convolution with a mollifier and the harmonic mean value property provide a standard proof.

If $u$ is harmonic on $B_r(x_0)$, then for every integer $m\ge 1$,The first-order estimate follows by differentiating the ball mean-value formula and applying the divergence theorem; iteration gives higher orders.

If an entire harmonic function on ${\mathbb{R}}^n$ grows like $O(1+|x{|}^{k+\alpha })$ with $k\in \mathbb{N}$ and $0<\alpha <1$, then it is a harmonic polynomial of degree at most $k$. Apply the interior derivative estimate of order $k+1$ on expanding balls.

A harmonic function on all of ${\mathbb{R}}^n$ with a finite global Hölder seminorm of exponent $0<\mu <1$ is constant. Indeed $|w(x)-w(0)|\le C|x{|}^{\mu }$, and the interior derivative estimate for a harmonic function on balls of radius $R$ bounds $|\nabla w(0)|$ by $C{R}^{\mu -1}$. Letting $R\to \infty$ makes this gradient zero, and translating the centre proves the assertion everywhere.

A real-valued function $u$ on a simply connected domain is harmonic exactly whenfor some holomorphic function $f$. One construction observes that $u_x-i{u}_y$ is holomorphic and takes its holomorphic primitive.

A twice differentiable function is strictly subharmonic where $\Delta u>0$. It cannot have an interior local maximum because the Hessian at such a maximum is negative semidefinite.

A continuous subharmonic function on the closure of a bounded domain attains its maximum on the boundary. If it is twice differentiable, the strict condition $\Delta u>0$ rules out an interior maximum immediately because the Hessian matrix there would be negative semidefinite.

For a harmonic function $w$ smooth inside a bounded domain and $C^1$ on its closure, $\Delta |Dw{|}^2=2|D^{2}w{|}^2\ge 0$. The weak maximum principle for elliptic operators and continuity of the gradient imply ${\sup }_{\Omega }|Dw|={\sup }_{\partial \Omega }|Dw|$.

A continuous function harmonic on a bounded domain attains its maximum and minimum on the boundary. Apply the strictly subharmonic result to $u+ϵ|x{|}^2$ and let $ϵ\downarrow 0$, then apply the same argument to $-u$.

A bounded harmonic function on the complex plane is constant. More generally, an entire harmonic function bounded on either side is constant: write it as $\mathrm{Re}f$ by harmonic function as the real part of a holomorphic function, then apply the Liouville theorem to $e^f$ or $e^{-f}$.

A bounded entire harmonic function satisfies $u(x)={\mathbb{E}}_{x}u(B_t)$ because $u(B)$ is a bounded local martingale. The Gaussian heat kernel has a directional derivative with $L^1$ norm $\sqrt{2/\pi }/\sqrt{t}$. Consequently $|u(x)-u(y)|\le \Vert u{\Vert }_{\infty }\sqrt{2/\pi }|x-y|/\sqrt{t}$; letting $t\to \infty$ proves the harmonic Liouville theorem in every dimension.

A harmonic function on ${\mathbb{R}}^2\setminus \{0\}$ bounded on one side is constant. Shift and change sign to make it nonnegative. Its circular mean has the form $a\log r+b$; positivity for every $r>0$ forces $a=0$. For $|x|=r$, the mean value property for harmonic functions bounds its value by the mean integral over $r/2<|z|<3r/2$, giving at most $8b$. A removable singularity for a bounded harmonic function fills the origin, and the harmonic Liouville theorem finishes the proof. The dimensional distinction matters: $|x{|}^{2-n}$ is a positive nonconstant harmonic function on punctured ${\mathbb{R}}^n$ for $n\ge 3$.

For a bounded harmonic function $f$ on ${\mathbb{R}}^d$, the processes $f(B_t^x)$ and $f(B_t^y)$ are bounded martingales. Couple the Brownian motions from $x$ and $y$ so that they coalesce in finite time almost surely. Thenwhich proves that $f$ is constant in every dimension.

Every nonconstant harmonic function $u:\mathbb{C}\to \mathbb{R}$ is unbounded above and below by the harmonic Liouville theorem. The intermediate value theorem then gives $u(\mathbb{C})=\mathbb{R}$.

Let ${\wp }_{\Lambda }$ be the Weierstrass elliptic function forBecause $2\pi i$ is a period, the formulais independent of the branch of the complex logarithm. It is a nonconstant harmonic function onand its other period gives $u(qz)=u(z)$.

A harmonic polynomial is a polynomial annihilated by the Laplacian.

If ${\mathcal{P}}_{\ell }$ is the space of degree-$\ell$ homogeneous polynomials and ${\mathcal{H}}_{\ell }$ its harmonic polynomials, then ${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$. Under the Fischer inner product, the second summand is orthogonal to the kernel of the Laplacian. Iteration expresses every polynomial restricted to the unit sphere as a sum of spherical harmonics.

A harmonic conjugate v makes u+iv holomorphic; it exists globally on simply connected domains.

On a disc, the closed one-form $-u_{y}dx+u_{x}dy$ has a potential $v$ when $\Delta u=0$. The Cauchy--Riemann equations then make $u+iv$ holomorphic.

The function $\log |z|$ is harmonic on the punctured complex plane, but it is not globally the real part of a holomorphic function there. Such a function would have derivative $1/z$, whose integral around the unit circle is $2\pi i$, whereas the integral of a derivative around a closed curve is zero.

Every $C^2$ harmonic function is locally the real part of a holomorphic function and is therefore infinitely differentiable.

Harmonic functions admit locally convergent power-series expansions.

A harmonic function on a connected domain that vanishes on a nonempty open subset vanishes everywhere.