Harmonic Function

A harmonic function is a twice continuously differentiable function on an open subset of Euclidean space whose Laplacian vanishes. In coordinates, a function $u$ is harmonic when $\Delta u=\sum_{j=1}^{n}\partial^2u/\partial x_j^2=0$. Harmonicity also has an averaging interpretation: the value at the center of every ball contained in the domain equals the average over its boundary sphere, and also over the ball itself. These equivalent differential and averaging descriptions connect local equations with global behavior. (web.math.princeton.edu)

Harmonic functions describe equilibrium phenomena, including steady temperatures in homogeneous materials without heat sources and electric potentials in charge-free regions. Their mathematical importance rests on strong structural properties: they are real analytic, obey maximum principles, and are closely related to holomorphic functions in two dimensions. Their theory encompasses boundary-value problems, energy minimization, spherical expansions, and singularities, with further interpretations through Brownian motion and extensions to curved spaces and discrete networks. (web.stanford.edu)

Definition, Domain, and Elementary Examples

A classical real-valued harmonic function on an open set $\Omega\subseteq\mathbb R^n$ is a function $u:\Omega\to\mathbb R$ belonging to $C^2(\Omega)$—its partial derivatives through order two exist and are continuous—and satisfying Laplace’s equation

Here $\Delta$ is the Laplacian, with the convention $\Delta=\operatorname{div}\nabla$. The definition imposes an equation at every interior point, not a boundary condition. (web.stanford.edu)

A domain usually means a nonempty, open, connected subset of Euclidean space. Connectedness is not required to define harmonicity: on a disconnected open set, the equation holds independently on each connected component. For a complex-valued function $u=v+iw$, harmonicity is interpreted componentwise. Since $\Delta u=\Delta v+i\Delta w$, it is equivalent to harmonicity of both real-valued functions $v$ and $w$. (axler.net)

Constants and affine functions $u(x)=b+\sum_{j=1}^{n}a_jx_j$ are harmonic because all their second derivatives vanish. (web.stanford.edu) Direct differentiation supplies nonlinear examples in two dimensions:

By contrast, $\Delta(x^2+y^2)=2+2=4$, so $x^2+y^2$ is not harmonic on any nonempty open subset of the plane. These calculations show that harmonicity requires cancellation in the sum of pure second derivatives, rather than their separate vanishing. The same examples work in higher dimensions when the functions are independent of the remaining coordinates.

In one dimension the equation becomes $u''=0$; integrating twice gives $u(x)=ax+b$ on each interval component of the open set. In dimensions $n\geq2$, the equation permits much richer families. For example, direct calculation for $u(x,y)=e^{ax}\cos(ay)$, with any real parameter $a$, gives $u_{xx}=a^2u$ and $u_{yy}=-a^2u$. Thus nonlinear, nonpolynomial harmonic functions already occur on the whole plane, and extend to $\mathbb R^n$ by ignoring additional coordinates.

Harmonic functions on a fixed open set form a vector space: linearity gives $\Delta(au+bv)=a\Delta u+b\Delta v$. Their derivatives are harmonic as well. Although the definition assumes only $C^2$ regularity, harmonic functions are automatically infinitely differentiable; commuting derivatives then yields $\Delta(\partial_j u)=\partial_j(\Delta u)=0$, and likewise for higher derivatives. (axler.net) Products need not remain harmonic: the product rule gives

For instance, $x$ is harmonic but $x^2$ has Laplacian $2$.

Translations, orthogonal transformations—including reflections—and uniform dilations preserve harmonicity. (axler.net) More explicitly, let $T(x)=a+sQx$, where $s>0$ and $Q^{\mathsf T}Q=I$. The chain rule gives

so $u\circ T$ is harmonic on $T^{-1}(\Omega)$. Uniform scaling is essential here: stretching the coordinates unequally changes the relative weights of the second derivatives. For example, composing $x^2-y^2$ with $(x,y)\mapsto(2x,y)$ produces $4x^2-y^2$, whose Laplacian is $6$.

Physical Origins and Equilibrium Models

The study of harmonic functions developed from eighteenth-century celestial mechanics and nineteenth-century potential theory. In 1782, Pierre-Simon Laplace showed that the Newtonian gravitational potential satisfies, away from attracting matter, the equation now called Laplace’s equation, $\Delta u=0$. Representing an inverse-square force through a scalar potential made it possible to study the field through a single differential equation rather than three separate force components. The nineteenth-century extension to electricity and magnetism is exemplified by George Green’s 1828 An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. These applications established potential theory as a common mathematical framework for otherwise different physical phenomena. (cambridge.org)

In electrostatics, the electric field and electric potential are related by $\mathbf E=-\nabla V$. In vacuum, Gauss’s law gives $\nabla\cdot\mathbf E=\rho_e/\varepsilon_0$, where $\rho_e$ is charge density and $\varepsilon_0$ is vacuum permittivity. Substitution yields Poisson’s equation,

Thus $V$ is harmonic wherever the charge density vanishes, even when charges outside that region produce a nonzero field inside it. The potential is a scalar; the field is its negative gradient, not the potential itself. In a dielectric with spatially varying scalar permittivity, the corresponding equation is instead $\nabla\cdot(\varepsilon(x)\nabla V)=-\rho_{\mathrm{free}}$. A charge-free region therefore need not have a harmonic potential unless the constitutive assumptions reduce this equation to a constant multiple of $\Delta V=0$. (web.mit.edu)

For Newtonian gravity, adopt the convention $\mathbf g=-\nabla\Phi$, with $\Phi$ denoting gravitational potential per unit mass. The gravitational flux law is $\nabla\cdot\mathbf g=-4\pi G\rho_m$, so

where $G$ is the gravitational constant and $\rho_m$ is mass density. Consequently, $\Phi$ is harmonic in regions containing no attracting matter. For a point mass $M$, $\Phi(x)=-GM/|x|$ is harmonic for $|x|>0$, but singular at the mass. This distinction matters in geophysics: the Earth’s exterior gravitational potential obeys Laplace’s equation, whereas its interior generally obeys Poisson’s equation. “Source-free” is a local condition on density, not a claim that the field vanishes or that no sources exist elsewhere. (www-gpsg.mit.edu)

For heat conduction in a stationary solid, local energy balance reads $\rho c\,\partial_tT=-\nabla\cdot\mathbf q+s$, where $T$ is temperature, $\rho c$ is volumetric heat capacity, $\mathbf q$ is heat flux, and $s$ is heat production per unit volume. Fourier’s law gives $\mathbf q=-k\nabla T$ for isotropic thermal conductivity $k$. With constant $k>0$, these relations yield the heat equation $\rho c\,\partial_tT=k\Delta T+s$. At steady state, with no internal heat production, $\Delta T=0$. Steady state does not require uniform temperature or zero heat flux: different boundary temperatures can sustain continuous heat transport. If conductivity varies spatially, the source-free equilibrium equation is $\nabla\cdot(k(x)\nabla T)=0$; anisotropic conductivity replaces $k$ by a conductivity tensor. Neither equation generally reduces to Laplace’s equation. (web.stanford.edu)

In ideal fluid models, irrotationality means $\nabla\times\mathbf v=0$, allowing a velocity potential $\phi$ with $\mathbf v=\nabla\phi$, locally, or globally on a simply connected flow region. Incompressibility supplies $\nabla\cdot\mathbf v=0$, hence $\Delta\phi=0$. These two kinematic conditions produce harmonicity; inviscidity belongs to the usual ideal-flow model but is not itself sufficient. The potential may depend on time while remaining harmonic in the spatial variables at each instant. By contrast, steady compressible mass conservation gives $\nabla\cdot(\rho\mathbf v)=0$, and therefore $\nabla\cdot(\rho\nabla\phi)=0$: variable density prevents the direct reduction to Laplace’s equation. (ocw.mit.edu)

Mean-Value Properties and Maximum Principles

Let $u$ be real-valued and harmonic on an open set $\Omega\subset\mathbb R^n$, with $n\ge2$. For every ball whose closure $\overline{B(a,r)}$ lies in $\Omega$, the mean-value properties state that

Here $dS$ is surface measure, $dx$ is volume measure, and the denominators normalize each measure to total mass one. If $\omega_n=|B(0,1)|$, these denominators are $n\omega_n r^{n-1}$ and $\omega_n r^n$, respectively. Thus the value at the center equals both the spherical average and the solid-ball average, at every admissible radius—not merely in the limit as the radius tends to zero. (web.math.princeton.edu)

The divergence theorem explains why these averages are independent of radius. Set

Writing this integral on the unit sphere and differentiating gives

where $\nu$ is the outward unit normal. Continuity gives $M(r)\to u(a)$ as $r\to0$, establishing the spherical formula. Integrating over concentric spherical shells establishes the ball formula. The calculation uses only harmonicity inside the ball; no regularity of the boundary of $\Omega$ is required. (web.math.princeton.edu)

Conversely, a continuous function on $\Omega$ is harmonic if, at every point, it satisfies either mean-value formula for every sufficiently small radius. No differentiability need be assumed. One proof smooths the function by convolution with a smooth radial averaging kernel: the mean-value identities make this convolution equal to the original function locally, thereby establishing smoothness. For a smooth function, the spherical-average expansion

then forces $\Delta u(a)=0$. The averaging condition is therefore a characterization of harmonicity, rather than merely a consequence of Laplace’s equation. (axler.net)

The weak maximum and minimum principles concern bounded open sets: if $u\in C^2(\Omega)\cap C(\overline\Omega)$ is harmonic, then

Connectedness is unnecessary for these boundary statements. The strong principles, by contrast, require connectedness but neither boundedness nor boundary continuity: if $u$ attains its supremum or infimum over $\Omega$ at an interior point, it is constant. Indeed, if $u(a)=M$ is the maximum, averaging values no greater than $M$ can give $M$ only when every value in a sufficiently small ball equals $M$. The set where $u=M$ is consequently both open and relatively closed; connectedness makes it all of $\Omega$. Applying the argument to $-u$ gives the minimum principle. (mathsite.math.berkeley.edu)

Comparison follows by applying the weak principle to $u-v$: on a bounded open set, two harmonic functions continuous on the closure satisfy $u\le v$ throughout whenever $u\le v$ on the boundary. Consequently, prescribed continuous boundary values $f$ bound any solution between $\min_{\partial\Omega}f$ and $\max_{\partial\Omega}f$, and a Dirichlet boundary-value problem has at most one such solution; this does not establish existence. Unbounded domains require control at infinity. In the half-space $\{x_n>0\}$, both $0$ and $x_n$ are harmonic and vanish on the boundary, yet differ inside. A sufficient additional comparison condition is

together with $u\le v$ on the finite boundary and continuity there. Applying the bounded-domain principle on expanding truncated domains then yields $u\le v$. (axler.net)

Interior Regularity, Positivity, and Convergence

A classical harmonic function $u\in C^2(\Omega)$ automatically belongs to $C^\infty(\Omega)$, and is in fact real analytic. Smoothness follows by averaging against a smooth radial kernel supported in a sufficiently small ball: the mean-value property makes this average equal to $u$, while differentiation of the kernel gives derivatives of every order. Analyticity is stronger: near each $a\in\Omega$, a convergent Taylor series represents the function,

Here $\alpha$ is a multi-index, with $\alpha!=\prod_i\alpha_i!$. These conclusions are strictly interior statements; they do not imply a continuous extension, or differentiability, at the boundary. (axler.net)

Interior estimates quantify how derivatives depend on the function’s size and the available distance to the boundary. If $\overline{B_R(a)}\subset\Omega$, then for every multi-index $\alpha$,

where $C_{n,\alpha}$ is independent of $u$, $a$, and $R$. Repeated application of first-derivative estimates on nested balls gives the higher-order bounds. In particular, for bounded $u$ and $d(x)=\operatorname{dist}(x,\partial\Omega)$,

The factor $R^{-|\alpha|}$ reflects the scaling of derivatives under dilation; its growth as the boundary approaches explains why interior control alone gives no uniform boundary derivative bound. (math.ucdavis.edu)

For nonnegative harmonic functions, Harnack’s inequality compares values rather than derivatives. On a connected domain $\Omega$, every nonempty compact set $K\subset\Omega$ admits a constant $C=C(K,\Omega)\ge1$ such that

for every nonnegative harmonic $u$. Crucially, $C$ depends on the geometry, not on the function. Local comparisons on balls propagate through a Harnack chain: a finite sequence of overlapping balls, lying inside the domain, connecting the points being compared. Connectedness permits such chains, and compactness permits finitely many comparisons to control $K$. Consequently, a nonnegative harmonic function is either identically zero or strictly positive throughout a connected domain. (math.stanford.edu)

Harnack’s convergence principle states that a pointwise increasing sequence $u_j$ of harmonic functions on a connected domain has exactly two possibilities: it converges uniformly on every compact subset to a finite harmonic function, or $u_j(x)\to+\infty$ at every point. Boundedness of $u_j(a)$ at just one interior point rules out the second possibility. The mechanism is to apply Harnack’s inequality to the nonnegative differences $u_k-u_j$: convergence at $a$ then gives uniform Cauchy bounds on compact sets. No nonnegativity assumption on the original sequence is necessary, since one can replace it by $u_j-u_1$. (axler.net)

Liouville’s theorem states that a bounded harmonic function on all of $\mathbb R^n$ is constant: the first-derivative estimate on arbitrarily large balls forces its gradient to vanish as $R\to\infty$. One-sided boundedness also suffices. After changing sign and adding a constant, such a function becomes nonnegative; Harnack comparisons on expanding balls force its values at any two fixed points to agree. More generally, locally uniform convergence preserves harmonicity. The limit is continuous and inherits the mean-value property by passing limits through integrals on compactly contained balls; the converse mean-value theorem then applies. Interior estimates applied to differences further show locally uniform convergence of every derivative. (math.ucdavis.edu)

Harmonic Conjugates and Complex Analysis

Let $f=u+iv$ be a holomorphic function on a planar domain $D\subset\mathbb C$, with $z=x+iy$. The Cauchy–Riemann equations give $u_x=v_y$ and $u_y=-v_x$. Holomorphic functions have smooth components, so mixed partial derivatives commute:

Thus both components are harmonic. A real-valued function $v$ for which $u+iv$ is holomorphic is called a harmonic conjugate of $u$; merely requiring both functions to be harmonic is insufficient. For example, $u(x,y)=xy$ has conjugate $v(x,y)=(y^2-x^2)/2$, giving $f(z)=-iz^2/2$. (math.berkeley.edu)

Conversely, given a real-valued harmonic function $u$, finding its conjugate means solving $v_x=-u_y$, $v_y=u_x$. Introduce the differential form $\omega=-u_y\,dx+u_x\,dy$. The compatibility condition is precisely harmonicity: the derivative with respect to $y$ of the coefficient of $dx$ equals the derivative with respect to $x$ of the coefficient of $dy$, since $-u_{yy}=u_{xx}$. On a sufficiently small disk, this makes the line integral

independent of the path within the disk, and differentiation recovers the required equations. The same construction works throughout a simply connected domain. On any connected domain, two conjugates of the same $u$ differ by a real constant, because both partial derivatives of their difference vanish. (math.berkeley.edu)

On a domain that is not simply connected, the exact global criterion is that every period vanish:

for every closed piecewise smooth curve $\gamma$ in the domain. This is the path-independence condition for the preceding construction. For $u(z)=\log|z|=\tfrac12\log(x^2+y^2)$ on $\mathbb C\setminus\{0\}$, the form becomes

Along the counterclockwise unit circle $z=e^{it}$, its integral is $\int_0^{2\pi}dt=2\pi$, excluding a single-valued global conjugate. Locally the conjugate is an argument of $z$, and $u+iv$ is a branch of the complex logarithm. A slit plane admits such a branch, whereas continuing the argument once around the origin changes it by $2\pi$. Non-simple connectivity permits this obstruction but does not force it for every harmonic function. (math.ucla.edu)

Planar harmonicity is also preserved under holomorphic substitution. If $\phi:D\to G$ is holomorphic and $u$ is twice continuously differentiable on $G$, the chain rule and Cauchy–Riemann equations yield

Consequently, harmonic $u$ gives harmonic $u\circ\phi$, even where $\phi'$ vanishes. When $\phi$ is a bijective conformal map, its inverse is holomorphic as well, so harmonic functions transfer in both directions. This allows planar boundary-value problems to be moved to a disk or half-plane: boundary data are transported by the inverse map, the transformed problem is solved there, and its solution is pulled back by composition. Continuous boundary attainment requires appropriate boundary extension of the map; interior conformality alone supplies only the interior correspondence. (math.toronto.edu)

Boundary-Value Problems and Integral Representations

The Dirichlet boundary-value problem on a bounded domain $\Omega\subset\mathbb R^n$ asks, for prescribed $f\in C(\partial\Omega)$, for a function $u\in C^2(\Omega)\cap C(\overline\Omega)$ satisfying

The maximum principle guarantees uniqueness, but existence with continuous boundary attainment depends on the domain. For the ball $B_R(0)$, the solution is given explicitly by the Poisson kernel:

where $\sigma_{n-1}$ is the surface area of the unit sphere and $dS$ is unnormalized surface measure. This positive kernel integrates to one; as $x$ approaches a boundary point, its mass concentrates near that point, yielding the prescribed continuous boundary value. (web.stanford.edu)

On the unit disk, writing $x=re^{i\theta}$, the formula becomes

For continuous $f$, this defines a harmonic function extending continuously to the closed disk; moreover, the radial boundary values converge uniformly to $f$ as $r\uparrow1$. For example, $f(e^{it})=\cos t$ gives $u(re^{i\theta})=r\cos\theta$, the Cartesian coordinate function $u(x,y)=x$. Boundary continuity is sufficient here: differentiability of $f$ is not required. (math.berkeley.edu)

For a bounded domain with sufficiently smooth boundary, a Dirichlet Green’s function encodes the geometry in a more general integral representation. Using the sign convention

where $\delta_x$ denotes a unit point source, Green’s identity gives, for a sufficiently regular harmonic $u$,

Here $\nu_\xi$ is the outward unit normal. Thus the domain’s Poisson kernel is $-\partial_{\nu_\xi}G_\Omega$. One constructs $G_\Omega(x,y)=\Phi(y-x)-h_x(y)$, where $\Phi$ is a fundamental solution of $-\Delta$ and $h_x$ is harmonic with boundary values $\Phi(\xi-x)$. The correction $h_x$, unlike the universal singularity $\Phi$, depends on the boundary’s shape. (web.stanford.edu)

Oskar Perron’s method, introduced in 1923, constructs solutions without an explicit kernel. A continuous subharmonic function satisfies the mean-value inequality: its value at a ball’s center does not exceed its spherical average. Perron takes the supremum of such competitors $v\in C(\overline\Omega)$ satisfying $v\le f$ on $\partial\Omega$. Harmonic replacement on interior balls shows that the resulting envelope is harmonic inside $\Omega$; boundary attainment is a separate question. A barrier at $\zeta$ is a continuous superharmonic function, positive on $\overline\Omega\setminus\{\zeta\}$ and zero at $\zeta$. Barriers characterize regular boundary points, where the Perron solution attains every continuous datum. An exterior tangent ball meeting $\overline\Omega$ only at $\zeta$ supplies a barrier. By contrast, on $0<|z|<1$, the continuous boundary data zero on the circle and one at the isolated puncture admit no continuous harmonic extension: boundedness makes the puncture removable, and uniqueness on the full disk forces the extension to be zero. (math.mcgill.ca)

Dirichlet conditions prescribe $u$; Neumann conditions prescribe its outward normal derivative $\partial_\nu u=g$. For a bounded connected domain with smooth boundary, harmonicity imposes the compatibility condition

by the divergence theorem. With sufficiently regular compatible data, the Neumann solution exists and is unique up to an additive constant; a normalization such as $\int_\Omega u\,dx=0$ fixes that constant. Mixed conditions prescribe values on one boundary portion and normal derivatives on another. A Dirichlet portion of positive surface measure removes the constant ambiguity, although changes of boundary-condition type can reduce regularity near their junctions. (sites.math.duke.edu)

Dirichlet Energy and Weak Harmonicity

For a real-valued function $u$ on an open set $\Omega\subset\mathbb R^n$, the Dirichlet energy is

Its natural finite-energy setting is the Sobolev space $H^1(\Omega)$, consisting of square-integrable functions whose first weak derivatives are square-integrable. For a smooth $u$, consider variations $u+t\varphi$, where $\varphi\in C_c^\infty(\Omega)$ is a smooth, compactly supported test function. Differentiating and applying integration by parts gives

Thus stationarity under every such variation is equivalent to $\Delta u=0$: Laplace’s equation is the Euler–Lagrange equation of this energy. Compact support eliminates the boundary term, so these variations leave the prescribed boundary values unchanged. (ocw.mit.edu)

Harmonicity implies not merely stationarity but global minimization within the appropriate boundary class. Suppose $u\in H^1(\Omega)$ is weakly harmonic and a competitor has the form $v=u+w$, with $w\in H_0^1(\Omega)$, the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$. The weak identity extends to $w$ by density, giving the exact decomposition

Consequently $E[v]\geq E[u]$. On a bounded domain, equality forces $w=0$ through the Poincaré inequality, establishing uniqueness. On a bounded Lipschitz domain, $w\in H_0^1(\Omega)$ means precisely that $u$ and $v$ have the same Sobolev boundary trace, rather than necessarily identical pointwise boundary values. (ocw.mit.edu)

A function $u\in H_{\mathrm{loc}}^1(\Omega)$ is weakly harmonic when

This weak formulation requires only first weak derivatives. A distinct, initially broader formulation assumes merely $u\in L_{\mathrm{loc}}^1(\Omega)$ and asks that

The latter states that $\Delta u=0$ as a distribution, without assuming an integrable gradient. For functions in $H_{\mathrm{loc}}^1$, the two formulations agree by the definition of weak derivatives. (wwwp2.math.uni-hamburg.de)

Weyl’s lemma, established by Hermann Weyl in 1940, states that any locally integrable distributionally harmonic function agrees almost everywhere with a smooth, classically harmonic function. The qualification matters: altering values on a set of measure zero does not change the integral identity, but may destroy continuity of the chosen pointwise representative. A proof uses mollification: convolving $u$ with a smooth approximate identity gives $u_\varepsilon$, which satisfies $\Delta u_\varepsilon=0$ wherever its convolution kernel remains inside $\Omega$. For radial kernels, the mean-value property leaves smooth harmonic functions unchanged by convolution. Commuting two such convolutions shows that sufficiently small mollifications coincide locally; their local $L^1$ convergence to $u$ then identifies the smooth harmonic representative. This is an interior result and imposes no boundary continuity. (math.mcgill.ca)

Variational existence requires a nonempty admissible class and control of minimizing sequences—not simply that $E\geq0$. For any bounded open $\Omega$ and prescribed extension $g\in H^1(\Omega)$, minimize over $g+H_0^1(\Omega)$. The Poincaré inequality controls the $L^2$ norm of $u-g$ by its gradient, yielding boundedness in $H^1$; weak compactness and weak lower semicontinuity of the energy produce a unique minimizer. Its first variation vanishes, and Weyl’s lemma supplies a classical harmonic representative. No boundary smoothness is needed for this affine-space construction. A Lipschitz boundary is a standard sufficient hypothesis for interpreting the condition through boundary traces. Arbitrary continuous boundary data need not admit an $H^1$ extension: Gantumur’s notes recount Friedrich Prym’s 1871 example on a disk where continuous data have no finite-energy extension. (math.mcgill.ca)

Harmonic Polynomials and Spherical Expansions

A polynomial $P$ on $\mathbb R^n$, with $n\geq2$, is homogeneous of degree $\ell$ if $P(tx)=t^\ell P(x)$ for every $t>0$; equivalently, every nonzero monomial has total degree $\ell$. If also $\Delta P=0$, it is a homogeneous harmonic polynomial. Its restriction $Y=P|_{S^{n-1}}$ to the unit sphere is a spherical harmonic of degree $\ell$, and homogeneity gives $P(r\omega)=r^\ell Y(\omega)$. Constants have degree zero, coordinate functions have degree one, and $x_1^2-x_2^2$ and $x_1x_2$ provide degree-two examples. The term “spherical harmonic” does not generally mean a function satisfying the intrinsic Laplace equation on the sphere: smooth intrinsically harmonic functions on the entire sphere are constant, whereas spherical harmonics of positive degree have nonzero Laplacian eigenvalues. (axler.net)

With the convention $\Delta=\operatorname{div}\nabla$, the Euclidean Laplacian in polar coordinates $x=r\omega$ is

where $\Delta_{S^{n-1}}$ is the sphere’s Laplace–Beltrami operator. Substituting $r^\ell Y(\omega)$ yields

Thus degree-$\ell$ spherical harmonics satisfy

These are angular eigenfunctions; the negative sign reflects the stated operator convention. (cis.upenn.edu)

Different degrees are orthogonal for the inner product $\langle f,g\rangle=\int_{S^{n-1}}f\overline g\,dS$: integration by parts makes the spherical Laplacian self-adjoint, and its eigenvalues are distinct across degrees. Choosing an orthonormal basis $Y_{\ell,j}$, $1\leq j\leq d_\ell$, in each degree gives every $f\in L^2(S^{n-1})$ the expansion

Convergence here is in the $L^2$ norm, not necessarily pointwise. Completeness follows from the decomposition of homogeneous polynomials into terms $|x|^{2k}H_{\ell-2k}(x)$, with each $H_{\ell-2k}$ harmonic, together with density of polynomial restrictions on the sphere. (axler.net)

For a harmonic function in a ball centered at the origin, the corresponding interior expansion is

with uniform convergence on compact subsets of the ball. Each degree therefore acquires the radial factor $r^\ell$. If $u$ extends continuously to the boundary of a radius-$R$ ball and its boundary values $f(R\omega)$ have coefficients $c_{\ell,j}$, then $a_{\ell,j}=R^{-\ell}c_{\ell,j}$. The factors $(r/R)^\ell$ suppress higher angular degrees away from the boundary. (axler.net)

In two dimensions, the angular basis consists of $1$, $\cos(\ell\theta)$, and $\sin(\ell\theta)$, so the expansion becomes a radially weighted Fourier series:

In three dimensions, each degree has $2\ell+1$ basis functions, conventionally $Y_{\ell m}$ with $-\ell\leq m\leq\ell$. Separation of variables gives radial solutions $r^\ell$ and $r^{-\ell-1}$; regularity at the center excludes the latter. Outside a sphere enclosing a localized charge distribution, a potential tending to zero at infinity instead has a multipole expansion in $r^{-\ell-1}Y_{\ell m}$. Degrees zero, one, and two are respectively the monopole, dipole, and quadrupole terms, decaying as $r^{-1}$, $r^{-2}$, and $r^{-3}$. (cis.upenn.edu)

Kelvin Transform, Reflection, and Isolated Singularities

For $n\ge2$, inversion in the unit sphere is the map $I(x)=x/|x|^2$, defined for $x\ne0$. It satisfies $I(I(x))=x$ and $|I(x)|=|x|^{-1}$, exchanging the punctured unit ball with the exterior of the closed unit ball. If $u$ is defined on an open set $\Omega\subset\mathbb R^n\setminus\{0\}$, its Kelvin transform is

For $u\in C^2(\Omega)$, differentiation gives the transformation law

Thus $K$ preserves harmonicity and is its own inverse. In dimension two the prefactor is $1$; in higher dimensions it is essential. For example, when $n=3$, $K[1](x)=1/|x|$, so inversion converts a constant harmonic function into one with an isolated singularity. (matthewhr.wordpress.com)

The Schwarz reflection principle extends harmonic functions across a flat boundary by odd reflection. Write $x=(x',t)$, and suppose $u$ is harmonic in the upper half-ball $B_R^+=\{x:|x|<R,\ t>0\}$, continuous up to its flat face, and zero there. Then

is harmonic throughout $B_R$. These hypotheses require neither prescribed normal derivatives nor prior differentiability across the boundary: the theorem supplies interior smoothness of the extension. The zero boundary condition makes the odd extension continuous; odd reflection of arbitrary nonzero boundary data does not have this property. (axler.net)

An isolated singularity at $a$ is removable if a function harmonic on $B_R(a)\setminus\{a\}$ extends harmonically to $B_R(a)$. Local boundedness near $a$, meaning $|u(x)|\le M$ in some punctured neighborhood, is sufficient. More generally, removability holds under the weaker growth conditions

The extended value is determined by the limiting behavior of $u$, rather than by an arbitrary assignment at the missing point. (math.uni-hamburg.de)

The exceptional growth rates arise directly from radial solutions of Laplace’s equation. Writing $u(x)=f(r)$, with $r=|x-a|>0$, gives

and hence

When $B\ne0$, these functions are harmonic away from $a$ but have nonremovable singularities there. In particular, $1/r$ in three dimensions and $\log(1/r)$ in two dimensions demonstrate why harmonicity on a punctured domain does not by itself imply extendibility. (math.uni-hamburg.de)

Bôcher’s theorem, named for Maxime Bôcher, shows that nonnegativity excludes more complicated isolated singularities. If $u\ge0$ is harmonic on $B_R(a)\setminus\{a\}$, there are a constant $c\ge0$ and a function $h$ harmonic on the whole ball such that

Changing the reference radius in the logarithm only changes an additive constant in $h$. The singularity is removable exactly when $c=0$; otherwise its leading term is radial despite the possible angular dependence of $h$. (axler.net)

Boundary Limits and Harmonic Function Spaces

Let $B=\{x\in\mathbb R^n:|x|<1\}$, with $n\ge2$, and let $S=\partial B$. A radial limit at $\zeta\in S$ is $\lim_{r\uparrow1}u(r\zeta)$, which tests a single approach path. A nontangential limit requires the same limiting value as $x\to\zeta$ within each region

These regions exclude approaches that become arbitrarily tangential to the sphere. Nontangential convergence therefore implies radial convergence, but radial convergence alone is weaker. Neither follows merely from interior smoothness: boundary behavior needs additional size or integrability hypotheses. This distinction also applies to bounded harmonic functions, whose almost-everywhere boundary values need not form a continuous boundary function. (axler.net)

With normalized surface measure $d\sigma$, so that $\sigma(S)=1$, the harmonic Hardy space $h^p(B)$, for $1\le p<\infty$, consists of harmonic functions satisfying

For $p=\infty$, the condition is boundedness on $B$, with norm $\sup_B|u|$. Unlike holomorphic Hardy spaces on the disk, these spaces impose no holomorphicity requirement. Their defining bound controls entire families of spherical slices, rather than individual boundary paths. (axler.net)

The Poisson integral gives a precise description of these spaces. For $1<p\le\infty$, every $u\in h^p(B)$ has a unique representation

and $\|u\|_{h^p}=\|f\|_{L^p}$. At $p=1$, the correct boundary object is instead a finite signed or complex Borel measure $\mu$:

Thus an $h^1$ function need not be the Poisson integral of an integrable boundary function. (axler.net)

Representative Fatou-type results assert that Poisson integrals of finite measures have finite nontangential limits almost everywhere with respect to surface measure. If $d\mu=f\,d\sigma+d\mu_s$, where $\mu_s$ is singular, the limit is $f$; the singular part is invisible to these almost-everywhere values. On the disk, for example, the point mass at $1$ produces

Its boundary limit is zero except at $1$, although $u(0)=1$, and $u(r)=(1+r)/(1-r)$ diverges there. This directly illustrates both failure of continuous extension and failure to reconstruct a general $h^1$ function from its almost-everywhere boundary values alone. (fac.iitg.ac.in)

The harmonic Bergman space $b^p(B)$ instead uses the volume norm $\|u\|_{b^p}=(\int_B|u|^p\,dx)^{1/p}$. For $p=2$, it is a closed subspace of $L^2(B)$, hence a Hilbert space. The mean-value property and Cauchy–Schwarz inequality give

so point evaluation is bounded. Consequently, $b^2(B)$ is a reproducing-kernel Hilbert space: a unique kernel section $R_B(x,\cdot)$ reproduces each value through

This kernel represents evaluation using interior volume integration, rather than the boundary integration used by the Poisson kernel. (library.slmath.org)

Brownian Motion and Harmonic Measure

Standard Brownian motion $B_t$ in $\mathbb R^n$, started at $x$, has independent coordinate increments with mean zero and variance $t$. Its infinitesimal generator is $\tfrac12\Delta$: for a smooth, compactly supported test function $v$,

The factor $\tfrac12$ comes from the variance normalization. Thus Laplace’s equation identifies functions with zero infinitesimal expected change along Brownian trajectories. More explicitly, Itô’s formula gives, while the path remains inside the domain,

For harmonic $u$, the drift term vanishes, leaving only the stochastic-integral term. (cims.nyu.edu)

Let $D\subset\mathbb R^n$ be a domain and let $\tau_D=\inf\{t\geq0:B_t\notin D\}$ be its exit time. If $u$ is harmonic in $D$, then $u(B_t)$ is a local martingale before $\tau_D$; this statement does not require a boundary extension of $u$. To obtain an ordinary martingale, one can first stop at the exit time $\tau_U$ of an open set $U$ whose closure is compactly contained in $D$. The process $u(B_{t\wedge\tau_U})$ is then bounded, and optional stopping yields

For a bounded domain $D$, if $u\in C(\overline D)$ is harmonic in $D$, boundedness and almost-sure finite exit allow passage to the limit, giving $u(x)=\mathbb E_x[u(B_{\tau_D})]$. Without boundedness or suitable uniform integrability, a local-martingale assertion alone does not justify this identity. (math.uchicago.edu)

For bounded $D$, continuous paths ensure that $B_{\tau_D}\in\partial D$ almost surely. The harmonic measure with pole $x\in D$ is the boundary probability distribution

For bounded Borel boundary data $f$, define

This function is harmonic inside $D$, even when the boundary is irregular. Indeed, the strong Markov property, applied at the exit time from any ball centered at $x$ and contained in $D$, expresses $u(x)$ as the average of $u$ at that ball’s exit point. Rotational invariance makes this exit distribution uniform on the sphere, recovering the mean-value characterization of harmonicity. (math.uchicago.edu)

Harmonic measure weights the boundary according to exit probabilities, not generally according to surface area. In the unit disk $\mathbb D$, its density is the Poisson kernel:

At $z=0$, the exit angle is uniform, so an arc of angular length $\alpha$ has probability $\alpha/(2\pi)$. Away from the center, the density is larger near the boundary point closest to $z$. The usual Poisson integral is therefore precisely an average of boundary values against a Brownian exit distribution. (math.uchicago.edu)

Boundary regularity determines whether this stochastic extension attains prescribed continuous data. At a regular boundary point $\xi$, exit distributions concentrate near $\xi$ as the starting point approaches it from inside:

Consequently, $u(x)\to f(\xi)$ for continuous $f$. An exterior cone with vertex $\xi$ is a sufficient geometric condition for regularity. If every boundary point of a bounded domain is regular, the expectation formula supplies the unique solution in $C(\overline D)$ of the Dirichlet boundary-value problem. At an irregular point, interior harmonicity still holds, but convergence to the assigned boundary value can fail. (math.uchicago.edu)

Curved Spaces, Discrete Networks, and Computation

On a Riemannian manifold $(M,g)$, a twice continuously differentiable function is harmonic when $\Delta_g u=0$, where the Laplace–Beltrami operator is defined intrinsically as $\operatorname{div}_g\nabla_g$. In local coordinates,