Laplace's equation $\Delta u = 0$ is the prototype elliptic PDE. Every major result in elliptic theory — mean value properties, maximum principles, regularity, Harnack inequalities — appears first and most explicitly here, before being generalised to variable-coefficient operators on the [second-order elliptic equations page](/page/Second-Order%20Elliptic%20Equations). The Laplacian's full rotational symmetry forces the fundamental solution to be radial, enabling explicit representation formulas that are unavailable for general operators. [motivation] ## Motivation ### Physical Origins Laplace's equation arises whenever a physical quantity reaches equilibrium. If $u$ is the steady-state temperature in a region $\Omega$ with no heat sources, Fourier's law ($q = -k\nabla u$) and conservation of energy ($\nabla \cdot q = 0$) give $\Delta u = 0$. The same equation governs the gravitational potential outside a mass distribution, the electrostatic potential in a charge-free region, and the velocity potential of an irrotational incompressible flow. ### Why Explicit Formulas Matter For a general uniformly elliptic operator $Lu = -\sum a_{ij} \partial_{ij} u + \sum b_i \partial_i u + cu$, the theory relies on abstract tools — Sobolev spaces, Lax–Milgram, Fredholm theory — developed on the [elliptic page](/page/Second-Order%20Elliptic%20Equations). These tools are necessary because the variable coefficients $a_{ij}(x)$ destroy the symmetries that make explicit computation possible. For the Laplacian, the rotational invariance of $\Delta$ yields a radial fundamental solution, from which representation formulas, mean value identities, and the Poisson kernel follow by direct calculation. ### The Mean Value Property as the Organising Principle For the [heat equation](/page/Heat%20Equation), the mean value formula is a *consequence* of the fundamental solution, derived after considerable work. For Laplace's equation, the mean value property follows immediately from the Green's representation formula, and the entire qualitative theory — maximum principle, smoothness, Liouville, Harnack — flows from it by short arguments. This article is structured around that flow. [/motivation] ## Definitions The equation $\Delta u = 0$ appears throughout mathematics and physics, but its solutions have a distinctive character — they are infinitely smooth, satisfy averaging identities, and are completely determined by their [boundary](/page/Boundary) values. To study these solutions systematically, we need a name for them and a precise specification of the regularity we assume. [definition: Harmonic Function] Let $\Omega \subseteq \mathbb{R}^n$ be an [open set](/page/Open%20Set). A function $u \in C^2(\Omega)$ is **harmonic** in $\Omega$ if \begin{align*} \Delta u(x) := \sum_{i=1}^n \partial_{x_i}^2 u(x) = 0 \quad \text{for all } x \in \Omega. \end{align*} [/definition] The inhomogeneous version $-\Delta u = f$ is **Poisson's equation**. When $f \neq 0$, the source term breaks harmonicity, and the solution is constructed by [convolution](/page/Convolution) with the fundamental solution. ## Fundamental Solution To solve the Dirichlet problem $\Delta u = 0$ in $\Omega$ with $u = g$ on $\partial\Omega$, or the Poisson problem $-\Delta u = f$, we need a way to invert the Laplacian — to go from the data ($g$ or $f$) to the solution $u$. In one dimension, inverting $-u'' = f$ amounts to integrating twice, and the Green's function is built from piecewise linear functions. In higher dimensions, the role of the Green's function is played by the **fundamental solution** $\Phi$, the response to a point source $\delta_0$. Without it, we have no explicit representation of solutions and must resort to purely abstract existence arguments. The key to explicit formulas is that the rotational symmetry of $\Delta$ forces $\Phi$ to be radial: writing $\Phi(x) = v(|x|)$ and imposing $\Delta v = 0$ for $r > 0$ gives the ODE $(r^{n-1} v')' = 0$, whose solutions are $v(r) = ar^{2-n} + b$ for $n \geq 3$ and $v(r) = a\log r + b$ for $n = 2$. The normalisation constant is fixed by requiring $-\Delta\Phi = \delta_0$ in distributions. [quotetheorem:566] The [distributional](/page/Distribution) identity $-\Delta\Phi = \delta_0$ is established by testing against $\varphi \in C_c^\infty(\mathbb{R}^n)$, excising a ball $B(0, \varepsilon)$, applying Green's identity on the complement, and evaluating the boundary integrals as $\varepsilon \to 0$. ## Green's Formulas and Representation The fundamental solution $\Phi$ solves $-\Delta\Phi = \delta_0$, but a physical problem asks for a solution in a bounded domain $\Omega$ with prescribed boundary values — not on all of $\mathbb{R}^n$ with a point source. To connect $\Phi$ to boundary value problems, we need a formula that expresses the value of a function at an interior point in terms of boundary data and the Laplacian. Green's second identity ([integration by parts](/theorems/210) applied twice) provides exactly this, with $\Phi$ serving as the integration kernel. [quotetheorem:567] For harmonic functions ($\Delta u = 0$), the volume integral vanishes, and $u(x)$ is determined entirely by boundary data — the mathematical expression of the physical principle that a steady-state field in $\Omega$ is fully determined by what happens on $\partial\Omega$. When $\Delta u = f \neq 0$ on all of $\mathbb{R}^n$ (with no boundary), the representation simplifies further. The boundary [integrals](/page/Integral) disappear entirely, and the solution is a pure convolution with $\Phi$. This gives the most direct route to solving Poisson's equation — but only on the whole space, where there is no boundary to worry about. On bounded domains, the boundary integrals persist and require the Green's function machinery of a later section. [quotetheorem:568] [example: Poisson's Equation In Three Dimensions] Let $n = 3$ and $f(x) = \mathbb{1}_{B(0,1)}(x)$ (extended smoothly). The solution to $-\Delta u = f$ on $\mathbb{R}^3$ is $u(x) = \frac{1}{4\pi} \int_{B(0,1)} \frac{1}{|x - y|} \, d\mathcal{L}^3(y)$. For $|x| > 1$, the integral equals $\frac{|B(0,1)|}{4\pi|x|} = \frac{1}{3|x|}$ by Newton's shell theorem — the potential outside a uniform ball looks like a point mass. For $|x| < 1$, the computation requires splitting the integral at $|y| = |x|$ and yields $u(x) = \frac{1}{6}(1 - |x|^2/3)$, a quadratic whose Laplacian is indeed the constant $-1$. [/example] ## Mean-Value Formulas The Green's representation formula expresses $u(x)$ as a complicated integral involving both $u$ and $\partial_\nu u$ on $\partial\Omega$, weighted by $\Phi$ and $\partial_\nu \Phi$. For a general domain, this is the best we can do. But when $\Omega$ is a ball centred at $x$ and $u$ is harmonic, everything simplifies dramatically: the fundamental solution is constant on $\partial B(x, r)$ (because $|x - y| = r$ there), the normal derivative integral involving $\partial_\nu u$ vanishes by the divergence theorem ($\int \partial_\nu u = \int \Delta u = 0$), and the remaining integral collapses to a pure average of $u$ over the sphere. This simplification is the mean value property, and it is the single most powerful tool in the classical theory. [quotetheorem:31] The converse is also true: a [continuous](/page/Continuity) function satisfying the ball mean value property for every ball is necessarily $C^\infty$ and harmonic. This characterisation provides a way to verify harmonicity without computing second [derivatives](/page/Derivative). [quotetheorem:910] [example: Mean Value Property Of A Radial Function] Let $n = 3$ and $u(x) = 1/|x|$ for $x \neq 0$. This is harmonic on $\mathbb{R}^3 \setminus \{0\}$. Fix $x_0 = (2, 0, 0)$ and $r = 1$. The sphere mean value property asserts $u(x_0) = 1/2$ equals the average of $1/|y|$ over $\partial B(x_0, 1)$. This is not geometrically apparent — the sphere includes points where $|y| = 1$ (so $u = 1$) and $|y| = 3$ (so $u = 1/3$) — but the weighted average comes out exactly to $1/2$. [/example] ## Properties of Harmonic [Functions](/page/Function) The logical chain: mean value → maximum principle → uniqueness, and independently mean value → smoothness → derivative estimates → Liouville, Harnack. ### Maximum Principle Given a harmonic function on a bounded domain, where does it achieve its largest value? For a generic continuous function, the maximum could be anywhere — interior or boundary. But the mean value property imposes a severe constraint: if a harmonic function achieves its maximum at an interior point $x_0$, then $u(x_0)$ equals the average of $u$ over every ball centred at $x_0$, and an average cannot equal the maximum unless the function is constant throughout the ball. This forces the maximum onto the boundary. [quotetheorem:32] The practical consequence is that bounding a harmonic function on $\overline{\Omega}$ reduces to bounding it on $\partial\Omega$ — the foundation of *a priori* estimates in elliptic theory. ### Uniqueness The Dirichlet problem asks: given boundary data $g$ on $\partial\Omega$ and a source $f$ in $\Omega$, is there a unique solution? Without uniqueness, the problem is ill-posed — different solutions could give different predictions from the same data. The maximum principle resolves this immediately: if two solutions share the same boundary data, their difference is harmonic with zero boundary values, and the maximum principle forces the difference to vanish. [quotetheorem:33] The same two-line argument (difference → maximum principle → zero) generalises to variable-coefficient operators on the [elliptic page](/page/Second-Order%20Elliptic%20Equations). ### Regularity The definition of harmonic requires $u \in C^2$, but nothing more. Is a harmonic function always smoother than the minimum $C^2$ we assumed? The answer is dramatically yes: harmonicity forces infinite differentiability and even real-analyticity. The mechanism is the mean value property: mollifying $u$ with a radial kernel and using the MVP shows that $u_\varepsilon = u$ exactly (not approximately), so $u$ inherits the infinite smoothness of the [mollifier](/page/Standard%20Mollifier). [quotetheorem:36] Real-analyticity is a strictly stronger conclusion: [quotetheorem:40] The proof uses the [derivative estimates](/theorems/37) combined with Stirling's formula: the bound $|D^\alpha u(x_0)| \le C^{|\alpha|+1}\,\alpha!$ on a definite ball is exactly the Cauchy-type condition that guarantees convergence of the Taylor [series](/page/Series). No other second-order PDE has this property — solutions to the [heat equation](/page/Heat%20Equation) are $C^\infty$ but not real-analytic. ### Derivative Estimates Knowing that harmonic functions are smooth is qualitative. For compactness arguments, Schauder estimates, and the proof of Liouville's theorem, we need *quantitative* control: how large can the $k$-th derivative be at a point, given only information about the function values? Without such bounds, we cannot pass to [limits](/page/Limit) in [sequences](/page/Sequence) of harmonic functions or extract convergent subsequences. The mean value property gives exactly the needed estimate by expressing $u(x_0)$ (and hence, by differentiating, $D^\alpha u(x_0)$) as an integral over a ball. [quotetheorem:37] The estimate $|D^\alpha u(x_0)| \leq C_k r^{-k} \|u\|_{L^\infty(B(x_0,r))}$ (the $L^\infty$ form) is the elliptic analogue of Cauchy's inequality in complex analysis; it is the engine behind both Liouville's theorem and the convergence theory for sequences of harmonic functions. ### Liouville's Theorem Can a non-constant harmonic function be defined on all of $\mathbb{R}^n$? Certainly — $u(x) = x_1$ is harmonic and non-constant. But it is unbounded. If we additionally require boundedness, the derivative estimates from the previous section force $\nabla u$ to vanish: the bound $|\nabla u(x)| \leq C r^{-1} \|u\|_{L^\infty}$ holds for *every* $r > 0$, and sending $r \to \infty$ makes the right side zero. The conclusion is that entire bounded harmonic functions must be constant. [quotetheorem:38] [example: Sharpness Of The Boundedness Hypothesis] The function $u(x, y) = xy$ is harmonic on $\mathbb{R}^2$ ($\partial_{xx}(xy) + \partial_{yy}(xy) = 0$) and non-constant. There is no contradiction with Liouville's theorem because $u$ is unbounded. More generally, any harmonic polynomial of degree $\geq 1$ is unbounded and non-constant. One can sharpen Liouville: a harmonic function on $\mathbb{R}^n$ with polynomial growth of degree $\leq k$ must be a harmonic polynomial of degree $\leq k$. [/example] ### Harnack's Inequality The maximum principle bounds a harmonic function by its boundary values, but it says nothing about how the function varies *within* the interior. A non-negative harmonic function could, in principle, be very large at one interior point and very small at a nearby point. Harnack's inequality rules this out: on any compact subset, the ratio $\sup u / \inf u$ is bounded by a universal constant depending only on the geometry and dimension. This oscillation control is essential for convergence theorems (e.g., showing that a locally bounded sequence of harmonic functions has a convergent subsequence). [quotetheorem:41] The generalisation to variable-coefficient operators (Moser's Harnack inequality, via the De Giorgi–Nash–Moser technique) is one of the deepest results in elliptic theory; see the [elliptic page](/page/Second-Order%20Elliptic%20Equations). ## Green's Function and the Poisson Kernel The Green's representation formula requires both $u$ and $\partial_\nu u$ on $\partial\Omega$, but the Dirichlet problem specifies only $u$. To eliminate $\partial_\nu u$, one constructs a **Green's function** $G(x, y) = \Phi(y - x) - \phi^x(y)$, where the corrector $\phi^x$ is harmonic in $y$ and chosen so that $G(x, \cdot) = 0$ on $\partial\Omega$. The normal derivative of $G$ on the boundary gives the **Poisson kernel**. [quotetheorem:909] The symmetry $G(x,y) = G(y,x)$ is not obvious from the definition (which treats $x$ and $y$ asymmetrically), and the proof requires Green's second identity on a doubly-excised domain. Part (iv) shows that the Green's function simultaneously handles both the source term ($f$) and the boundary data ($g$): setting $f = 0$ gives the Poisson integral, while setting $g = 0$ gives the volume potential. For a general domain, the corrector exists but has no closed form. For the ball $B(0, r)$, it is constructed explicitly via Kelvin reflection: the image of $x$ under inversion in $\partial B(0, r)$ is $\tilde{x} = r^2 x / |x|^2$. [quotetheorem:576] The Poisson kernel $K(x, y) = -\frac{\partial G}{\partial\nu_y}(x,y) = \frac{r^2 - |x|^2}{n\omega_n r\,|y - x|^n}$ is positive, integrates to $1$, and concentrates as $x \to \partial B$ — the properties of an approximate identity. [example: Poisson Integral On The Disk] For $n = 2$ and $r = 1$, parametrise $\partial B(0, 1)$ by $y = e^{i\theta}$ and write $x = \rho e^{i\phi}$ with $\rho < 1$. The Poisson kernel becomes $K(\rho, \theta - \phi) = \frac{1 - \rho^2}{2\pi(1 - 2\rho\cos(\theta - \phi) + \rho^2)}$. If $g(\theta) = \cos(k\theta)$, the Poisson integral evaluates to $u(\rho, \theta) = \rho^k \cos(k\theta)$ — the real part of $z^k$, connecting the Poisson integral to complex analysis and [Fourier series](/page/Fourier%20Series) on the circle. [/example] ## Energy Methods Everything so far has been based on the pointwise mean value property. But there is a completely different way to characterise harmonic functions: as minimisers of the **Dirichlet energy** $D[v] = \frac{1}{2}\int_\Omega |\nabla v|^2$. Among all functions with the same boundary values, the harmonic one has the smallest energy — its gradient field is, in a precise sense, the least "wasteful." This variational characterisation is historically the origin of the weak formulation of elliptic equations: Dirichlet proposed it in the 19th century, and the modern theory of [Sobolev spaces](/page/Sobolev%20Space) was built in large part to make the minimisation rigorous. [quotetheorem:577] The [Sobolev space](/page/Sobolev%20Space) framework makes the minimisation rigorous by replacing $C^2$ competitors with $W^{1,2}$ functions, and the [Lax-Milgram theorem](/theorems/91) generalises the result to non-symmetric operators — see the [elliptic page](/page/Second-Order%20Elliptic%20Equations) and the [calculus of variations page](/page/Calculus%20of%20Variations%20(PDEs)). ## References - Evans, L. C. (2010). *Partial Differential Equations* (2nd ed.). American Mathematical Society. - Gilbarg, D. and Trudinger, N. S. (2001). *Elliptic Partial Differential Equations of Second Order*. Springer. - Axler, S., Bourdon, P., and Ramey, W. (2001). *Harmonic Function Theory*. Springer.