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Harmonic function

In mathematics, a harmonic function is a twice continuously differentiable real-valued function that satisfies Laplace's equation, \nabla^2 u = 0, in an open domain of Euclidean space.^[1] These functions model equilibrium states in physical systems, such as steady-state temperature distributions in heat conduction, electrostatic potentials, and gravitational fields.^[2] Harmonic functions possess several fundamental properties that distinguish them from general smooth functions. They satisfy the mean value property, stating that the value at any interior point equals the average value over any sphere (or ball) centered at that point contained within the domain.^[1] Additionally, the maximum principle asserts that a non-constant harmonic function on a bounded domain attains its maximum and minimum values only on the boundary, implying that interior extrema indicate constancy.^[2] In the context of complex analysis, the real and imaginary parts of holomorphic (analytic) functions are harmonic, and conversely, every harmonic function in a simply connected domain is the real part of a holomorphic function unique up to an imaginary constant.^[3] The study of harmonic functions forms a cornerstone of potential theory, where they solve boundary value problems like the Dirichlet problem using tools such as the Poisson integral formula.^[3] Applications extend to diverse fields, including solving the heat equation in the steady-state limit, fluid dynamics for incompressible irrotational flows, and eigenvalue problems in geometry, such as those related to the Poincaré conjecture.^[2] Advanced properties, like Harnack's inequality, provide bounds on the growth of positive harmonic functions and ensure continuity up to the boundary under certain conditions.^[3]

Definition and Etymology

Etymology of the Term "Harmonic"

The term "harmonic" traces its linguistic roots to the ancient Greek word harmonia (ἁρμονία), denoting "harmony," "agreement," or "concord of sounds," particularly the pleasing fitting together of musical tones. This etymology stems from harmos (ἁρμός), meaning "joint" or "fitting," derived from the Proto-Indo-European root ar-, "to fit together." In the context of music, the concept gained mathematical significance through Pythagoras in the 6th century BCE, who is credited with discovering that consonant intervals produced by vibrating strings or other instruments correspond to simple integer ratios of their lengths or frequencies—such as 2:1 for the octave, 3:2 for the perfect fifth, and 4:3 for the perfect fourth—thus linking numerical proportions to auditory harmony.^[4]^[5] In the 18th century, the term entered mathematical discourse through studies of vibrations and wave propagation, where solutions often decomposed into superpositions of sinusoidal components analogous to musical overtones. Jean d'Alembert adopted harmonic notions in his 1747 memoir on the vibrating string, presenting the first published derivation of the one-dimensional wave equation and its general solution as a superposition of traveling waves, drawing implicit parallels to acoustic phenomena. Similarly, Leonhard Euler explored these ideas in works like his 1739 Tentamen novae theoriae musicae, which systematically analyzed musical intervals and consonance using ratios and vibrations, and extended them in the 1740s to fluid dynamics and acoustics, treating oscillatory motions in terms of harmonic components. In potential theory, the term was applied to functions satisfying Laplace's equation because they can be superposed like harmonic components in Fourier analysis, as developed by Fourier for heat problems and extended by Gauss to gravitational potentials.^[6]^[7] The transition to potential theory occurred in the 19th century, as mathematicians associated the term with functions exhibiting additive superposition properties akin to musical harmonics. Joseph Fourier's 1822 Théorie analytique de la chaleur employed trigonometric series expansions—later termed harmonic analysis—to solve heat conduction problems, highlighting the decomposition of arbitrary functions into harmonic-like terms. Carl Friedrich Gauss advanced this in his 1839–1840 investigations of terrestrial magnetism and gravitational attraction, studying functions satisfying Laplace's equation through their integral representations and superposition, which mirrored the composable nature of overtones. The specific phrase "harmonic function" for solutions to Laplace's equation appears in William Thomson's (later Lord Kelvin) works in the 1860s, with early uses in his 1863 paper on elastic spheroids and formalized in the 1867 Treatise on Natural Philosophy by Thomson and P. G. Tait.^[8]

Mathematical Definition

In mathematics, a harmonic function is a real-valued function u: \Omega \to \mathbb{R} defined on an open set \Omega \subset \mathbb{R}^n (with n \geq 2) that is twice continuously differentiable and satisfies Laplace's equation \Delta u = 0 in \Omega.^[9]^[10] The domain \Omega is required to be open, ensuring the equation holds locally in a neighborhood of each point in \Omega.^[11]^[12] The Laplace operator \Delta, also known as the Laplacian, is defined as \Delta u = \div([\nabla u](/page/Gradient)) = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}, where \nabla u denotes the gradient of u and \div its divergence; this assumes familiarity with basic vector calculus.^[11]^[1] In two dimensions, with coordinates (x, y), Laplace's equation takes the explicit form

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.

In three dimensions, with coordinates (x, y, z), it becomes

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.

These forms highlight the elliptic nature of the partial differential equation governing harmonic functions.^[9]^[10] Laplace's equation was first introduced by Pierre-Simon Laplace in 1782 within his work on celestial mechanics, where it arose in modeling gravitational potentials.^[13] This partial differential equation serves as the cornerstone for the theory of harmonic functions in the broader context of elliptic partial differential equations.^[11]

Examples

Classical Examples in Euclidean Space

In Euclidean space, the simplest examples of harmonic functions are the linear functions. In \mathbb{R}^2, consider u(x,y) = ax + by for constants a, b \in \mathbb{R}. To verify it is harmonic, compute the Laplacian:

\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 + 0 = 0.

This holds trivially since the second partial derivatives vanish.^[14] Next, quadratic harmonic functions in \mathbb{R}^2 arise as the real and imaginary parts of holomorphic functions, such as z^2 = (x + iy)^2. The real part is u(x,y) = x^2 - y^2, and the imaginary part is v(x,y) = 2xy. For u(x,y) = x^2 - y^2, the partial derivatives are \partial u / \partial x = 2x and \partial u / \partial y = -2y, so

\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2 + (-2) = 0.

Similarly for v(x,y) = 2xy, \partial v / \partial x = 2y and \partial v / \partial y = 2x, yielding \Delta v = 0 + 0 = 0. These are the degree-2 homogeneous harmonic polynomials in two variables.^[1] Another class of examples in \mathbb{R}^2 involves exponential functions, such as u(x,y) = e^x \cos y, which is the real part of the holomorphic function e^z = e^{x+iy}. The partial derivatives are \partial u / \partial x = e^x \cos y and \partial u / \partial y = -e^x \sin y, so the second derivatives are \partial^2 u / \partial x^2 = e^x \cos y and \partial^2 u / \partial y^2 = -e^x \cos y. Thus,

\Delta u = e^x \cos y + (-e^x \cos y) = 0.

The imaginary part v(x,y) = e^x \sin y satisfies \Delta v = 0 analogously.^[15] The logarithmic potential provides a fundamental example in \mathbb{R}^2, given by u(x,y) = \log \sqrt{x^2 + y^2} = \frac{1}{2} \log (x^2 + y^2), which is the fundamental solution to Laplace's equation away from the origin. In polar coordinates r = \sqrt{x^2 + y^2}, this simplifies to u(r) = \log r. The Laplacian in polar form is \Delta u = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}. Since u is radial, the \theta-term vanishes, and \partial u / \partial r = 1/r, so r \partial u / \partial r = 1 and \partial / \partial r (1) = 0, yielding \Delta u = 0 for r > 0.^[16] In \mathbb{R}^3, a fundamental example is the Newtonian potential u(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}} for (x,y,z) \neq (0,0,0), which satisfies [\Delta](/page/Delta) u = 0 away from the origin. In spherical coordinates with r = \sqrt{x^2 + y^2 + z^2}, u(r) = 1/r. The radial Laplacian is \Delta u = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right), where \partial u / \partial r = -1/r^2, so r^2 \partial u / \partial r = -1 and \partial / \partial r (-1) = 0, yielding \Delta u = 0 for r > 0.^[3] Linear functions such as u(x,y,z) = x are also harmonic in \mathbb{R}^3. The partial derivatives are \partial u / \partial x = 1, \partial u / \partial y = 0, and \partial u / \partial z = 0, so

\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0 + 0 + 0 = 0.

More generally, constant functions (degree 0) and linear functions (degree 1) restricted to the unit sphere S^2 yield the spherical harmonics Y_0^0 and Y_1^m (m = -1,0,1), which are harmonic as restrictions of homogeneous harmonic polynomials. These form the basis for higher-degree spherical harmonics solving Laplace's equation in spherical coordinates.^[17]^[18]

Examples from Physics and Applications

Harmonic functions arise naturally in physics as solutions to Laplace's equation, ∇²Φ = 0, which governs equilibrium states in various conservative fields without sources or sinks.^[19] In electrostatics, the electric potential Φ in charge-free regions is harmonic, satisfying Laplace's equation due to the absence of charge density.^[20] A classic example is the potential outside a grounded conducting sphere in the presence of a point charge, where the boundary condition of zero potential on the sphere leads to a harmonic solution expressed in spherical harmonics.^[21] In Newtonian gravity, the gravitational potential Φ in vacuum regions away from masses also obeys Laplace's equation, making it harmonic and enabling the use of multipole expansions for distant field approximations.^[22] This property ensures that gravitational fields in empty space are irrotational and divergence-free, facilitating analytical solutions for planetary or stellar configurations.^[10] Steady-state heat conduction in isotropic media without internal heat sources is modeled by the temperature distribution T satisfying Laplace's equation, where T is harmonic.^[23] Such problems often involve solving Dirichlet boundary value problems, prescribing fixed temperatures on boundaries to determine the equilibrium heat flow.^[24] For irrotational, incompressible, and inviscid fluid flows, the velocity potential φ is harmonic, as it satisfies ∇²φ = 0 derived from the continuity and irrotationality conditions.^[25] This framework applies to ideal fluid dynamics, such as airflow around streamlined bodies, where φ provides a scalar description of the velocity field ∇φ.^[26] Physical applications of harmonic functions frequently involve boundary value problems to match experimental conditions. In the Dirichlet problem, the harmonic function takes prescribed values on the boundary, as in specifying surface potentials for electrostatic shielding.^[24] The Neumann problem, conversely, prescribes the normal derivative on the boundary, corresponding to specified fluxes like heat or electric field normals in conduction or capacitance setups.^[27] In modern contexts, harmonic functions support image processing through techniques like harmonic inpainting, which smoothly fills missing regions by solving Laplace's equation with Dirichlet conditions from surrounding pixels, preserving overall image continuity.^[28] Similarly, in computer graphics, harmonic mappings enable smooth interpolation between shapes, using harmonic coordinates to deform meshes while minimizing distortion and ensuring C∞ smoothness for realistic animations.^[29]

Fundamental Properties

Mean Value Property

The mean value property is a fundamental characterizing feature of harmonic functions. If u: \Omega \to \mathbb{R} is harmonic on an open set \Omega \subset \mathbb{R}^n, then for every x \in \Omega and every radius r > 0 such that the closed ball \overline{B_r(x)} \subset \Omega,

u(x) = \frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)} u(y) \, d\sigma(y) = \frac{1}{|B_r(x)|} \int_{B_r(x)} u(y) \, dy,

where B_r(x) denotes the open ball of radius r centered at x, \partial B_r(x) its boundary sphere, |\cdot| denotes the respective surface measure or volume, d\sigma is the surface element, and dy is the Lebesgue measure. This property links the value of the function at the center of a ball to its average over the ball or its boundary sphere, reflecting the smoothing effect of solutions to Laplace's equation. The mean value property for harmonic functions, particularly over spheres, was first established by Carl Friedrich Gauss in 1840 as part of his work on potential theory.^[30] Gauss's theorem originally applied to the arithmetic mean of potentials on spherical surfaces, providing an early insight into the symmetry and averaging behavior inherent to harmonic functions. To outline a proof of the mean value property using the divergence theorem, consider the volume integral form first. Since u is harmonic, \Delta u = 0 in B_r(x). Apply the divergence theorem to the vector field u \nabla u:

\int_{B_r(x)} \nabla \cdot (u \nabla u) \, dy = \int_{\partial B_r(x)} u \frac{\partial u}{\partial n} \, d\sigma,

where n is the outward unit normal. The left side expands as \int_{B_r(x)} (u \Delta u + |\nabla u|^2) \, dy = \int_{B_r(x)} |\nabla u|^2 \, dy, since \Delta u = 0. For the sphere average, define m(r) = \frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)} u \, d\sigma. Differentiating under the integral and using the divergence theorem on \nabla u yields m'(r) = 0, implying m(r) is constant for $0 < r < R, and by continuity at r = 0, m(r) = u(x). The ball version follows by integrating the sphere average radially.^[24]^[11] The sphere and ball versions of the mean value property are equivalent for C^2 functions, but the ball version highlights greater symmetry in the averaging process, as it involves volume integrals that capture the full interior behavior without restricting to boundaries.^[31] This equivalence underscores the rotational invariance of harmonic functions in Euclidean space. Conversely, if a continuous function u: \Omega \to \mathbb{R} satisfies the mean value property over balls (or spheres) in \Omega, then u is harmonic (i.e., C^\infty and \Delta u = 0) in \Omega. The proof proceeds by showing that the mean value property implies \Delta u = 0 in the sense of distributions, and by elliptic regularity, u is C^2 and satisfies Laplace's equation pointwise. Specifically, for small balls, the property forces the second derivatives to balance such that the Laplacian vanishes, with continuity ensuring no singularities.^[32]^[31] This equivalence provides a characterization of harmonicity without directly invoking the Laplacian, useful in contexts like potential theory.

Maximum Principle

The strong maximum principle for harmonic functions states that if u is a non-constant harmonic function defined on a bounded connected open set \Omega \subset \mathbb{R}^n, then u cannot attain its maximum value at any interior point of \Omega; if it does, u must be constant throughout \Omega.^[33] This principle highlights the "smoothing" effect of harmonicity, preventing isolated peaks or valleys in the interior without the function being flat everywhere. A related weak maximum principle asserts that for a continuous real-valued harmonic function u on a bounded domain \Omega, the supremum of u over \Omega equals the maximum of u on the boundary \partial \Omega, i.e., \sup_{\Omega} u = \max_{\partial \Omega} u.^[34]^[33] A sketch of the proof for the strong maximum principle relies on the mean value property of harmonic functions. Suppose u attains its maximum M at an interior point x_0 \in \Omega. By the mean value property, u(x_0) equals the average of u over any sufficiently small ball B_r(x_0) \subset \Omega. Since u \leq M everywhere and u(x_0) = M, the average over B_r(x_0) is also M, implying u \equiv M on B_r(x_0) by the properties of averages for harmonic functions. By connectedness of \Omega and continuity of u, this constancy extends to all of \Omega, contradicting the assumption that u is non-constant.^[33] The minimum principle follows dually by applying the maximum principle to -u, which is also harmonic, ensuring that non-constant harmonic functions cannot attain interior minima either.^[34] These principles have key applications, particularly in establishing uniqueness for solutions to the Dirichlet problem. If u_1 and u_2 are two harmonic functions on a bounded domain \Omega that agree on \partial \Omega, then their difference w = u_1 - u_2 is harmonic with zero boundary values. By the weak maximum principle, \sup_{\Omega} |w| = \max_{\partial \Omega} |w| = 0, so w \equiv 0 and u_1 = u_2.^[34]^[33] To illustrate that these properties are specific to harmonic functions, consider non-harmonic examples like u(x) = x^2 on the interval [-1, 1]; while harmonic functions in one dimension are linear and satisfy the maximum principle, this quadratic function has second derivative 2 (not zero, so not harmonic) and attains its minimum at an interior point, but the negative -x^2 attains a maximum interiorly, violating the principle.^[35]

Advanced Analytic Properties

Regularity Theorem

The regularity theorem for harmonic functions asserts that any weak solution u to Laplace's equation \Delta u = 0 in an open set \Omega \subset \mathbb{R}^n is in fact smooth and real analytic in \Omega. Specifically, if u \in L^1_{\mathrm{loc}}(\Omega) satisfies \int_\Omega u \Delta \phi \, dx = 0 for all test functions \phi \in C_c^\infty(\Omega), then u \in C^\infty(\Omega) and u is equal almost everywhere to a real analytic function.^[36] This result, known as Weyl's lemma, highlights the hypoellipticity of the Laplace operator, ensuring that solutions are smooth even under minimal integrability assumptions.^[37] Proofs of this theorem typically bootstrap regularity through approximation techniques. One approach convolves u with standard mollifiers to produce smooth functions u_\varepsilon = u * \rho_\varepsilon that approximate u and satisfy \Delta u_\varepsilon = (\Delta u) * \rho_\varepsilon = 0 exactly, since \Delta u = 0 in the distributional sense; the mean value property then extends to these approximations, implying higher-order derivatives are also harmonic and controlled by local bounds on u. Iterating this process elevates regularity from L^1_{\mathrm{loc}} to C^\infty.^[11] Alternatively, the mean value property directly yields that derivatives of harmonic functions are harmonic, allowing differentiation under the integral representation to establish infinite differentiability.^[36] Schauder estimates provide quantitative Hölder bounds tailored to the Laplace equation, stating that for any integer k \geq 0 and $0 < \alpha < 1, there exists a constant C = C(n, k, \alpha, r) such that

\|D^k u\|_{C^\alpha(B_r(x_0))} \leq C \|u\|_{L^\infty(B_{2r}(x_0))}

for any ball B_{2r}(x_0) \subset \Omega, where D^k denotes spatial derivatives of order k. These estimates, adapted from general elliptic theory, exploit the explicit Green's function for the Laplacian to control oscillations of derivatives via potential theory. They underscore the uniform ellipticity of the operator, yielding global Hölder norms from mere boundedness without needing right-hand side data.^[38] Historically, the analyticity of entire harmonic functions was established by Sergei Bernstein in 1912 using series expansions, with extensions to local regularity by Hermann Weyl and others in the 1930s and 1940s through distributional frameworks.^[39] Unlike parabolic or hyperbolic PDEs, where solutions may develop singularities, the Laplace equation's strict ellipticity and lack of variable coefficients ensure this bootstrapping to analyticity holds universally for weak solutions, preventing loss of regularity.^[36]

Harnack's Inequality

Harnack's inequality provides quantitative estimates on the variation of positive harmonic functions within balls, offering precise control over their growth and oscillation. Named after the German mathematician Axel Harnack, who first derived it in his 1887 treatise on potential theory, the inequality states that if u is a positive harmonic function on the open ball B_R(a) \subset \mathbb{R}^n with n \geq 2, and if x \in B_r(a) for some $0 < r < R, then letting d = |x - a|,

\frac{R - d}{(R + d)^{n-1}} R^{n-2} \, u(a) \leq u(x) \leq \frac{R + d}{(R - d)^{n-1}} R^{n-2} \, u(a).

For a uniform bound over the entire inner ball B_r(a), the constants are evaluated at d = r:

\frac{R - r}{(R + r)^{n-1}} R^{n-2} \, u(a) \leq u(x) \leq \frac{R + r}{(R - r)^{n-1}} R^{n-2} \, u(a).

^[40] ^[33] A more refined pointwise version, applicable when u is positive and harmonic in B_r(x_0), bounds u(y) for y \in B_r(x_0) letting d = |y - x_0| as

\frac{r - d}{(r + d)^{n-1}} r^{n-2} \, u(x_0) \leq u(y) \leq \frac{r + d}{(r - d)^{n-1}} r^{n-2} \, u(x_0),

where the exponents reflect the dimensional dependence, with explicit constants derived from the Poisson kernel representation.^[33] The proof typically proceeds via the Poisson integral formula for the ball, which expresses u in terms of its boundary values, combined with estimates on the kernel's monotonicity, or alternatively by applying the mean value property over concentric annuli to bound ratios of averages.^[33] For general (signed) harmonic functions, estimates analogous to Harnack's inequality can be obtained by decomposing into positive and negative parts or by shifting with a suitable constant to apply the positive case. Specifically, if u is harmonic in B_r(x_0), then for large enough M > \sup_{B_r} |u|, the function v = u + M is positive harmonic, and applying the inequality to v yields bounds on u(y) relative to u(x_0) and the supremum norm of u on the ball, such as |u(y)| \leq C(n) \frac{r + |y - x_0|}{(r - |y - x_0|)^{n-1}} r^{n-2} \sup_{B_r} |u|, where C(n) is a dimension-dependent constant.^[41] This extension leverages the structure of harmonic functions without requiring subharmonicity of the parts directly. Harnack's theorem on chains extends these local estimates to global control in connected domains. By covering a compact subset K \subset \Omega with a finite chain of overlapping balls where consecutive balls have controlled radius ratios, the inequalities chain together to yield a uniform constant C_K > 0 such that for any positive harmonic u on \Omega, \sup_K u / \inf_K u \leq C_K.^[33] In particular, for connected domains that are "thin"—such as long narrow tubes or domains with bounded Harnack chain length—positive harmonic functions must be constant, as the chaining forces the ratio to approach 1. This principle underpins applications like bounding the diameter of the image of compact sets under harmonic maps (where \diam u(K) \leq (C_K - 1) \inf_K u for positive u) and establishing continuity up to the boundary by applying the estimates to sequences of shrinking balls approaching boundary points.^[33]

Liouville's Theorem

Liouville's theorem asserts that if u is a harmonic function on \mathbb{R}^n that is bounded, say |u(x)| \leq M for some constant M > 0 and all x \in \mathbb{R}^n, then u must be constant.$$] This result holds for any dimension n \geq 1. A standard proof exploits the mean value property of harmonic functions. Suppose u is bounded by M. Fix x \in \mathbb{R}^n and consider balls B(0, r) and B(x, r) of radius r > |x|. By the mean value property, [ u(0) = \frac{1}{V(B(0,r))} \int_{B(0,r)} u(y) , dV(y), \quad u(x) = \frac{1}{V(B(x,r))} \int_{B(x,r)} u(y) , dV(y),

where $V$ denotes volume. The difference satisfies

|u(x) - u(0)| \leq \frac{V(B(0,r) \Delta B(x,r))}{V(B(x,r))} \cdot 2M,

with $\Delta$ the [symmetric difference](/page/Symmetric_difference). As $r \to \infty$, the ratio of volumes tends to 0, implying $u(x) = u(0)$ for all $x$, so $u$ is [constant](/page/Constant).$$\] This argument, avoiding [complex analysis](/page/Complex_analysis), traces to a concise proof by [Nelson](/page/Nelson).\[$$ Extensions address growth conditions beyond boundedness. If $|u(x)| \leq C(1 + |x|^k)$ for constants $C > 0$ and $k \geq 0$, then $u$ is a [harmonic](/page/Harmonic) polynomial of degree at most $k$.$$\] For instance, in $\mathbb{R}^3$ ($n=3$), linear growth ($k=1 = n-2$) yields linear polynomials, while slower growth ($k < n-2$) restricts to lower-degree polynomials, including constants for $k < 1$. The theorem parallels Liouville's 1853 result in complex analysis: every bounded entire holomorphic function on $\mathbb{C}$ is constant.\[$$ In $\mathbb{R}^2$, the connection is direct, as bounded harmonic functions are real parts of bounded holomorphic functions, hence constant. The boundedness result for harmonic functions in higher dimensions was generalized by Bernstein in 1912.$$\] Non-constant examples include unbounded harmonics like linear functions $u(x) = \mathbf{a} \cdot x$, which satisfy $\Delta u = 0$ but grow without bound.\[$$ ### Removable Singularities A fundamental result concerning isolated singularities of harmonic functions is the removable singularity theorem, which states that if $u$ is a harmonic function on an open set $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) excluding an isolated point $p \in \Omega$, and $u$ is bounded in a neighborhood of $p$, then there exists a harmonic extension $\tilde{u}$ of $u$ to the entire domain $\Omega$.[](https://www.axler.net/HFT.pdf) This theorem asserts that the singularity at $p$ is removable, meaning $u$ can be redefined at $p$ to yield a harmonic function on $\Omega$.[](https://www.math.unipd.it/~acesar/removable.pdf) The proof relies on the mean value property of harmonic functions. Specifically, for a small ball $B(p, r) \subset \Omega$, define $\tilde{u}(p)$ as the limit of the average values of $u$ over spheres centered at $p$ with radii approaching zero; boundedness ensures this limit exists and is finite. Extending $\tilde{u}$ harmonically via the Poisson integral formula on balls around $p$, one verifies that $\tilde{u} = u$ almost everywhere on $\Omega \setminus \{p\}$, and thus $\tilde{u}$ is the desired extension.[](https://www.axler.net/HFT.pdf) This approach leverages the symmetry and averaging properties inherent to solutions of [Laplace's equation](/page/Laplace's_equation). A stronger version, known as Weyl's lemma, extends removability to cases where $u$ is merely locally integrable ($L^1_{\mathrm{loc}}$) near the singularity while satisfying $\Delta u = 0$ in the distributional sense on $\Omega$. [Weyl's lemma](/page/Weyl's_lemma) establishes that any such $u$ is in fact infinitely differentiable and harmonic on the whole domain, implying that $L^1_{\mathrm{loc}}$ singularities are removable for harmonic functions.[](https://www.axler.net/HFT.pdf) This regularity result underscores the robustness of harmonic functions, as weak solutions automatically smooth out across potential singularities. Singularities are not always removable; counterexamples arise when boundedness fails. In two dimensions ($n=2$), the fundamental solution $\frac{1}{2\pi} \log |x|$ is harmonic on $\mathbb{R}^2 \setminus \{0\}$ but unbounded near the origin, rendering the singularity non-removable.[](https://www.axler.net/HFT.pdf) Similarly, in higher dimensions ($n \geq 3$), the fundamental solution $c_n |x|^{2-n}$ (where $c_n = \frac{1}{(n-2) \omega_n}$ and $\omega_n$ is the surface area of the unit sphere) is harmonic away from the origin but diverges as $|x| \to 0$, so it does not extend harmonically; however, the boundedness condition in the theorem ensures removability in such settings.[](https://www.axler.net/HFT.pdf) The origins of these ideas trace back to Bernhard Riemann's work in the 1850s on removable singularities for holomorphic functions, which inspired analogous results for harmonic functions as solutions to Laplace's equation.[](https://www.math.unipd.it/~acesar/removable.pdf) These concepts were formalized and extended in the context of potential theory by Hermann Weyl in the 1940s, particularly through his contributions to regularity and removability in higher dimensions.[](https://www.axler.net/HFT.pdf) ## Connections to Complex Analysis ### Harmonic Functions and Analytic Functions In two dimensions, a real-valued $ C^2 $ function $ u $ defined on an open set $ \Omega \subset \mathbb{R}^2 $ is harmonic if and only if there exists a holomorphic function $ f: \Omega \to \mathbb{C} $ such that $ u = \operatorname{Re} f $ (or, equivalently, $ u = \operatorname{Im} f $ up to multiplication by $ -i $).[](https://www.axler.net/HFT.pdf) This local equivalence arises from the natural identification of $ \mathbb{R}^2 $ with the complex plane $ \mathbb{C} $, allowing harmonic functions to be expressed as components of analytic functions.[](https://www.math.purdue.edu/~eremenko/dvi/harm.pdf) The forward implication—that the real part of a holomorphic function is harmonic—follows from the Cauchy-Riemann equations. If $ f(z) = u(x,y) + i v(x,y) $ with $ z = x + i y $ is holomorphic on $ \Omega $, then $ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $ and $ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $. Differentiating the first equation with respect to $ x $ gives $ \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 v}{\partial y \partial x} $, and differentiating the second with respect to $ y $ yields $ \frac{\partial^2 u}{\partial y^2} = -\frac{\partial^2 v}{\partial x \partial y} $; by equality of mixed partials, $ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $, so $ \Delta u = 0 $.[](https://www.axler.net/HFT.pdf) A similar computation shows $ v $ is also harmonic.[](https://www.axler.net/HFT.pdf) Conversely, if $ u $ is harmonic on $ \Omega $, a harmonic conjugate $ v $ exists locally such that $ f = u + i v $ is holomorphic. The conjugate $ v $ satisfies the Cauchy-Riemann equations $ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $ and $ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $, which ensure $ f $ is analytic.[](https://www.math.purdue.edu/~eremenko/dvi/harm.pdf) To construct $ v $ locally, consider the 1-form $ \omega = -\frac{\partial u}{\partial y} \, dx + \frac{\partial u}{\partial x} \, dy $; its exterior derivative is $ d\omega = \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) dy \wedge dx = 0 $ since $ \Delta u = 0 $, making $ \omega $ closed and thus locally exact.[](https://www.math.purdue.edu/~eremenko/dvi/harm.pdf) Fixing a base point $ (x_0, y_0) \in \Omega $, define

v(x,y) = v(x_0, y_0) + \int_{(x_0, y_0)}^{(x,y)} -\frac{\partial u}{\partial y}(\xi, \eta) , d\xi + \frac{\partial u}{\partial x}(\xi, \eta) , d\eta,

where the path integral is independent of the path within a simply connected neighborhood of $ (x_0, y_0) $.[](https://www.math.purdue.edu/~eremenko/dvi/harm.pdf) An explicit formula, assuming the base point is $ (0,0) $ and $ u $ extends appropriately, is

v(x,y) = \int_0^y \frac{\partial u}{\partial x}(x, t) , dt - \int_0^x \frac{\partial u}{\partial y}(s, 0) , ds + c

for some constant $ c $.[](https://www.axler.net/HFT.pdf) In multiply connected domains, the harmonic conjugate may be multi-valued, requiring branch cuts for single-valuedness. For instance, $ u(x,y) = \log |z| = \frac{1}{2} \log(x^2 + y^2) $ is harmonic on $ \mathbb{C} \setminus \{0\} $, with conjugate $ v(x,y) = \arg z $; here, $ f(z) = \log z $ is holomorphic on $ \mathbb{C} \setminus (-\infty, 0] $, but $ v $ jumps by $ 2\pi $ across the branch cut, reflecting the topology of the punctured plane.[](http://ramanujan.math.trinity.edu/rdaileda/teach/s20/m4364/lectures/harmonic_handout.pdf) This representation is peculiar to two dimensions, enabled by the complex structure of $ \mathbb{C} $. In dimensions $ n > 2 $, no analogous global or direct local decomposition of harmonic functions as real parts of holomorphic functions exists, owing to the absence of a compatible complex [multiplication](/page/Multiplication); however, smooth harmonic functions can be locally expressed as graphs over hypersurfaces via the [implicit function theorem](/page/Implicit_function_theorem) applied to level sets.[](https://www.axler.net/HFT.pdf) ### Conjugate Harmonic Functions In the [plane](/page/Plane), given a harmonic function $u(x, y)$ defined on an open domain $\Omega \subseteq \mathbb{R}^2$, a function $v(x, y)$ is called a [harmonic conjugate](/page/Harmonic_conjugate) of $u$ if the complex-valued function $f(z) = u(x, y) + i v(x, y)$ is holomorphic on $\Omega$.[](http://ramanujan.math.trinity.edu/rdaileda/teach/s20/m4364/lectures/harmonic_handout.pdf) Both $u$ and $v$ are harmonic on $\Omega$, and they satisfy the Cauchy-Riemann equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$.[](https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf) If a harmonic conjugate $v$ exists for $u$, then it is unique up to an additive real constant; that is, any other [harmonic conjugate](/page/Harmonic_conjugate) $v'$ satisfies $v' = v + c$ for some constant $c \in \mathbb{R}$.[](http://www.math.utah.edu/~korevaar/4200fall19/sept13.pdf) This uniqueness follows from the fact that if $u + i v$ and $u + i v'$ are both holomorphic, their difference $i(v - v')$ is holomorphic with real part zero, implying it is constant.[](http://www.math.utah.edu/~korevaar/4200fall19/sept13.pdf) However, the existence of a global harmonic conjugate depends on the topology of the domain: it is guaranteed if $\Omega$ is simply connected, but may fail in non-simply connected domains, such as the punctured plane $\mathbb{C} \setminus \{0\}$, where the conjugate may be multi-valued or undefined globally.[](http://ramanujan.math.trinity.edu/rdaileda/teach/s20/m4364/lectures/harmonic_handout.pdf)[](https://mathworld.wolfram.com/HarmonicConjugateFunction.html) To construct the harmonic conjugate $v$ of a given [harmonic](/page/Harmonic) $u$ on a simply connected [domain](/page/Domain), fix a point $(x_0, y_0) \in \Omega$ and define

v(x, y) = \int_{(x_0, y_0)}^{(x, y)} -\frac{\partial u}{\partial y}(s, t) , ds + \frac{\partial u}{\partial x}(s, t) , dt,

where the [integral](/page/Integral) is taken along any [path](/page/Path) in $\Omega$ from $(x_0, y_0)$ to $(x, y)$.[](http://ramanujan.math.trinity.edu/rdaileda/teach/s20/m4364/lectures/harmonic_handout.pdf) This [line integral](/page/Line_integral) is path-independent (hence well-defined) precisely because $u$ is [harmonic](/page/Harmonic), ensuring the [differential form](/page/Differential_form) $-\frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x} dy$ is closed; the resulting $v$ then satisfies the Cauchy-Riemann equations and is [harmonic](/page/Harmonic).[](http://ramanujan.math.trinity.edu/rdaileda/teach/s20/m4364/lectures/harmonic_handout.pdf) Harmonic conjugates find applications in two-dimensional physics, particularly in modeling irrotational, incompressible fluid flows, where the velocity potential $\phi$ (a [harmonic](/page/Harmonic) function) has a [harmonic conjugate](/page/Harmonic_conjugate) known as the stream function $\psi$, forming the complex potential $\phi + i \psi$.[](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/07%3A_Two_Dimensional_Hydrodynamics_and_Complex_Potentials/7.05%3A_Stream_Functions) The level curves of $\phi$ ([equipotential](/page/Equipotential) lines) and $\psi$ ([streamlines](/page/Stream_function)) are orthogonal, reflecting the fact that the velocity [field](/page/Field) is tangent to streamlines and normal to equipotentials.[](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/07%3A_Two_Dimensional_Hydrodynamics_and_Complex_Potentials/7.05%3A_Stream_Functions) This orthogonality holds generally for any harmonic conjugates $u$ and $v$: their gradients satisfy $\nabla u \cdot \nabla v = \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} = 0$ by the Cauchy-Riemann equations (assuming $\nabla u \neq 0$), so the level curves, being perpendicular to the gradients, intersect at right angles.[](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/06%3A_Harmonic_Functions/6.06%3A_Orthogonality_of_Curves) ## Generalizations ### Subharmonic and Superharmonic Functions A function $u$ defined on an [open set](/page/Open_set) $\Omega \subseteq \mathbb{R}^n$ is subharmonic if it is upper semicontinuous and satisfies $\Delta u \geq 0$ in the weak (distributional) sense. Equivalently, for $C^2$ functions, subharmonicity holds if the Laplacian $\Delta u \geq 0$ [pointwise](/page/Pointwise). A function $u$ is superharmonic if $-u$ is subharmonic, which corresponds to $\Delta u \leq 0$ in the weak sense. Subharmonic functions satisfy a [mean value inequality](/page/Mean): for any [ball](/page/Ball) $B(x, r) \subset \Omega$, $u(x) \leq \frac{1}{|B(x, r)|} \int_{B(x, r)} u(y) \, dy$, where $|B(x, r)|$ denotes the volume of the ball. This inequality also holds for the spherical mean over the boundary $\partial B(x, r)$. Harmonic functions achieve equality in this inequality, distinguishing them as both subharmonic and superharmonic. Non-constant subharmonic functions obey a maximum principle: they cannot attain a local maximum in the interior of $\Omega$ unless constant on the connected component containing that point. If a subharmonic function is bounded above on $\Omega$, its maximum is attained on the boundary $\partial \Omega$. This principle extends the maximum principle for harmonic functions by replacing equality with inequality. Examples of subharmonic functions include convex functions on $\mathbb{R}^n$, since their Hessians are positive semidefinite, implying $\Delta u \geq 0$. In the complex plane, $|z|^2$ is subharmonic because $\Delta |z|^2 = 4 > 0$. More generally, for a holomorphic function $f$ on a domain in $\mathbb{C}$, $|f(z)|^2$ is subharmonic, as $\Delta |f|^2 = 4 |f'|^2 \geq 0$. Perron's method constructs solutions to the [Dirichlet problem](/page/Dirichlet_problem) by taking the supremum of all subharmonic functions on $\Omega$ that are bounded above by the boundary data $\phi$ on $\partial \Omega$; this supremum yields a [harmonic](/page/Harmonic) function equal to $\phi$ continuously on the [boundary](/page/Boundary) under suitable conditions on $\Omega$. This approach leverages the [maximum principle](/page/Maximum_principle) for subharmonics to ensure the solution's uniqueness and properties. The uniform limit of a locally uniformly convergent sequence of subharmonic functions is subharmonic. If a function is both subharmonic and superharmonic, it is [harmonic](/page/Harmonic). Thus, [harmonic](/page/Harmonic) functions can arise as limits of subharmonic functions in appropriate settings. ### Harmonic Functions on Manifolds In the setting of Riemannian manifolds, a smooth real-valued [function](/page/Function) $u$ defined on a [Riemannian manifold](/page/Riemannian_manifold) $(M, g)$ is harmonic if it satisfies the equation $\Delta_M u = 0$, where $\Delta_M$ denotes the Laplace-Beltrami operator associated to the metric $g$.[](https://www.math.mcgill.ca/toth/spectral%20geometry.pdf) This generalizes the classical notion of harmonic functions in [Euclidean space](/page/Euclidean_space), where the Laplace-Beltrami operator reduces to the standard Laplacian.[](https://www.math.mcgill.ca/toth/spectral%20geometry.pdf) In local coordinates $(x^i)$ on $M$, the Laplace-Beltrami operator acts on $u$ as

\Delta u = \frac{1}{\sqrt{|g|}} \partial_i \left( \sqrt{|g|} , g^{ij} \partial_j u \right),

\tau(\phi)^k = g^{ij} \left( \frac{\partial^2 \phi^k}{\partial x^i \partial x^j} - \Gamma^l_{ij} \frac{\partial \phi^k}{\partial x^l} + \Gamma^k_{\alpha \beta}(\phi) \frac{\partial \phi^\alpha}{\partial x^i} \frac{\partial \phi^\beta}{\partial x^j} \right),

where $\Gamma$ are the [Christoffel symbols](/page/Christoffel_symbols) of $g$ and $h$, respectively; the equation $\tau(\phi) = 0$ forms a nonlinear elliptic system of PDEs.[](https://math.jhu.edu/~js/Math748/helein.harmonic.pdf) A key property is that when $\dim M = 1$, harmonic maps reduce to geodesics in $N$: a map from a curve to $N$ is harmonic if and only if it is a geodesic, as the tension field then coincides with the geodesic equation, and such maps minimize energy in their homotopy class.[](https://math.jhu.edu/~js/Math646/eells-sampson.pdf) More generally, harmonic maps are stationary points of the energy functional and, under suitable conditions, minimize energy within homotopy classes. Examples include the identity map on any Riemannian manifold, which is trivially harmonic since $d\phi$ is isometric and $\tau(\mathrm{id}) = 0$. Another prominent class consists of holomorphic maps between Kähler manifolds: if $\phi$ is holomorphic from a Kähler domain $(M, g, J)$ to a Kähler target $(N, h, J')$, then $\phi$ is harmonic because its $(1,0)$-part aligns with the complex structure, making $d\phi$ isotropic and satisfying $\tau(\phi) = 0$.[](https://arxiv.org/pdf/2010.03545) Existence and uniqueness for harmonic maps are addressed via the Dirichlet problem, which seeks a harmonic map with prescribed boundary values on $\partial M$; regularity results ensure that solutions are smooth in the interior under mild boundary conditions, as established by boundary estimates and the maximum principle for the tension field.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-2/Boundary-regularity-and-the-Dirichlet-problem-for-harmonic-maps/10.4310/jdg/1214437663.full) The Eells-Sampson theorem guarantees existence: for compact Riemannian manifolds $M$ and $N$ with $N$ of non-positive sectional curvature, every homotopy class of maps from $M$ to $N$ contains a harmonic representative that minimizes energy, obtained as a limit of the harmonic map heat flow.[](https://math.jhu.edu/~js/Math646/eells-sampson.pdf) Uniqueness holds in specific cases, such as when the target $N$ is strictly negatively curved, by convexity of the energy functional along geodesics.[](https://math.jhu.edu/~js/Math646/eells-sampson.pdf) Recent applications of harmonic maps appear in Teichmüller theory, where they parametrize the [Teichmüller space](/page/Teichmüller_space) of Riemann surfaces via the Hopf differential (the $(2,0)$-part of the pullback metric), enabling studies of extremal length and quasiconformal mappings; for instance, harmonic maps from a fixed surface to varying hyperbolic targets encode the geometry of moduli spaces. This connection has led to rigidity results, such as geometric superrigidity for representations of surface groups.[](https://www.math.umd.edu/~raw/papers/teich.pdf)

References

[1]
[PDF] 18.04 S18 Topic 5: Introduction to harmonic functions
Harmonic functions appear regularly and play a fundamental role in math, physics and engineering. In this topic we'll learn the definition, some key properties ...
[2]
None
### Summary of Introduction to Harmonic Functions
[3]
[PDF] Harmonic functions - Purdue Math
Mar 18, 2024 · In any simply connected region in the plane, every harmonic function is the real part of an analytic function f. This f is defined up to.
[4]
Harmonic - Etymology, Origin & Meaning
### Summary of Etymology of "Harmonic"
[5]
Pythagoras (Stanford Encyclopedia of Philosophy)
Summary of each segment:
[6]
Jean d'Alembert - Biography
### Summary of d'Alembert's Work on the Wave Equation and Vibrating Strings in the 1740s
[7]
Leonhard Euler (1707 - 1783) - Biography - MacTutor
... harmonic series and Euler's constant and other results on series. In ... equation, and the Euler equations for the motion of an inviscid incompressible fluid.
[8]
Earliest Known Uses of Some of the Words of Mathematics (H)
Aug 7, 2018 · HARMONIC FUNCTION. According to Kline (p. 685) William Thomson (later called Lord Kelvin) introduced the term in 1850. The OED's earliest ...Missing: etymology | Show results with:etymology
[9]
Harmonic Function -- from Wolfram MathWorld
Any real function u(x,y) with continuous second partial derivatives which satisfies Laplace's equation, del ^2u(x,y)=0, (1) is called a harmonic function.
[10]
[PDF] V7. Laplace's Equation and Harmonic Functions 1. The Laplace ...
Definition. A function w(x, y) which has continuous second partial derivatives and solves Laplace's equation (1) is called a harmonic function.
[11]
[PDF] Chapter 2: Laplace's equation - UC Davis Math
A solution of Laplace's equation is called a harmonic function. Laplace's equation is a linear, scalar equation. It is the prototype of an elliptic.
[12]
Harmonic Functions
The main computational tool is the harmonic extension algorithm which allows us to compute the values of harmonic functions at points within a cell if the cell ...Missing: mathematics | Show results with:mathematics
[13]
Laplace equation - Encyclopedia of Mathematics
Jan 9, 2024 · P.S. Laplace, Hist. Acad. Sci. Paris (1782) (1785). [4], P.S. Laplace, "Celestial mechanics" , 2 , Chelsea, reprint (1966) (Translated from ...
[14]
[PDF] 23. Harmonic functions Recall Laplace's equation ∆u ... - UCSD Math
Harmonic functions in one variable are easy to describe. The general solution of uxx = 0 is u(x) = ax + b, for constants a and b.
[15]
[PDF] spherical harmonics and homogeneous har- monic polynomials
Corollary 1 means any polynomial of degree n in three variables can be expressed uniquely as a linear combination of terms of the form: r2k times a harmonic ...
[16]
Math 3423 - Nikola Petrov
... ex[cos(y)+isin(y)]; Cartesian and ... Reading/thinking assignment: The harmonic function v(x,y) is said to be a harmonic conjugate of the harmonic function ...
[17]
[PDF] An Introduction to Applied Partial Differential Equations Marek Z. El ...
Feb 1, 2023 · fundamental solution. Laplace's equation and the wave equation are dealt with ... v1(r)=1, v2(r) = log r. (3.2.26). Since we require our ...<|separator|>
[18]
[PDF] Harmonic Functions on Domains in Rn Topics
These notes present material on harmonic functions on domains in Euclidean space. They have some overlap with results presented in Chapters 3 and 5 of [T],.
[19]
[PDF] Lesson 35. Potential theory, Electrostatic fields - Purdue Math
Potential theory is the theory of harmonic functions, that is, solutions to Laplace's equation ∇2. Φ = 0. In applications, electrostatic and gravitational ...
[20]
[PDF] 18.02SC MattuckNotes: Relation to Physics Parts 2 to 3
3. Harmonic functions in space. For example, the potential function for an electrostatic field E is harmonic in any region of space which is free of ...
[21]
20. Green's Functions in Spherical Coordinates - Galileo and Einstein
As an example of using spherical harmonics in electrostatics, we'll take another look at the old favorite of a point charge outside a grounded conducting sphere ...
[22]
[PDF] Lesson 39. Potential theory - Purdue Math
Potential theory is the theory of harmonic functions, that is, solutions to Laplace's equation ∇2 Φ = 0. In applications, electrostatic and gravitational ...
[23]
[PDF] Lecture 30: Harmonic functions
Lecture 30: Harmonic functions. I. Motivation. Harmonic functions crop up in thermodynamics, electrostatics, fluid mechanics, and many other "natural ...
[24]
[PDF] HARMONIC FUNCTIONS, GREEN'S FUNCTIONS and POTENTIALS.
udA. In words: the value of a harmonic function at a point equals its average value on any sphere centered at that point. As a consequence, the strong maximum ...
[25]
[PDF] CHAPTER 3 High Speed flows 3.1 Irrotational flows of ... - MIT
Dec 1, 2002 · ... irrotational vector can be expressed as the gradient of a scalar potential. Thus we define the velocity potential φ by q = ∇φ. (3.1.1). An ...
[26]
[PDF] Potential Flow - Research
Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous effects become negligible, for example ...
[27]
[PDF] Chapter 6 Partial Differential Equations
There are three broad classes of boundary conditions: a) Dirichlet boundary conditions: The value of the dependent vari- able is specified on the boundary. b) ...
[28]
[PDF] Error Analysis for Image Inpainting - UCLA Mathematics
For general images, harmonic inpainting is not appropriate since it tends to smooth out the edges.
[29]
[PDF] Bounded Distortion Harmonic Shape Interpolation
We present a planar shape interpolation method that is designed to produce C∞ harmonic mappings, which are the most widely used type of mappings in graphics ...
[30]
[PDF] Harmonic functions
Oct 8, 2013 · In this section, we will prove the mean value theorem of Gauss, and derive some of its direct consequences. Theorem 2 (Gauss 1840). Let Ω ...
[31]
[PDF] 3 Laplace's Equation
In this section, we prove a mean value property which all harmonic functions satisfy. ... By the mean value property, u(x0) = −. Z. B(x0,r) u(y)dy for all B ...
[32]
[PDF] Laplace's Equation
This is the mean-value property for harmonic functions; we will give another proof in ... Theorem 3.12 (Converse to mean-value property). Let u ∈ C2(Ω) ...<|control11|><|separator|>
[33]
[PDF] Harmonic Function Theory - Sheldon Axler
u(y) u(x) ≤ C for all points x and y in K and all positive harmonic functions u on Ω. Proof: We will prove that there is a constant C ∈ (1,∞) such that u(y)/u(x) ...
[34]
Elliptic Partial Differential Equations of Second Order - SpringerLink
Book Title · Elliptic Partial Differential Equations of Second Order ; Authors · David Gilbarg, Neil S. Trudinger ; Series Title · Classics in Mathematics ; DOI ...Missing: URL | Show results with:URL
[35]
[PDF] The Maximum Principle - Trinity University
Moral. A nonconstant harmonic function on a domain cannot attain an absolute maximum value there. Proof. Let S = u−1({M}) = {z ∈ Ω|u(z) = M}.
[36]
[PDF] Regularity Theory for Elliptic PDE - UB
Regularity in PDEs, especially elliptic ones, is a basic question about whether solutions are smooth, linked to Hilbert's XIXth problem.
[37]
[PDF] Weyl's Lemma
A continuous function is harmonic if and only if it is weakly-harmonic. Proof. (Harmonic implies weakly-harmonic) Suppose f is harmonic, then by definition.
[38]
[PDF] Schauder estimates for elliptic and parabolic equations
Introduction. The Schauder estimate for the Laplace equation was traditionally built upon the New- ton potential theory. Different proofs were found later ...
[39]
On new results in the theory of minimal surfaces - Project Euclid
Of course, in most cases these measures only lead to sufficient conditions. The first such discussion was carried out by S. Bernstein in 1912 ... harmonic ...
[40]
The Harnack Inequality - SpringerLink
... first obtained by Axel Harnack in 1887 [45], page 62. Download to read the full chapter text. Chapter PDF. Explore related subjects. Discover the latest ...Missing: original Harnack's
[41]
[PDF] Maximum principles, Harnack inequality for classical solutions
The Harnack inequality is classically used to prove solutions to elliptic equa- tions are Hölder continuous. Let us first recall what happens for harmonic.
[42]
[PDF] Removable singularities of harmonic functions. - Math-Unipd
This theorem is the analogous of Riemann theorem on removable singularities for holo- morfic functions. Proof. Let r > 0 such that B(x0,r) ⊂ Ω. Let ˜u be the ...
[43]
[PDF] Harmonic Functions - Trinity University
Harmonic Functions. Page 16. Definition and Examples. Harmonic Conjugates. Existence of Conjugates. Corollary 1. Harmonic conjugates always exist locally.
[44]
[PDF] harmonic functions, harmonic conjugates; 1.6 differentiation and ...
Sep 13, 2019 · A where A is an open simply connected domain. (A domain is ... Then there exists a harmonic conjugate v x, y to u x, y , unique up ...
[45]
Harmonic Conjugate Function -- from Wolfram MathWorld
The harmonic conjugate to a given function u(x,y) is a function v(x,y) such that f(x,y)=u(x,y)+iv(x,y) is complex differentiable (i.e., satisfies the ...
[46]
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff
[47]
None
Summary of each segment:
[48]
[math/9902052] Harmonic functions on the real hyperbolic ball I - arXiv
Feb 8, 1999 · We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions.
[49]
GREEN'S FUNCTIONS, HARMONIC FUNCTIONS, AND VOLUME ...
By the estimate of heat kernels, these estimates for the minimal positive Green's function have been obtained independently by. Grigor'yan [14] and Saloff-Coste ...
[50]
[PDF] Eigenfunctions of the Laplacian of Riemannian manifolds Updated
Aug 15, 2017 · These lecture notes are an expanded version of the author's CBMS ten Lectures at the University of Kentucky in June 20-24, 2011.
[51]
harmonic differential form in nLab
### Summary of Harmonic Differential Form from nLab
[52]
[PDF] L2 harmonic forms on non-compact Riemannian manifolds
L2-harmonic k-forms are L2 k-forms which are closed and coclosed, meaning dα = δα = 0.
[53]
[PDF] The Hodge Theorem In this section we assume that M is an oriented ...
Hodge theorem. An arbitrary de Rham cohomology class of an oriented compact. Riemannian manifold can be represented by a unique harmonic form. In other words ...
[54]
[PDF] HODGE DECOMPOSITION Contents 1. Introduction 1 2. Laplace ...
Finding harmonic forms is related to solving equations of the form ∆ω = 0, or more generally equa- tions of the form ∆ω = α. Thus, we will be interested in ...
[55]
[PDF] William Vallance Douglas Hodge. 17 June 1903 -- 7 July 1975
Jul 7, 1975 · The theory of harmonic forms thus provides a remarkably rich and detailed structure for the cohomology of algebraic manifolds. This 'Hodge ...
[56]
[PDF] Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
Nov 15, 2013 · The forms in Hk(M,g) are called harmonic forms with respect to the metric g. The Hodge theorem implies that. H k(M,g). ∼. = H k. DR(M). ∼. = H k ...
[57]
[PDF] Harmonic Mappings of Riemannian Manifolds
Thus f is harmonic if and only if at each pair of points P, f(P) there are such coordinates in which f satisfies the Euclidean-Laplace equation at P. Page 11 ...
[58]
[PDF] helein.harmonic.pdf - Johns Hopkins University
4. Harmonic maps to a Euclidean space. A smooth map φ : (M,g) → Rn is harmonic if and only if each of its components is a harmonic function on (M,g), as in ...
[59]
[PDF] Harmonic maps from Kähler manifolds - arXiv
Oct 7, 2020 · This report attempts a clean presentation of the theory of harmonic maps from complex and. Kähler manifolds to Riemannian manifolds.
[60]
Boundary regularity and the Dirichlet problem for harmonic maps
1983 Boundary regularity and the Dirichlet problem for harmonic maps. Richard Schoen, Karen Uhlenbeck. DOWNLOAD PDF + SAVE TO MY LIBRARY.
[61]
[PDF] Harmonic Maps and Teichmüller Theory Georgios D. Daskalopoulos ...
Teichmüller theory is rich in applications to topology and physics. By way ... Geometric superrigidity uses harmonic maps to prove results of this type.