Harmonic functions are twice continuously differentiable functions that satisfy . They play a key role in potential theory, appearing in , , and heat conduction. These functions have unique properties that make them essential in mathematical physics.
The and are fundamental characteristics of harmonic functions. These properties allow us to understand their behavior and solve boundary value problems. and the provide powerful tools for analyzing and constructing harmonic functions in various domains.
Definition of harmonic functions
Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation
Play a fundamental role in potential theory, a branch of mathematics that studies the behavior of functions satisfying certain partial differential equations
Arise naturally in various contexts, such as in the study of electrostatics, fluid dynamics, and heat conduction
Laplace's equation
Solutions in various dimensions
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In one dimension, Laplace's equation reduces to dx2d2u=0, and its solutions are linear functions of the form u(x)=ax+b
In two dimensions, Laplace's equation is ∂x2∂2u+∂y2∂2u=0, and its solutions include harmonic polynomials and logarithmic functions
In three dimensions, Laplace's equation is ∂x2∂2u+∂y2∂2u+∂z2∂2u=0, and its solutions include spherical harmonics and Coulomb potentials
Solutions to Laplace's equation in higher dimensions have applications in quantum mechanics and string theory
Mean value property
Integral formulas
The mean value property states that the value of a at any point is equal to the average of its values over any sphere or ball centered at that point
For a harmonic function u defined on a domain Ω⊂Rn, the mean value property can be expressed as u(x)=∣∂B(x,r)∣1∫∂B(x,r)u(y)dS(y), where B(x,r) is a ball of radius r centered at x
The mean value property can also be expressed using volume integrals: u(x)=∣B(x,r)∣1∫B(x,r)u(y)dy
These integral formulas provide a way to reconstruct a harmonic function from its boundary values
Maximum principle
Consequences and applications
The maximum principle states that a non-constant harmonic function cannot attain its maximum or minimum value at an interior point of its domain
A consequence of the maximum principle is that if two harmonic functions agree on the boundary of a domain, they must agree everywhere inside the domain
The maximum principle is useful in proving uniqueness results for boundary value problems involving Laplace's equation
The maximum principle has applications in the study of elliptic partial differential equations and in the theory of Markov processes
Harnack's inequality
Harnack's convergence theorem
Harnack's inequality provides an estimate for the ratio of the values of a positive harmonic function at two points in terms of the distance between the points and their distance to the boundary of the domain
For a positive harmonic function u defined on a domain Ω⊂Rn, Harnack's inequality states that u(y)u(x)≤C(d(x,∂Ω)∣x−y∣)α, where C and α are positive constants that depend only on the dimension n
Harnack's convergence theorem states that a bounded sequence of harmonic functions on a domain has a subsequence that converges uniformly on compact subsets to a harmonic function
Harnack's inequality and convergence theorem are powerful tools in the study of the boundary behavior of harmonic functions and in the development of a potential theory for more general elliptic operators
Poisson integral formula
Dirichlet problem for a disk
The Poisson integral formula provides a solution to the Dirichlet problem for Laplace's equation on a disk in terms of the boundary values of the function
For a continuous function f defined on the unit circle ∂D, the Poisson integral of f is the function u(r,θ)=2π1∫02π1−2rcos(θ−t)+r21−r2f(eit)dt, which is harmonic on the unit disk D and satisfies u(eiθ)=f(eiθ) for all θ
The Dirichlet problem for a disk asks to find a harmonic function on the disk that takes prescribed values on the boundary circle
The Poisson integral formula provides an explicit solution to the Dirichlet problem for a disk, demonstrating the existence and uniqueness of solutions to this boundary value problem
Green's functions
Fundamental solution of Laplace's equation
Green's functions are a powerful tool in the study of linear partial differential equations, particularly in solving boundary value problems
For Laplace's equation, the G(x,y) is a function of two variables that satisfies ΔxG(x,y)=δ(x−y), where δ is the Dirac delta function
The fundamental solution of Laplace's equation in Rn is given by Φ(x)={−2π1log∣x∣,(n−2)ωn1∣x∣2−n,n=2n≥3, where ωn is the surface area of the unit sphere in Rn
Green's functions can be used to represent harmonic functions in terms of their boundary values, providing an alternative approach to the Poisson integral formula
Representation of harmonic functions
Poisson integral vs Green's function
Harmonic functions on a domain can be represented using either the Poisson integral formula or Green's functions
The Poisson integral formula expresses a harmonic function in terms of its boundary values, while the Green's function representation involves the values of the function and its normal derivative on the boundary
For a harmonic function u on a domain Ω with boundary ∂Ω, the Green's function representation is given by u(x)=∫∂Ω(u(y)∂ν∂G(x,y)−G(x,y)∂ν∂u(y))dS(y), where ν is the outward unit normal to ∂Ω
The choice between the Poisson integral and Green's function representation depends on the specific problem and the available information about the harmonic function and its boundary values
Harmonic conjugates
Cauchy-Riemann equations
Harmonic conjugates are pairs of harmonic functions that are related by the Cauchy-Riemann equations
If u(x,y) is a harmonic function, then there exists a harmonic function v(x,y), called the harmonic conjugate of u, such that f(z)=u(x,y)+iv(x,y) is an analytic function of the complex variable z=x+iy
The Cauchy-Riemann equations for a pair of harmonic functions u and v are given by ∂x∂u=∂y∂v and ∂y∂u=−∂x∂v
Harmonic conjugates play a crucial role in the connection between harmonic functions and complex analysis, allowing for the application of powerful tools from complex analysis to the study of harmonic functions
Subharmonic and superharmonic functions
Characterizations and properties
Subharmonic functions are upper semicontinuous functions that satisfy the sub-mean value property, meaning that the value of the function at any point is less than or equal to the average of its values over any ball centered at that point
Superharmonic functions are lower semicontinuous functions that satisfy the super-mean value property, with the inequality reversed
Harmonic functions are both subharmonic and superharmonic
The maximum principle holds for subharmonic functions: a cannot attain its maximum value at an interior point of its domain unless it is constant
The minimum principle holds for superharmonic functions: a superharmonic function cannot attain its minimum value at an interior point of its domain unless it is constant
Subharmonic and superharmonic functions arise in the study of more general elliptic partial differential equations and in the theory of viscosity solutions
Boundary behavior of harmonic functions
Fatou's theorem
The boundary behavior of harmonic functions is a central topic in potential theory, as it relates to the solvability of boundary value problems and the regularity of solutions
Fatou's theorem states that if u is a positive harmonic function on the unit ball B in Rn, then u has a non-tangential limit at almost every point of the boundary ∂B
A non-tangential limit at a point x∈∂B is defined as the limit of u(y) as y approaches x from within a cone with vertex at x and aperture less than π/2
Fatou's theorem has been generalized to harmonic functions on more general domains and to solutions of other elliptic partial differential equations
Liouville's theorem
Applications in complex analysis
states that every bounded harmonic function on the entire space Rn must be constant
In the context of complex analysis, Liouville's theorem implies that every bounded entire function (a function analytic on the entire complex plane) must be constant
Liouville's theorem is a powerful result with many applications in complex analysis, such as in the proof of the fundamental theorem of algebra and in the classification of conformal mappings between domains
Generalizations of Liouville's theorem, such as Picard's theorems, play a crucial role in the study of the behavior of analytic functions near singularities and in value distribution theory