8.4 Bounded harmonic functions

Bounded harmonic functions are a key concept in potential theory. They satisfy Laplace's equation and are bounded by a constant value. These functions have important applications in complex analysis, partial differential equations, and geometry.

The maximum and minimum principles characterize bounded harmonic functions. These principles state that such functions attain their extreme values on the boundary of their domain, leading to powerful results like Liouville's theorem and Harnack's inequality.

Definition of bounded harmonic functions

• Bounded harmonic functions are a key concept in potential theory that play a crucial role in understanding the behavior of harmonic functions and their properties
• A function $u(x)$ is said to be harmonic in a domain $\Omega \subset \mathbb{R}^n$ if it satisfies Laplace's equation $\Delta u = 0$ in $\Omega$
• A harmonic function $u(x)$ is called bounded if there exists a constant $M > 0$ such that $|u(x)| \leq M$ for all $x \in \Omega$
• Bounded harmonic functions have important applications in various areas of mathematics, including complex analysis, partial differential equations, and geometry

Maximum principle for bounded harmonic functions

• The maximum principle is a fundamental result in potential theory that characterizes the behavior of bounded harmonic functions
• It states that if $u(x)$ is a bounded harmonic function in a domain $\Omega$, then $u(x)$ attains its maximum value on the boundary $\partial \Omega$
• Consequently, if $u(x)$ attains its maximum value at an interior point of $\Omega$, then $u(x)$ must be constant throughout $\Omega$
• The maximum principle has far-reaching implications in the study of harmonic functions and is used to prove various properties and theorems

Minimum principle for bounded harmonic functions

• The minimum principle is the counterpart of the maximum principle and describes the behavior of bounded harmonic functions in terms of their minimum values
• It states that if $u(x)$ is a bounded harmonic function in a domain $\Omega$, then $u(x)$ attains its minimum value on the boundary $\partial \Omega$
• Similar to the maximum principle, if $u(x)$ attains its minimum value at an interior point of $\Omega$, then $u(x)$ must be constant throughout $\Omega$
• The minimum principle is often used in conjunction with the maximum principle to establish properties of bounded harmonic functions

Liouville's theorem for bounded harmonic functions

• Liouville's theorem is a powerful result in potential theory that characterizes entire bounded harmonic functions
• It states that if $u(x)$ is a bounded harmonic function defined on the entire space $\mathbb{R}^n$, then $u(x)$ must be a constant function

Entire bounded harmonic functions are constant

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• Liouville's theorem implies that any bounded harmonic function defined on the entire space $\mathbb{R}^n$ is necessarily constant
• This result highlights the rigidity of bounded harmonic functions and their behavior on unbounded domains
• The proof of Liouville's theorem relies on the maximum and minimum principles and the properties of harmonic functions

Harnack's inequality

• Harnack's inequality is a fundamental estimate in potential theory that quantifies the growth of positive harmonic functions
• It states that if $u(x)$ is a positive harmonic function in a domain $\Omega$, then for any compact subset $K \subset \Omega$, there exists a constant $C > 0$ such that $\sup_{x \in K} u(x) \leq C \inf_{x \in K} u(x)$
• Harnack's inequality provides a quantitative relationship between the maximum and minimum values of a positive harmonic function on a compact set

Applications of Harnack's inequality

• Harnack's inequality has numerous applications in potential theory and related fields
• It is used to prove the Harnack principle, which states that a sequence of positive harmonic functions that is uniformly bounded above converges uniformly on compact subsets to a harmonic function
• Harnack's inequality is also employed in the study of elliptic partial differential equations, heat equations, and other areas of analysis

Bounded harmonic functions in half-spaces

• The study of bounded harmonic functions in half-spaces is of particular interest in potential theory
• A half-space is a domain of the form $\mathbb{H}^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_n > 0\}$
• Bounded harmonic functions in half-spaces exhibit special properties and can be represented using the Poisson integral formula

Poisson integral representation

• The Poisson integral formula provides a representation of bounded harmonic functions in half-spaces
• It states that if $u(x)$ is a bounded harmonic function in the half-space $\mathbb{H}^n$, then it can be represented as $u(x) = \int_{\mathbb{R}^{n-1}} P(x, y) f(y) dy$, where $P(x, y)$ is the Poisson kernel and $f(y)$ is a bounded function on $\mathbb{R}^{n-1}$
• The Poisson integral formula establishes a correspondence between bounded harmonic functions in half-spaces and bounded functions on the boundary

Boundary values of bounded harmonic functions

• The boundary values of bounded harmonic functions in half-spaces play a crucial role in their study
• The Poisson integral representation implies that a bounded harmonic function in a half-space is uniquely determined by its boundary values
• The boundary values of a bounded harmonic function can be understood in terms of the limit of the function as it approaches the boundary
• The relationship between bounded harmonic functions and their boundary values is a fundamental aspect of potential theory

Bounded harmonic functions in balls

• Bounded harmonic functions in balls, or spherical domains, exhibit special properties due to the symmetry of the domain
• A ball in $\mathbb{R}^n$ is a domain of the form $B(x_0, r) = \{x \in \mathbb{R}^n : |x - x_0| < r\}$, where $x_0$ is the center and $r > 0$ is the radius

Mean value property

• The mean value property is a characteristic property of harmonic functions in balls
• It states that the value of a harmonic function at the center of a ball is equal to the average of its values on the surface of the ball
• Mathematically, if $u(x)$ is a harmonic function in a ball $B(x_0, r)$, then $u(x_0) = \frac{1}{|\partial B(x_0, r)|} \int_{\partial B(x_0, r)} u(x) dS(x)$, where $|\partial B(x_0, r)|$ denotes the surface area of the ball
• The mean value property is a consequence of the maximum and minimum principles for harmonic functions

Representation using spherical harmonics

• Bounded harmonic functions in balls can be represented using spherical harmonics
• Spherical harmonics are special functions defined on the unit sphere that form an orthonormal basis for the space of square-integrable functions on the sphere
• Any bounded harmonic function in a ball can be expanded as a series of spherical harmonics, providing a useful representation for studying their properties
• The coefficients of the spherical harmonic expansion are related to the boundary values of the harmonic function on the sphere

Bounded harmonic functions vs bounded holomorphic functions

• Bounded harmonic functions and bounded holomorphic functions are closely related concepts in complex analysis and potential theory
• A holomorphic function is a complex-valued function that is differentiable in a certain sense, satisfying the Cauchy-Riemann equations
• Bounded holomorphic functions are holomorphic functions that are bounded in absolute value throughout their domain
• The real and imaginary parts of a bounded holomorphic function are bounded harmonic functions, establishing a connection between the two concepts
• However, not every bounded harmonic function is the real or imaginary part of a bounded holomorphic function, highlighting the differences between the two classes of functions

Bounded harmonic functions in unbounded domains

• The study of bounded harmonic functions in unbounded domains is of particular interest in potential theory
• Unbounded domains are domains that extend to infinity in at least one direction, such as the entire space $\mathbb{R}^n$ or half-spaces

Behavior at infinity

• The behavior of bounded harmonic functions at infinity is a crucial aspect of their study in unbounded domains
• Liouville's theorem states that a bounded harmonic function defined on the entire space $\mathbb{R}^n$ must be constant
• In other unbounded domains, such as half-spaces or exterior domains, the behavior of bounded harmonic functions at infinity is characterized by their boundary values or asymptotic properties
• Understanding the behavior of bounded harmonic functions at infinity is essential for studying their global properties and for solving boundary value problems

Phragmén-Lindelöf principle

• The Phragmén-Lindelöf principle is a powerful result in complex analysis and potential theory that relates the growth of a function at infinity to its boundary values
• It states that if a holomorphic function is bounded in a sector of the complex plane and has a certain growth rate at infinity, then it must be bounded throughout the sector
• The Phragmén-Lindelöf principle has analogues for harmonic functions in unbounded domains, providing insights into their behavior at infinity
• The principle is used to establish uniqueness results and to study the asymptotic behavior of solutions to partial differential equations in unbounded domains

Bounded harmonic functions on Riemannian manifolds

• The study of bounded harmonic functions extends beyond Euclidean spaces to more general settings, such as Riemannian manifolds
• A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which allows for the measurement of distances, angles, and curvature

Generalizations of maximum and minimum principles

• The maximum and minimum principles for bounded harmonic functions have natural generalizations to the setting of Riemannian manifolds
• These generalizations take into account the geometry of the manifold and the properties of the Laplace-Beltrami operator, which is the analogue of the Laplacian on Riemannian manifolds
• The generalized maximum and minimum principles provide insights into the behavior of bounded harmonic functions on Riemannian manifolds and are used to establish various properties and theorems

Cheng's Liouville theorem

• Cheng's Liouville theorem is a generalization of Liouville's theorem for bounded harmonic functions to the setting of Riemannian manifolds
• It states that if a Riemannian manifold has nonnegative Ricci curvature and admits a bounded harmonic function, then the manifold must be isometric to Euclidean space and the harmonic function must be constant
• Cheng's Liouville theorem highlights the interplay between the geometry of the manifold and the existence of bounded harmonic functions
• The theorem has important implications in geometric analysis and the study of harmonic functions on Riemannian manifolds

Bounded harmonic functions in potential theory

• Bounded harmonic functions play a central role in potential theory, which is the study of harmonic functions and their applications to various problems in mathematics and physics

Capacity and equilibrium measures

• Capacity is a fundamental concept in potential theory that quantifies the size or thickness of a set in terms of its ability to support certain measures
• The capacity of a set is defined using the notion of equilibrium measures, which are probability measures that minimize a certain energy functional
• Bounded harmonic functions are closely related to capacity and equilibrium measures, as they can be used to characterize the behavior of these objects
• The study of capacity and equilibrium measures provides insights into the fine properties of sets and their relationship to harmonic functions

Riesz decomposition theorem

• The Riesz decomposition theorem is a fundamental result in potential theory that decomposes a subharmonic function into the sum of a potential and a harmonic function
• A subharmonic function is a function that satisfies a certain inequality related to the Laplacian, generalizing the concept of harmonic functions
• The Riesz decomposition theorem states that any subharmonic function $u(x)$ can be written as $u(x) = p(x) + h(x)$, where $p(x)$ is a potential and $h(x)$ is a harmonic function
• The potential $p(x)$ captures the singular behavior of the subharmonic function, while the harmonic function $h(x)$ represents the smooth or regular part
• The Riesz decomposition theorem has important applications in potential theory, complex analysis, and partial differential equations, providing a way to study the structure and properties of subharmonic functions