🔢Potential Theory Unit 7 – Subharmonic Functions & Maximum Principle

Definition and Basic Properties

• Subharmonic functions are upper semicontinuous functions that satisfy the sub-mean value property
• For a function $u$ to be subharmonic on a domain $\Omega$, it must satisfy $u(x) \leq \frac{1}{|B_r(x)|} \int_{B_r(x)} u(y) dy$ for all balls $B_r(x) \subset \Omega$
  • This means the value of $u$ at any point is less than or equal to the average value of $u$ over any ball centered at that point
• Subharmonic functions are not necessarily continuous, but they are upper semicontinuous
  • Upper semicontinuity means that for any point $x_0$ and $\varepsilon > 0$, there exists a neighborhood $U$ of $x_0$ such that $u(x) < u(x_0) + \varepsilon$ for all $x \in U$
• The set of subharmonic functions on a domain $\Omega$ is a convex cone, meaning that if $u$ and $v$ are subharmonic, then $au + bv$ is also subharmonic for any $a, b \geq 0$
• Subharmonic functions satisfy the maximum principle, which states that a subharmonic function cannot attain its maximum value at an interior point of its domain unless it is constant

Subharmonic Functions: Key Concepts

• Subharmonic functions are closely related to the Laplace operator $\Delta = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$
  • A twice continuously differentiable function $u$ is subharmonic if and only if $\Delta u \geq 0$
• The family of subharmonic functions includes important classes of functions such as convex functions and plurisubharmonic functions
  • Convex functions are subharmonic, and the sum of two convex functions is also convex (and thus subharmonic)
  • Plurisubharmonic functions are subharmonic functions defined on complex domains that satisfy additional properties
• Subharmonic functions have important applications in potential theory, complex analysis, and partial differential equations
• The Riesz Decomposition Theorem states that any subharmonic function can be decomposed into the sum of a potential and a harmonic function
  • This decomposition is unique up to an additive constant
• Subharmonic functions satisfy the strong maximum principle, which states that if a subharmonic function attains its maximum value at an interior point, then it must be constant on the entire connected component containing that point

Relationship to Harmonic Functions

• Harmonic functions are smooth functions that satisfy the Laplace equation $\Delta u = 0$
• Every harmonic function is also subharmonic, but the converse is not true
  • This means that the set of harmonic functions is a proper subset of the set of subharmonic functions
• The difference between a subharmonic function and a harmonic function is called a potential
  • If $u$ is subharmonic and $h$ is harmonic, then $u - h$ is a potential
• Subharmonic functions can be approximated by smooth subharmonic functions using mollification techniques
  • This allows for the extension of properties of smooth subharmonic functions to general subharmonic functions
• The Poisson integral formula expresses a harmonic function as the integral of its boundary values against the Poisson kernel
  • This formula can be extended to subharmonic functions, providing a way to reconstruct a subharmonic function from its boundary values

Maximum Principle for Subharmonic Functions

• The maximum principle is a fundamental property of subharmonic functions that constrains their behavior
• The weak maximum principle states that a subharmonic function $u$ on a domain $\Omega$ satisfies $\sup_\Omega u = \sup_{\partial \Omega} u$
  • In other words, the maximum value of $u$ on the closure of $\Omega$ is attained on the boundary $\partial \Omega$
• The strong maximum principle states that if a subharmonic function $u$ attains its maximum value at an interior point of $\Omega$, then $u$ must be constant on the entire connected component containing that point
• A consequence of the maximum principle is the uniqueness of solutions to the Dirichlet problem for subharmonic functions
  • If $u$ and $v$ are subharmonic functions on $\Omega$ that agree on $\partial \Omega$, then $u \equiv v$ on $\Omega$
• The maximum principle can be used to prove the Phragmén-Lindelöf principle, which provides growth estimates for subharmonic functions on unbounded domains
• Viscosity solutions of certain partial differential equations, such as the Monge-Ampère equation, satisfy a maximum principle that is analogous to the one for subharmonic functions

Applications in Potential Theory

• Subharmonic functions play a crucial role in potential theory, which studies the behavior of functions that arise from physical phenomena such as electrostatics and gravitation
• The Riesz Decomposition Theorem allows for the representation of subharmonic functions as the sum of a potential and a harmonic function
  • This decomposition is used to study the fine properties of subharmonic functions and their associated Riesz measures
• Subharmonic functions are closely related to the concept of capacity, which measures the ability of a set to support a certain class of functions
  • Sets of capacity zero are precisely the polar sets of subharmonic functions
• The Poisson-Jensen formula is a generalization of the Poisson integral formula that expresses a subharmonic function in terms of its boundary values and the Riesz measure associated with its Laplacian
• Subharmonic functions are used to construct Green functions and Evans potentials, which are fundamental objects in potential theory
  • Green functions are used to solve boundary value problems, while Evans potentials are used to study the fine properties of Riesz measures

Boundary Value Problems

• Boundary value problems involve finding a function that satisfies a partial differential equation on a domain and takes prescribed values on the boundary of the domain
• The Dirichlet problem for subharmonic functions asks for a subharmonic function $u$ on a domain $\Omega$ that takes prescribed continuous values on the boundary $\partial \Omega$
  • The Perron method is a constructive approach to solving the Dirichlet problem using upper and lower envelopes of subharmonic functions
• The regularity of the solution to the Dirichlet problem depends on the regularity of the boundary data and the domain
  • For smooth boundary data and domains with sufficiently regular boundaries, the solution is smooth up to the boundary
• The Neumann problem involves finding a function that satisfies a partial differential equation on a domain and whose normal derivative takes prescribed values on the boundary
  • Subharmonic functions can be used to study the solvability and regularity of solutions to the Neumann problem
• The Poisson integral formula and its generalizations provide a way to represent the solution to the Dirichlet problem in terms of the boundary data
  • These representations are useful for studying the behavior of the solution near the boundary and for proving regularity results

Advanced Topics and Extensions

• Plurisubharmonic functions are a generalization of subharmonic functions to complex domains
  • A function $u$ is plurisubharmonic if it is upper semicontinuous and subharmonic on every complex line
• Plurisubharmonic functions have important applications in complex analysis and complex geometry
  • They are used to define the Levi form, which measures the curvature of the boundary of a complex domain
  • Plurisubharmonic functions are also used to study the complex Monge-Ampère equation and its applications to Kähler geometry
• Viscosity solutions provide a generalized notion of solution for partial differential equations that may not have classical solutions
  • Subharmonic functions are closely related to viscosity subsolutions of the Laplace equation
• The theory of subharmonic functions can be extended to more general elliptic partial differential equations, such as the $p$-Laplace equation and fully nonlinear equations
  • These extensions require more sophisticated tools from nonlinear potential theory and viscosity solution theory
• Subharmonic functions have applications in other areas of mathematics, such as harmonic analysis, geometric measure theory, and dynamical systems
  • For example, subharmonic functions are used to study the fine properties of harmonic measure and to construct invariant measures for dynamical systems

Practice Problems and Examples

• Verify that the function $u(x, y) = |x|$ is subharmonic on $\mathbb{R}^2$ but not harmonic
• Prove that the maximum of two subharmonic functions is also subharmonic
• Find the Riesz decomposition of the function $u(x, y) = -\log(x^2 + y^2)$ on the unit disk $\{(x, y) : x^2 + y^2 < 1\}$
• Solve the Dirichlet problem for the function $u(x, y) = x^2 - y^2$ on the square $[-1, 1] \times [-1, 1]$
  • Hint: Use the Poisson integral formula and the fact that $u$ is harmonic
• Prove that the function $u(z) = |z|^2$ is plurisubharmonic on $\mathbb{C}$
• Show that the function $u(x, y) = x^2 + y^2$ is a viscosity subsolution of the Laplace equation on $\mathbb{R}^2$
• Construct a subharmonic function on the unit disk that has a logarithmic pole at the origin
  • Hint: Use the fundamental solution of the Laplace equation
• Prove that the Dirichlet problem for subharmonic functions is solvable on any bounded domain with continuous boundary data