Green's functions are powerful tools in complex analysis, solving Poisson's equation with point sources. They're key to tackling inhomogeneous differential equations and converting them into integral equations, making them essential for solving boundary value problems.

Constructing Green's functions varies based on the domain. In unbounded spaces, we use fundamental solutions and the method of images. For bounded domains, techniques like eigenfunction expansion come into play. These methods help us solve Dirichlet problems and understand harmonic functions better.

Green's functions: Definition and properties

Definition and fundamental properties

• Green's functions solve Poisson's equation with a point source, represented by the Dirac delta function: $\Delta G(x,y) = \delta(x-y)$
• Exhibit symmetry: $G(x,y) = G(y,x)$, allowing the interchange of variables without changing the function value
• Unique up to the addition of a harmonic function, providing flexibility in their construction
• Normal derivative vanishes on the boundary: $\partial G/\partial n = 0$, ensuring compatibility with boundary conditions

Applications to differential equations

• Solve inhomogeneous differential equations with specified boundary conditions
  • Inhomogeneous term represents a source or sink in the problem domain
  • Green's function acts as a fundamental solution, capturing the response to a point source
• Enable the conversion of a differential equation problem into an integral equation
  • Integral equation involves the Green's function and the inhomogeneous term
  • Solution obtained by integrating the product of the Green's function and the source term

Constructing Green's functions

Green's functions in unbounded domains

• Whole space $\mathbb{R}^n$:
  • For $n=2$: $G(x,y) = -\frac{1}{2\pi} \log|x-y|$
  • For $n\geq3$: $G(x,y) = \frac{1}{n(2-n)\omega_n}|x-y|^{2-n}$, where $\omega_n$ is the volume of the unit $n$-sphere
• Half-space: Obtained using the method of images
  • G(x,y) = \Phi(x,y) - \Phi(x,y^*)$, where $y^* is the reflection of $y$ across the boundary
  • $\Phi$ is the fundamental solution in the unbounded domain

Green's functions in bounded domains

• Ball: Constructed using the Poisson integral formula and the method of images
  • Poisson integral formula expresses the solution in terms of the boundary values
  • Method of images accounts for the boundary conditions by introducing virtual sources
• Rectangle: Obtained using the method of eigenfunction expansion
  • Express $G$ as a series of eigenfunctions of the Laplacian operator
  • Eigenfunctions satisfy the boundary conditions and form a complete basis
  • Coefficients determined by imposing the point source condition

Green's functions for Dirichlet problems

Dirichlet problem formulation

• Find a harmonic function $u$ in a domain $\Omega$ with prescribed boundary values $\phi$ on $\partial\Omega$
• Harmonic functions satisfy Laplace's equation: $\Delta u = 0$ inside the domain
• Boundary condition: $u = \phi$ on $\partial\Omega$, specifying the function values on the boundary

Green's function solution

• Solution expressed using the Green's function: $u(x) = \int_{\partial\Omega} \phi(y) \frac{\partial G(x,y)}{\partial n_y} dS_y$
• Normal derivative $\frac{\partial G}{\partial n_y}$ taken with respect to $y$ in the outward direction on the boundary
• Green's function captures the influence of boundary values on the solution at interior points
• Boundary integral represents a weighted average of the boundary values, with weights determined by the Green's function

Numerical implementation

• Discretize the boundary integral by dividing the boundary into segments or elements
• Approximate the Green's function and its normal derivative using numerical methods (finite differences, boundary element method)
• Solve the resulting linear system to obtain the solution at interior points
• Refine the discretization to achieve desired accuracy and convergence

Green's functions vs Poisson kernel

Poisson kernel as a special case

• Poisson kernel is a Green's function for specific domains: unit disk and upper half-plane
• Unit disk: Poisson kernel in polar coordinates $(r,\theta)$: $P(r,\theta) = \frac{1-r^2}{1-2r \cos \theta + r^2}$
• Upper half-plane: Poisson kernel in Cartesian coordinates $(x,y)$ with $y>0$: $P(x,y) = \frac{y}{\pi(x^2+y^2)}$

Poisson integral formula

• Dirichlet problem solution on the unit disk: $u(r,\theta) = \frac{1}{2\pi} \int_0^{2\pi} \phi(e^{i\phi}) P(r,\theta-\phi) d\phi$
• Expresses the solution as a convolution of the boundary values with the Poisson kernel
• Poisson kernel acts as a weight function, determining the influence of boundary values on interior points

Relationship between Green's functions and Poisson kernels

• Green's functions can be used to derive the Poisson kernel and the Poisson integral formula
  • Poisson kernel obtained by taking the normal derivative of the Green's function on the boundary
  • Poisson integral formula derived by applying Green's identity and using the properties of the Green's function
• Poisson kernels provide explicit formulas for solving Dirichlet problems in specific domains
• Green's functions offer a more general framework for solving boundary value problems in various geometries