Preface

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This section presents the basic properties of solutions to the Laplace and Helmholtz equations.

Laplace's equation (also called the potential equation or harmonic equation) is a second-order partial differential equation named after Pierre-Simon Laplace who, beginning in 1782, studied its properties while investigating the gravitational attraction of arbitrary bodies in space. However, the equation first appeared in 1752 in a paper by Euler on hydrodynamics. Laplace's equation is often written as:

where \( \Delta = \nabla^2 = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \cdots + \frac{\partial^2}{\partial x_n^2} \) is the Laplace operator or Laplacian in n dimensional space. Here Ω is some region in ℝⁿ with smooth boundary. In two dimensions, the Laplace equation in rectangular coordinates becomes Correspondingly, in three dimensions, Laplace's equation is

Another very important version of the above equation is the so called the Helmholtz equation

named after the German physician and physicist Hermann von Helmholtz (1821--1894). If the region Ω extends to infinity, then it is assumed that a solution to the Laplace's or Helmholtz's equation satisfies additional constraint: it should approach zero at infinity for dimensions larger than 2. In two dimensions, it is sufficient to require soundness of the solution.

A nonhomogeneous version of the Laplace equation

where f is a given smooth function, is called the Poisson's equation. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson (1781--1840). The Poisson's equation in three dimensional case is where f is a given smooth function. A particular solution of the Poisson equation \( \Delta u = f \) is given by in two dimensions. For three dimensional space, we have a particular solution to the Poisson equation ∇²u = f: In general n-dimensional case, the following function gives a particular solution to the Poisson's equation for the Laplacian, where ω_n is the surface area of the unit sphere \( x_1^2 + x_2^2 + \cdots + x_n^2 =1 . \)

The Laplacian occurs in different situations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point. We summarize the standard examples of its applications in the following table.

Application	u	f
Incompressible irrotational flow in two or three dimensions	u = velocity potential, velocity = q = ∇u	Mass source strength/unit volume
Electrostatics	u = electrostatic potential; electric field = E = ∇u	Charge/unit volume
Steady state temperature in a solid	u = temperature	Heat source strength/unit volume
Two dimensional incompressible steady flow	u = stream function = ψ = velocity = q = ψxi - ψyj	-Vorticity
Static deflection of a thin membrane in two dimensions	u = deflection	Pressure
Newtonian gravitation	u = gravitational potential; force of gravity = F = - ∇u	Mass density

Laplace's or Helmholtz equations have infinite many solutions. To take out one of them, we need to impose some conditions that usually follow from physical counterpart. Since there is no time dependence in the Laplace's equation or Poisson's equation, there is no initial conditions to be satisfied by their solutions. However, there should be certain boundary conditions on the boundary curve or surface \( \partial\Omega \) of the region Ω in which the differential equation is to be solved. Typically, there are known three types of boundary conditions.

• The Dirichlet boundary conditions or boundary conditions of the first type:
  \[ \left. u \right\vert_{\partial\Omega} = f({\bf x}) \quad\mbox{for interior problem or}\quad {\bf x} \in \mathbb{R}^n \setminus \Omega \quad \mbox{for exterior problem}, \qquad {\bf x} \in \partial\Omega , \]
  where f is a given function, defined on the boundary \( \partial\Omega . \) The first-type boundary condition is named after the German mathematician Peter Gustav Lejeune Dirichlet (1805--1859).
• The Neumann or second-type boundary condition:
  \[ \left. \frac{\partial u}{\partial {\bf n}} \right\vert_{\partial\Omega} \equiv \left(\nabla u({\bf x}) \cdot {\bf n}({\bf x}) \right)_{\partial\Omega} = f({\bf x}), \qquad {\bf x} \in \partial\Omega , \]
  where n denotes the exterior unit normal vector to the boundary \( \partial\Omega . \) The second-type boundary condition is named after the German mathematician Carl (also Karl) Gottfried Neumann (1832--1925).
• The third type boundary condition is a type of boundary condition that combines two previously defined boundary conditions:
  \[ \left( a({\bf x})\,u({\bf x}) + b({\bf x})\,\frac{\partial u}{\partial {\bf n}} \right)_{\partial\Omega} \equiv \left( a({\bf x})\,u({\bf x}) + b({\bf x})\,\nabla u({\bf x}) \cdot {\bf n}({\bf x}) \right)_{\partial\Omega} = f({\bf x}), \qquad {\bf x} \in \partial\Omega , \]
  where a and b are nonzero functions or constants. The third boundary condition is variously designated, but sometimes it is called Robin's boundary condition, which is mistakenly associated with the French mathematical analyst Victor Gustave Robin (1855--1897) from the Sorbonne in Paris. Actually, Robin never used this boundary condition as it follows from the historical research article by Karl Gustafson and Takehisa Abe.
• The Wentzell boundary conditions
  \[ \left( a({\bf x})\,u({\bf x}) + b({\bf x})\,\frac{\partial u}{\partial {\bf n}} - c({\bf x})\,\nabla^2 u \right)_{\partial\Omega} \equiv \left( a({\bf x})\,u({\bf x}) + b({\bf x})\,\nabla u({\bf x}) \cdot {\bf n}({\bf x}) c c({\bf x}) \,\nabla^2 u \right)_{\partial\Omega} = f({\bf x}), \qquad {\bf x} \in \partial\Omega , \]
  were introduced in 1959 by the Russian mathematician Aleksander Dmitrievich Venttsel'     (born 1937) from Moscow University.

The problem of finding a solution of Laplace's equation that takes on given boundary values is known as a Dirichlet problem. However, if the values of the normal derivative are prescribed on the boundary, the problem is said to be a Neumann boundary value problem. Physically, it is plausible to expect that three types of boundary conditions will be sufficient to determine the solution completely. Indeed, it is possible to establish the existence and uniqueness of the solution of Laplace's (and Poisson's) equation under the first and third type boundary conditions, provided that the boundary \( \partial\Omega \) of the domain Ω is smooth (have no corner or edge). However, we avoid explicit statements and their proofs because the material is beyond the scope of the tutorial. Instead, we concentrate our attention on some typical problems that could be solved by means of separation of variables.

Actually, there are known two kinds of boundary value problems: internal or inner when we are after solutions inside the given domain Ω, and external or outer when solutions are to be found outside the given domain, namely in \( {\mathbb R}^n \setminus \Omega . \)

Note that the above uniqueness theorems are valid also for external boundary value problems. Often, the Dirichlet boundary value problems arise with discontinuous boundary conditions. If we assume that the function u satisfies Laplace's equation inside the domain Ω, contiguously adjoining the boundary values at the points of continuity of the latter, and bounded in neighborhoods of points of discontinuity, then the solution to the first type boundary value problem is unique.

It will be shown in next sections that boundary value problems can be solved without much difficulty for simple two-dimensional regions, such as rectangle, half-plane, circle, and so on, the problem is much more difficult for general domains. There are, of course, methods for approximating and solving boundary value problems such as the method of Balayage (see Tsuji's book), the Perron method, finite-difference methods, finite element method, Monte Carlo method, Galerkin method, and boundary element method to name some, but nevertheless the problem is difficult.

An important technique to solve boundary value problems is integral transformation. We will demonstrate later its applications in some relatively simple domain. For example, the Fourier integrals can be used to determine functions of Laplace's or Helmholtz operators in infinite space ℝⁿ. The Laplace operator Δ in the space ℝⁿ has only negative eigenvalues, so its negation will be a positive one. We can define its square root (using Fourier transformation) as where x ∈ ℝⁿ and C_n is a constant.

The real and imaginary part of any holomorphic function yield harmonic functions on ℝ² (these are said to be a pair of harmonic conjugate functions). the Laplace operator on the plane can be decomposed as \( \displaystyle \nabla^2 = \left( \frac{\partial}{\partial x} + {\bf j}\,\frac{\partial}{\partial y} \right) \left( \frac{\partial}{\partial x} - {\bf j}\,\frac{\partial}{\partial y} \right) \) where j is the unit vector in positive vertical direction on the complex plane ℂ, so j² = -1. Therefore, the solution of Laplace's equation in two dimensional domains is intimately related to the theory of analytic functions of a complex variable. This is immediately seen observing that, writing z = x + jy, the complex function g(z) = u_x − j u_y is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Hence, g has locally a primitive f, and u is the real part of f up to a constant, as u_x is the real part of f′ = g. In fact, this topic occupies a significant portion of texts on complex variables, and we are not after it.

Finally, we present two important methods to solve boundary value problems in arbitrary domain. We start with Green's function method, and then go to the reduction to integral equations---the technique used in finite element method.

 Green's function method  Reduction to integral equations

 

References

 

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