Section 19.3 PDE Theorems
Subsection The Main Idea: Initial Conditions
In physics situations, the classification and types of boundary conditions are typically straightforward: if there are two time derivatives, the equation is hyperbolic and we will need two initial conditions on the entire spatial region to make the solution unique; if there is only a single time derivative, the equation is parabolic and we will need only a single initial condition; if the equation has no time derivatives, the equation is elliptic and the solutions are qualitatively different from the previous two cases. It is easiest to understand the elliptic case from an explicit example.
Subsection The Main Idea: Spatial Boundary Conditions
In addition to initial conditions, we will need boundary conditions on the spatial variables. The three main type of boundary conditions encountered in physics are Dirichlet, when the value of the solution of the PDE goes to zero on a continuous portion of the boundary, Neumann, when the normal (to the boundary) derivative of the solution goes to zero on a continuous portion of the boundary, and periodic, when a spatial variable is periodic (for example, on a ring).
The theorems in the optional subsection below, give examples of when a PDE with boundary and initial conditions is guaranteed to have a unique solution. The theorems below consider cases when the (spatial) boundary conditions are either Dirichlet or Neumann on each piecewise smooth piece of the boundary. There are other theorems, not given here, that cover other cases, particularly periodic boundary conditions.
As a mathematical exercise, it would be easy to write down PDEs that do not satisfy a uniqueness theorem. Fortunately, for physicists, the universe provides evidence that unique solutions to the PDEs that are important to physics do exist. So, usually we are not concerned about using the theorems to prove that a solution exists, but rather we are interested in using the theorems to tell us how many of which kinds of boundary and initial conditions we need to specify to ensure that the solution is unique.
Subsection Optional Mathematical Details
Subsubsection Elliptic Equations
Example: Poisson’s Equation
\begin{equation}
\nabla^2 \psi(x_k) = f(x_k)\tag{19.3.1}
\end{equation}
Theorem: If \(\psi(x_k)\) satisfies Poisson’s equation throughout a closed, bounded region \(R\) and satisfies Dirichlet conditions on the the boundary \(\partial R\) of \(R\text{,}\) then \(\psi\) is unique.
Theorem: If \(\psi(x_k)\) satisfies Poisson’s equation throughout a closed, bounded region \(R\) and satisfies Neumann conditions on the the boundary \(\partial R\) of \(R\text{,}\) then \(\psi\) is unique up to an additive constant.
Corollary: If the boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece. If Dirichlet conditions are satisfied on at least one piece then \(\psi\) is unique.
Corollary: If the region \(R\) is unbounded (in some or all directions) but \(\psi = o(r^{-{1/2}})\) as \(r\rightarrow\infty\) (i.e. \(\psi\) falls off faster than \(r^{-1/2}\)) in the unbounded directions, then \(\psi\) is unique.
Subsubsection Parabolic Equations
Example: Inhomogeneous Diffusion Equation
\begin{equation}
\left({\partial \over \partial t} -k\nabla^2\right) \psi(t,x_k) = f(x_k)\tag{19.3.2}
\end{equation}
Theorem: If \(\psi(t, x_k)\) satisfies the inhomogeneous diffusion equation throughout a closed, bounded region \(R\) and satisfies either Dirichlet or Neumann conditions on the the boundary \(\partial R\) of \(R\text{,}\) and \(\psi\) satisfies the initial condition
\begin{equation}
\psi(t=0, x_k) = g(x_k)\tag{19.3.3}
\end{equation}
then \(\psi\) is unique.
Corollary: If the spatial boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece.
Corollary: If the region \(R\) is unbounded (in some or all spatial directions) but \(\psi = o(r^{-{1/2}})\) as \(r\rightarrow\infty\) (i.e. \(\psi\) falls off faster than \(r^{-1/2}\)) in the unbounded directions, then \(\psi\) is unique.
Subsubsection Hyperbolic Equations
Example: Inhomogeneous Wave Equation
\begin{equation}
\left({-1 \over v^2}{\partial^2 \over \partial t^2} +\nabla^2\right) \psi(t,x_k) = f(x_k)\tag{19.3.4}
\end{equation}
Theorem: If \(\psi(t, x_k)\) satisfies the inhomogeneous wave equation throughout a closed, bounded region \(R\) and satisfies either Dirichlet or Neumann conditions on the the boundary \(\partial R\) of \(R\text{,}\) and \(\psi\) satisfies the two initial conditions
\begin{equation}
\psi(t=0, x_k) = g(x_k)\tag{19.3.5}
\end{equation}
\begin{equation}
{\partial\psi \over \partial t}(t=0, x_k) = h(x_k)\tag{19.3.6}
\end{equation}
then \(\psi\) is unique.
Corollary: If the spatial boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece.
Corollary: If the region \(R\) is unbounded (in some or all spatial directions) but \(\psi = o(r^{-{1/2}})\) as \(r\rightarrow\infty\) (i.e. \(\psi\) falls off faster than \(r^{-1/2}\)) in the unbounded directions, then \(\psi\) is unique.
