The Energy Method: Uniqueness of Solutions to ODEs and PDEs

An ordinary differential equation (ODE) is an equation involving a scalar-valued function from the real line to the real line, and some of its derivatives. A partial differential equation (PDE) is a generalization of the ordinary differential equation to the event where the argument is in multiple dimensions. In this case we have partial derivatives in each direction. For example, if the input is an ordered pair, then the input belongs to R², so we may have partial derivatives in two different directions. No matter if we’re dealing with an ODE or a PDE, a solution will be a function that you can plug into the equation and satisfy it.

For any given ODE or PDE, there might be one solution; there might be multiple solutions; or there might not be any at all. The purpose of an energy method is to help answer this question. Here is an [admittedly very difficult] energy method problem from the qualifying exam I took at the University of Tennessee-Knoxville.

A problem from the August 2020 University of Tennessee PDE prelim. This can be solved using the infamous energy method.

Even if you don’t understand all the terminology and notation in the problem, the goal of an energy method should remain clear: investigate whether there can be multiple solutions to an ODE (or PDE), or not. Sometimes this is paired with finding an explicit solution, because of the following key idea: