Trace Operator - Application

Application

Consider the problem of solving Poisson's equation with zero boundary conditions:

$\begin{cases} -\Delta u = f \text{ in } \Omega\\ u_{|\partial \Omega} = 0. \end{cases}$

Here, is a given continuous function on

With the help of the concept of trace, define the subspace to be all functions in the Sobolev space (this space is also denoted ) whose trace is zero. Then, the equation above can be given the weak formulation

Find in such that

for all in

Using the Lax–Milgram theorem one can then prove that this equation has precisely one solution, which implies that the original equation has precisely one weak solution.

One can employ similar ideas to prove the existence and uniqueness of more complicated partial differential equations and with other boundary conditions (such as Neumann and Robin), with the notion of trace playing an important role in all such problems.

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