What the meaning of 'in the trace sense'?

Question

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There are always descriptions about the value of boundary. I want to know if this condition is necessary.

Answer 1

There are different ways you can require your solution to take a value on the boundary (say zero). So you can require that the limit of the solution at the boundary be equal to zero at each point (assuming you approach the point from inside the region). Or you can require that $Tr(u)=0$, which is a weaker requirement, since $Tr(u)$ is in a Lebesgue space (typically $L^{1/2}$). This is what the phrase "in the trace sense" means.