# The dependence of constant in a trace theorem on the diameter of domain

The trace theorem says for a nice domain, say bounded Lipschitz domain, $\Omega$, we have $$\left\Vert Tu \right\Vert_{H^{1/2}(\partial \Omega)} \leq C \left\Vert u \right\Vert_{H^1(\Omega)},$$ where $Tu$ is the trace of $u$. I would like to know how the constant $C$ depends on $\mathrm{diam}(\Omega)$.

• If you consider rescalings $t\Omega$ of the domain, you can easily derive from scaling the correct behaviour of $C(t)$. In general, the constant depends on more than just the diameter though... – Willie Wong Apr 14 '16 at 12:58
• Is there any good reference for the proof of the trace theorem with the $H^{1/2}$ trace space? – Engineer Apr 14 '16 at 18:24
• Adams and Fournier comes to mind, or Triebel's Theory of function spaces. But I am sure it is done in many books. – Willie Wong Apr 14 '16 at 19:30

Since $Tu=T(\chi u)$ for any smooth cutoff $\chi$ equal to 1 near the boundary, the norm of the trace operator should be independent of the size of the open set (maybe it could be influenced by the domain being too small). For a flat boundary the norm can be computed exactly without much difficulty, then an estimate of the norm in the general case could be obtained by estimating the norm of the change of variables necessary to straighten the boundary.