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On the traces of \(W^{2,p}(\Omega)\) for a Lipschitz domain. (English) Zbl 1029.46031

Summary: The object of this note is to give a characterization of the traces of the Sobolev space \(W^{2,p}(\Omega)\) when \(\Omega \subset \mathbb{R}^2\) is a connected bounded Lipschitz domain (possible not simply connected). Denote with \(\gamma_0\) the trace operator and with \(\partial_n\) the normal derivative on the boundary \(\Gamma\) of \(\Omega\). When \(\Gamma\) is smooth it is known that the range of operator \(\psi\mapsto (\gamma_0 (\psi)\), \(\partial_n\psi)\) is \(W^{2-1/p,p} (\Gamma)\times W^{1-1/p,p}\). In the Lipschitz case, G. Geymonat and F. Krasucki [C. R. Acad. Sci., Paris, Sér. I 330, 355-360 (2000; Zbl 0945.35065)] considered the case \(p=2\) and proved that \((g_0,g_1)\in H^1(\Gamma) \times L^2(\Gamma)\) is in the range of this map if and only if \((\partial_tg_0) {\mathbf n}-g_1 {\mathbf t}\in H^{1/2} (\Gamma)\), where \({\mathbf n}\) is the exterior unit normal and \({\mathbf t}= (-n_2,n_1)\) is the tangential unit vector on \(\Gamma\) which are defined almost everywhere since \(\Omega\) is Lipschitz.
In this work, the authors extend the result of Geymonat and Krasucki (loc. cit.) to the case \(1<p\). Their argument is different from that paper and is based on the existence of a continuous right inverse of the divergence operator on \(W^{1,p}_0(\Omega)\) which is a known nontrivial result.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.

Citations:

Zbl 0945.35065
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