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# A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function

• The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.
Mathematics Subject Classification: Primary: 74F10; Secondary: 46E35.

 Citation:

•  [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.doi: 10.1007/s00021-004-0121-y. [3] C. H. A. Cheng and S. Shkoller, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., 42 (2010), 1094-1155.doi: 10.1137/080741628. [4] S. Čanić and B. Muha, A nonlinear moving-boundary problem of parabolic-hyperbolic-hyperbolic type arising in fluid-multi-layered structure interaction problems, to appear in Proceedings of the Fourteenth International Conference on Hyperbolic Problems: Theory, Numerics and Applications, American Institute of Mathematical Sciences (AIMS) Publications. [5] Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proc. Amer. Math. Soc., 124 (1996), 591-600.doi: 10.1090/S0002-9939-96-03132-2. [6] C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.doi: 10.1137/070699196. [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [8] I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, DCDS-A, 32 (2012), 1355-1389.doi: 10.3934/dcds.2012.32.1355. [9] D. Lengeler and M. Ružička, Weak solutions for an incompressible newtonian fluid interacting with a linearly elastic koiter shell, Arch. Ration. Mech. Anal., 211 (2014), 205-255.doi: 10.1007/s00205-013-0686-9. [10] J. Lequeurre, Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation, J. Math. Fluid Mech., 15 (2013), 249-271.doi: 10.1007/s00021-012-0107-0. [11] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York, 1972. [12] B. Muha and S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919-968.doi: 10.1007/s00205-012-0585-5. [13] B. Muha and S. Čanić, Existence of a solution to a fluid-multi-layered-structure interaction problem, J. of Diff. Equations, 256 (2014), 658-706.doi: 10.1016/j.jde.2013.09.016.