# American Institute of Mathematical Sciences

March  2014, 9(1): 191-196. doi: 10.3934/nhm.2014.9.191

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

 1 Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb

Received  May 2013 Revised  July 2013 Published  April 2014

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.
Citation: Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks and Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191
##### References:
 [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.

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##### References:
 [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.
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