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

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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

Abstract
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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.

Keywords: non-Lipschitz domain., Trace Theorem, Sobolev spaces, fluid-structure interaction.

Mathematics Subject Classification: Primary: 74F10; Secondary: 46E3.

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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