American Institute of Mathematical Sciences

Advanced
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

Boris Muha 1,

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.
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 & 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, (1975).   Google Scholar

[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.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[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.  doi: 10.1137/080741628.  Google Scholar

[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, ().   Google Scholar

[5]

Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains,, Proc. Amer. Math. Soc., 124 (1996), 591.  doi: 10.1090/S0002-9939-96-03132-2.  Google Scholar

[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.  doi: 10.1137/070699196.  Google Scholar

[7]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).   Google Scholar

[8]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, DCDS-A, 32 (2012), 1355.  doi: 10.3934/dcds.2012.32.1355.  Google Scholar

[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.  doi: 10.1007/s00205-013-0686-9.  Google Scholar

[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.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[11]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I,, Translated from the French by P. Kenneth, (1972).   Google Scholar

[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.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.09.016.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[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.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[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.  doi: 10.1137/080741628.  Google Scholar

[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, ().   Google Scholar

[5]

Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains,, Proc. Amer. Math. Soc., 124 (1996), 591.  doi: 10.1090/S0002-9939-96-03132-2.  Google Scholar

[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.  doi: 10.1137/070699196.  Google Scholar

[7]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).   Google Scholar

[8]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, DCDS-A, 32 (2012), 1355.  doi: 10.3934/dcds.2012.32.1355.  Google Scholar

[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.  doi: 10.1007/s00205-013-0686-9.  Google Scholar

[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.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[11]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I,, Translated from the French by P. Kenneth, (1972).   Google Scholar

[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.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.09.016.  Google Scholar

[1]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633
[2]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199
[3]

Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787
[4]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355
[5]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[6]

George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations & Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563
[7]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[8]

Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks & Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397
[9]

George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817
[10]

Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269
[11]

Martina Bukač, Sunčica Čanić. Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation. Mathematical Biosciences & Engineering, 2013, 10 (2) : 295-318. doi: 10.3934/mbe.2013.10.295
[12]

Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
[13]

George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
[14]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
[15]

Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107
[16]

Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.

[17]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[18]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813
[19]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[20]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences