Skip to main content Accessibility help
×
Home

Embeddings for the space $LD_\gamma ^{p}$ on sets of finite perimeter

  • Nikolai V. Chemetov (a1) and Anna L. Mazzucato (a2)

Abstract

Given an open set with finite perimeter $\Omega \subset {\open R}^n$ , we consider the space $LD_\gamma ^{p}(\Omega )$ , $1\les p<\infty $ , of functions with pth-integrable deformation tensor on Ω and with pth-integrable trace value on the essential boundary of Ω. We establish the continuous embedding $LD_\gamma ^{p}(\Omega )\subset L^{pN/(N-1)}(\Omega )$ . The space $LD_\gamma ^{p}(\Omega )$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.

Copyright

References

Hide All
1Acosta, G., Armentano, M. G., Durán, R. G. and Lombardi, A. L.. Non-homogeneous Neumann problem for the Poisson equation in domains with an external cusp. J. Math. Anal. Appl. 310 (2005), 397411.
2Acosta, G., Durán, R. G. and López García, F.. Korn inequality and divergence operator: counterexamples and optimality of weighted estimates. Proc. Amer. Math. Soc. 141 (2013), 217232.
3Adams, R.. Sobolev spaces (Boston, MA: Academic Press, 1975).
4Ambrosio, L., Mortola, S. and Tortorelli, V. M.. Functionals with linear growth defined on vector valued BV functions. J. Math. Pures et Appt. 70 (1991), 269323.
5Ambrosio, L., Fusco, N. and Pallara, D.. Functions of bounded variation and free discontinuity problems (Oxford: Oxford Science publications, Clarendon press, 2000).
6Babadjian, J. F.. Traces of functions of bounded deformation. Indiana Univ. Math. J. 64 (2015), 12711290.
7Besov, O. V.. Integral estimates for differentiable functions on irregular domains. Doklady Mathematics 1 (2010), 8790 (published in Doklady Academii Nauk, 430, 5 (2010) 583–585).
8Bost, C., Cottet, G.-H. and Maitre, E.. Convergence analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid. SIAM J. Numer. Anal. 48 (2010), 1311337.
9Chemetov, N. V. and Nečasová, Š.. The motion of the rigid body in viscous fluid including collisions. Global solvability result. Nonlinear Anal.: Real World Appl. 34 (2017), 416445.
10Demengel, F. and Demengel, G.. Functional spaces for the theory of elliptic partial differential equations Translated from the 2007 French original by Reinie Erné. Universitext. (Springer, London; EDP Sciences, Les Ulis, 2012).
11Evans, L. C. and Gariepy, R. E.. Measure theory and fine properties of functions (CRC Press, 1991).
12Federer, H.. Geometric measure theory (Springer, 1969).
13Feireisl, E., Hillairet, M. and Nečasová, Š.. On the motion of several rigid bodies in an incompressible non-Newtonian fluid. Nonlinearity 21 (2008), 13491366.
14Gérard-Varet, D. and Hillairet, M.. Existence of weak solutions up to collision for viscous fluid-solid systems with slip. Comm. Pure Appl. Math. 67 (2014), 20222075.
15Gérard-Varet, D., Hillairet, M. and Wang, C.. The influence of boundary conditions on the contact problem in a 3D Navier-Stokes flow. J. Math. Pures Appl. 103 (2015), 138.
16Giusti, E.. Minimal surfaces and functions of bounded variation (Birkhttuser, 1984).
17Grisvard, P.. Problemes aux limites dans des domaines avec points de rebroussement. Ann. Fac. Sci. Toulouse 4 (1995), 561578.
18Gunzburger, M. D., Lee, H.-C. and Seregin, G. A.. Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2 (2000), 219266.
19Hesla, T. I.. Collision of smooth bodies in a viscous fluid: A mathematical investigation (2005), PhD Thesis – Minnesota.
20Hillairet, M.. Lack of collision between solid bodies in a 2D incompressible viscous flow. Comm. Partial Differ. Equ. 32 (2007), 13451371.
21Hoffmann, K.-H. and Starovoitov, V. N.. On a motion of a solid body in a viscous fluid. Two dimensional case. Adv. Math. Sci. Appl. 9 (1999), 633648.
22Judakov, N. V.. The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid. (Russian) Dinamika Splošn. Sredy Vyp. 18 (1974), 249253.
23Kilpelainen, T. and Maly, J.. Sobolev inequalities on sets with irregular boundaries. Z. Anal. Anwendungen 19 (2000), 369380. The correction of the proof in ‘A correction to: Sobolev inequalities on sets with irregular boundaries’.
24Labutin, D. A.. Definitiveness of Sobolev inequalities for a class of irregular domains. Proc. Steklov Inst. Math. 232 (2001), 211215.
25Leoni, G.. A first course in Sobolev spaces. Graduate studies in mathematics, vol. 105 (Providence, Rhode Island: AMS, 2009).
26Maz'ya, V. G.. Classes of domains and embedding theorems for function spaces. Soviet Math. Dokl. 133 (1960), 882885.
27Maz'ya, V. G. and Poborchi, S. V.. Differentiable functions on bad domains (River Edge, NJ: World Scientific Publishing Co., 1997).
28Neustupa, J. and Penel, P.. Existence of a weak solution to the Navier-Stokes equation with Navier's boundary condition around striking bodies. Comptes Rendus Mathematique 347 (2009), 685690.
29Neustupa, J. and Penel, P.. A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall. In The book: advances in Mathematical Fluid Mechanics: dedicated to Giovanni Paolo Galdi, Rannacher, R., Sequeira, A. (eds) (Berlin: Springer-Verlag, 2010),pp. 385408.
30Planas, G. and Sueur, F.. On the ‘viscous incompressible fluid + rigid body’ system with Navier conditions. Annales de l'I.H.P. Analyse non linÃⒸaire 31 (2014), 5580.
31San Martin, J. A., Starovoitov, V. and Tucsnak, M.. Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal. 161 (2002), 93112.
32Starovoitov, V. N.. Behavior of a rigid body in an incompressible viscous fluid near boundary. In The book: International Series of Numerical Mathematics (Basel: Birkhäuser), vol. 147 (2003),pp. 313327.
33Takahashi, T.. Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Advances in Differ. Equ. 8 (2003), 14991532.
34Temam, R.. Problèmes mathématique en plasticité' (Gauthier-Villars: Bordas, Paris, 1983).
35Temam, R. and Strang, G.. Functions of bounded deformation. Arch. Ration. Mech. Anal. 75 (1980), 721.
36Vol'pert, A. I.. The spaces BV and quasilinear equations. Mat. Sb. 73 (1967), 255302.
37Vol'pert, A. I. and Hudjaev, S. I.. Analysis in classes of discontinuous functions and equations of mathematical physics (Martinus Nijhoff Publishers, 1985).
38Weck, N.. Local compactness for linear elasticity in irregular domains. Math. Meth. Appl. Sci. 17 (1994), 107113.
39Ziemer, W.. Weakly differentiable functions (Springer, 1989).

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed