Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-22T10:19:13.857Z Has data issue: false hasContentIssue false
  • English
  • Français

EMBEDDING AND TRACE RESULTS FOR VARIABLE EXPONENT SOBOLEV AND MAZ'YA SPACES ON NON-SMOOTH DOMAINS

Part of: General theory of linear operators Elliptic equations and systems Special classes of linear operators

Published online by Cambridge University Press:  21 July 2015

ALEJANDRO VÉLEZ-SANTIAGO
ALEJANDRO VÉLEZ-SANTIAGO*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521-0135, USA e-mail: avelez@math.ucr.edu, alejandro.velez2@upr.edu, alejovelez32@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

MSC classification

Primary: 47A63: Operator inequalities
Secondary: 47B38: Operators on function spaces (general) 47A07: Forms (bilinear, sesquilinear, multilinear) 35J92: Quasilinear elliptic equations with $p$-Laplacian

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 471 - 489
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Acerbi, E. and Mingione, G., Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213–259.CrossRefGoogle Scholar
2.Achdou, Y. and Tchou, N., Trace results on domains with self-similar fractal boundaries, J. Math. Pures Appl. 89 (2008), 596–623.Google Scholar
3.Biegert, M., A priori estimate for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math. 133 (2010), 273–306.CrossRefGoogle Scholar
4.Biegert, M.On trace of Sobolev functions on the boundary of extension domains, Proc. Am. Math. Soc. 137 (2009), 4169–4176.CrossRefGoogle Scholar
5.Biegert, M. and Warma, M., Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad“ domains, Adv. Differ. Equ. 15 (2010), 893–924.Google Scholar
6.Bollt, E. M., Chartrand, R., Esedoglu, S., Schulz, P. and Vixie, K. R., Graduated adaptive image denoising: Local compromise between total variation and isotropic diffusion, Adv. Comput. Math. 31 (2009), 61–85.CrossRefGoogle Scholar
7.Bojarski, B., Remarks on Sobolev imbedding inequalities, in Proc. of the Conference on Complex Analysis (Joensu 1987), Lecture Notes in Math., vol. 1351 (Springer-Verlag, 1988), 52–68.Google Scholar
8.Chen, Y., Levine, S. and Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (4) (2006), 1386–1406.Google Scholar
9.Cruz-Uribe, D. V. and Fiorenza, A., Variable Lebesgue spaces, Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, Heidelberg, 2013).Google Scholar
10.Daners, D. and Drábek, P., A priori estimates for a class of quasi-linear elliptic equations, Trans. Am. Math. Soc. 361 (2009), 6475–6500.CrossRefGoogle Scholar
11.Danielli, D., Garofalo, N. and Nhieu, D.-H., Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc. 182 (2006).Google Scholar
12.Danielli, D., Garofalo, N. and Nhieu, D.-H., Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. Sc. Norm. Sup. Pisa. 27 (1998), 195–252.Google Scholar
13.Diening, L., Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. des Sci. Math. 129 (2005), 657–700.Google Scholar
14.Diening, L., Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces Lp(ċ) and Wk,p(ċ), Math. Nachr. 268 (2004), 31–43.Google Scholar
15.Diening, L., Harjulehto, P., Hästö, P. and Růžička, M., Lebesgue and Sobolev sapces with variable exponent, Lecture Notes in Mathematics (Springer-Verlag, Berlin Heidelberg, 2011).Google Scholar
16.Drábek, P. and Milota, J., Methods of nonlinear analysis. Applications to Differential Equations, Birkhäuser Adv. Texts (Birkhäuser, Basel, 2007).Google Scholar
17.Fan, X., Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), 1395–1412.Google Scholar
18.Fan, X., On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.Google Scholar
19.Fan, X., Shen, J. and Zhao, D., Sobolev embedding theorems for spaces Wm,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), 749–760.Google Scholar
20.Hajłasz, P., Koskela, P. and Tuominen, H., Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008), 1217–1234.Google Scholar
21.Jones, P. W., Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71–88.CrossRefGoogle Scholar
22.Kováčik, O. and Rákosník, J., On spaces Lp(x) and Wk,p(x), Czech. Math. J. 41 (1991), 592–618.Google Scholar
23.Maz'ya, V. G., Sobolev spaces (Springer-Verlag, Berlin, 1985).Google Scholar
24.Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034 (Springer-Verlag, Berlin, 1983).CrossRefGoogle Scholar
25.Natanson, I. P., Theory of functions of a real variable (GITTL, Moscow, 1950).Google Scholar
26.Nittka, R., Elliptic and parabolic problems with Robin boundary conditions on lipschitz domains, PhD Dissertation (Ulm, 2010).Google Scholar
27.Růžička, M., Electrorheological fluids: modeling and mathematical theoory (Springer-Verlag, Berlin, 2000).Google Scholar
28.Showalter, R. E., Monotone operators in banach space and nonlinear partial differential equations (Amer. Math. Soc., Providence, RI, 1997).Google Scholar
29.Vélez-Santiago, A. and Warma, M., A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions, J. Math. Anal. Appl. 372 (2010), 120–139.Google Scholar
30.Wallin, H., The trace to the boundary of Sobolev spaces on a snowflake, Manuscr. Math. 73 (1991), 117–125.CrossRefGoogle Scholar