Abstract
In this paper we provide a proof of the Sobolev–Poincaré inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), New York-London, 1975. Pure and Applied Mathematics, Vol. 65
Brenier Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987)
Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)
Cordero-Erausquin D., Nazaret B., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004)
Diening L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \({L^{p(\cdot)}}\) and \({W^{k,p(\cdot)}}\). Math. Nachr. 268, 31–43 (2004)
Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Volume 2017 of Lecture Notes in Mathematics. Springer, Heidelberg (2011)
Edmunds D.E., Rákosník J.: Sobolev embeddings with variable exponent. Stud. Math. 143(3), 267–293 (2000)
Edmunds D.E., Rákosník J.: Sobolev embeddings with variable exponent, II. Math. Nachr. 246/247, 53–67 (2002)
Escobar J.F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. (2) 136(1), 1–50 (1992)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Fan X.: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339(2), 1395–1412 (2008)
Harjulehto P., Hästö P.: Sobolev inequalities with variable exponent attaining the values 1 and n. Publ. Mat. 52(2), 347–363 (2008)
Kantorovich L.V.: On a problem of monge. Uspekhi Mat. Nauk. 3, 225–226 (1948) (in russian)
McCann R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)
Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, pp. 666–704 (1781)
Nazaret B.: Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65(10), 1977–1985 (2006)
Villani C.: Topics in Optimal Transportation, Volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)
Villani, C.: Optimal Transport, Volume 338 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer, Berlin (2009) (old and new)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Borthagaray, J.P., Fernández Bonder, J. & Silva, A. A mass transportation approach for Sobolev inequalities in variable exponent spaces. manuscripta math. 151, 133–146 (2016). https://doi.org/10.1007/s00229-016-0830-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-016-0830-6