An optimal limiting $2D$ Sobolev inequality
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- by Andrei Biryuk PDF
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Abstract:
The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Hölder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality \[ \|u\|_{L^{\infty }} \leqslant \frac {\|\nabla u\|_{L^2}}{\sqrt {2\pi \alpha }} \sqrt {\ln \Bigl (1+6\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}}{\|\nabla u\|_{L^{2}}} \sqrt {\ln (1+\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}} {\|\nabla u\|_{L^{2}}} )\!} \Bigr )} \] for functions $u\in W^{1,2}_0(B_1)$ on the unit disk $B_1$ in $\mathbb R^2$, $\alpha \in (0,1].$References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- A. Alvino, G. Trombetti, and P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal. 13 (1989), no. 2, 185–220. MR 979040, DOI 10.1016/0362-546X(89)90043-6
- J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762, DOI 10.1007/BF01212349
- H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), no. 4, 677–681. MR 582536, DOI 10.1016/0362-546X(80)90068-1
- Haïm Brézis and Stephen Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773–789. MR 579997, DOI 10.1080/03605308008820154
- Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
- Hans Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations 14 (1989), no. 4, 541–544. MR 989669, DOI 10.1080/03605302.1989.12088448
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
- S. Ibrahim, M. Majdoub, and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc. 135 (2007), no. 1, 87–97. MR 2280178, DOI 10.1090/S0002-9939-06-08240-2
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
- Elliott H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1976/77), no. 2, 93–105. MR 471785, DOI 10.1002/sapm197757293
- T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), no. 2, 259–269. MR 1317718, DOI 10.1006/jfan.1995.1012
- G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486, DOI 10.1515/9781400882663
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
Additional Information
- Andrei Biryuk
- Affiliation: Departamento de Matematica, Instituto Superior Técnico, Centro de Análise Mate- mática, Geometria e Sistemas Dinâmicos, Lisbon, Portugal
- Received by editor(s): March 3, 2009
- Received by editor(s) in revised form: August 10, 2009
- Published electronically: November 13, 2009
- Additional Notes: The author is supported in part by CAMGSD and FCT/POCTI-POCI/FEDER. Part of the research (Theorem 3) was done while the author was a member of McMaster University in 2004. The author was supported in part by a CRC postdoctoral fellowship of McMaster University.
- Communicated by: Walter Craig
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1461-1470
- MSC (2000): Primary 52A40, 46E35; Secondary 46E30, 26D10
- DOI: https://doi.org/10.1090/S0002-9939-09-10159-4
- MathSciNet review: 2578540