Scaling and variants of Hardy’s inequality
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- by Giséle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni and Silvia Romanelli PDF
- Proc. Amer. Math. Soc. 147 (2019), 1165-1172 Request permission
Abstract:
The two related one space dimensional singular linear parabolic equations (1), (2) studied by H. Brezis et al. [Comm. Pure Appl. Math. 24 (1971), pp. 395–416] have different scaling properties. These scaling properties lead to new variants of the Hardy and Caffarelli-Kohn-Nirenberg inequalities. These results are proved, and they imply some non-wellposedness results when the constant in the singular potential term is large enough.References
- Wolfgang Arendt, Gisèle Ruiz Goldstein, and Jerome A. Goldstein, Outgrowths of Hardy’s inequality, Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 51–68. MR 2259099, DOI 10.1090/conm/412/07766
- H. Brezis, W. Rosenkrantz, and B. Singer, On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math. 24 (1971), 395–416. MR 284717, DOI 10.1002/cpa.3160240305
- L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
- Alberto Cialdea and Vladimir Maz’ya, Semi-bounded differential operators, contractive semigroups and beyond, Operator Theory: Advances and Applications, vol. 243, Birkhäuser/Springer, Cham, 2014. MR 3235527, DOI 10.1007/978-3-319-04558-0
- Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, and Silvia Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr. 283 (2010), no. 4, 504–521. MR 2649366, DOI 10.1002/mana.200910086
- Gisele Ruiz Goldstein, Jerome A. Goldstein, and Ismail Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent, Appl. Anal. 84 (2005), no. 6, 571–583. MR 2151669, DOI 10.1080/00036810500047709
- Gisèle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni, and Silvia Romanelli, The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differential Equations 21 (2016), no. 3-4, 235–264. MR 3461294
- J. A. Goldstein, Semigroup of Linear Operators & Applications, second ed., Dover Publications, Inc., Mineola, New York, 2017.
- V. A. Liskevich and M. A. Perel′muter, Analyticity of sub-Markovian semigroups, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1097–1104. MR 1224619, DOI 10.1090/S0002-9939-1995-1224619-1
Additional Information
- Giséle Ruiz Goldstein
- Affiliation: Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, Tennessee 38152-3240
- MR Author ID: 333750
- Email: ggoldste@memphis.edu
- Jerome A. Goldstein
- Affiliation: Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, Tennessee 38152-3240
- MR Author ID: 74805
- Email: jgoldste@memphis.edu
- Rosa Maria Mininni
- Affiliation: Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy
- MR Author ID: 362106
- ORCID: 0000-0002-0230-1435
- Email: rosamaria.mininni@uniba.it
- Silvia Romanelli
- Affiliation: Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy
- MR Author ID: 237923
- Email: silvia.romanelli@uniba.it
- Received by editor(s): April 17, 2017
- Received by editor(s) in revised form: April 13, 2018, and June 17, 2018
- Published electronically: December 6, 2018
- Communicated by: Catherine Sulem
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1165-1172
- MSC (2010): Primary 35K65, 26D10, 47D06
- DOI: https://doi.org/10.1090/proc/14295
- MathSciNet review: 3896064