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Scaling and variants of Hardy’s inequality. (English) Zbl 1403.35140
The authors consider Hardy’s inequality and its important generalization by Caffarelli, Kohn, and Nirenberg We recall that these inequalities are used in connection with establishing wellposedness and illposedness for certain singular parabolic equations. The Hardy and Caffarelli, Kohn, and Nirenberg inequalities for functions on \(\mathbb R^+\) can be proved by scaling techniques. In this paper, the authors look at these issues from a reverse perspective. They start with two suitable equations that possess scaling properties and from them they derive an associated Hardy-type inequality. They apply their result to prove some non-wellposedness results when the constant in the singular potential term is large enough.
MSC:
35K65 Degenerate parabolic equations
26D10 Inequalities involving derivatives and differential and integral operators
47D06 One-parameter semigroups and linear evolution equations
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[1] Arendt, Wolfgang; Goldstein, Gis\`ele Ruiz; Goldstein, Jerome A., Outgrowths of Hardy’s inequality. Recent advances in differential equations and mathematical physics, Contemp. Math. 412, 51-68, (2006), Amer. Math. Soc., Providence, RI · Zbl 1113.26017
[2] Brezis, H.; Rosenkrantz, W.; Singer, B., On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math., 24, 395-416, (1971) · Zbl 0206.11203
[3] Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compositio Math., 53, 3, 259-275, (1984) · Zbl 0563.46024
[4] Cialdea, Alberto; Maz’ya, Vladimir, Semi-bounded differential operators, contractive semigroups and beyond, Operator Theory: Advances and Applications 243, xiv+252 pp., (2014), Birkh\"auser/Springer, Cham · Zbl 1317.47002
[5] Favini, Angelo; Goldstein, Gis\`ele Ruiz; Goldstein, Jerome A.; Obrecht, Enrico; Romanelli, Silvia, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283, 4, 504-521, (2010) · Zbl 1215.47035
[6] Goldstein, Gisele Ruiz; Goldstein, Jerome A.; Kombe, Ismail, Nonlinear parabolic equations with singular coefficient and critical exponent, Appl. Anal., 84, 6, 571-583, (2005) · Zbl 1071.35066
[7] Ruiz Goldstein, Gis\`ele; Goldstein, Jerome A.; Mininni, Rosa Maria; Romanelli, Silvia, The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differential Equations, 21, 3-4, 235-264, (2016) · Zbl 1341.47051
[8] b113 J. A. Goldstein, \em Semigroup of Linear Operators & Applications, second ed., Dover Publications, Inc., Mineola, New York, 2017.
[9] Liskevich, V. A.; Perel\cprime muter, M. A., Analyticity of sub-Markovian semigroups, Proc. Amer. Math. Soc., 123, 4, 1097-1104, (1995) · Zbl 0826.47030
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