Abstract
Motivated by a question of Brezis and Marcus, we show that the Lp–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an Lq norm q ≥ p, with a constant depending on the interior diameter of Ω.
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Barbatis, G., Filippas S., Tertikas, A.: A unified approach to improved Lp Hardy inequalities with best constants. Trans. Amer. Math. Soc. 356(6), 2169–2196 (2004)
Brezis, H., Marcus, M.: Hardyapos; inequalities revisited. Ann. Scuola Norm. Pisa 25, 217–237 (1997)
Brezis, H., Vázquez, J.-L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid 10, 443–469 (1997)
Cabré, X., Martel, Y.: Existence versus instantaneous blowup for linear heat equations with singular potentials. C.R. Acad. Sci. Paris Ser. I Math. 329, 973–978 (1999)
Dávila, J., Dupaigne, L.: Hardy-type inequalities. J. Eur. Math. Soc. 6(3), 335–365 (2004)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Math., CRC Press (1992)
Filippas, S., Mazapos;a, V., Tertikas, A.: A sharp Hardy Sobolev inequality. C. R. Acad. Paris Ser. I, 339, 483–486 (2004)
Filippas, S., Mazapos;a, V., Tertikas, A: Critical Hardy Sobolev Inequalities. (In preparation)
Filippas, S., Tertikas, A.: Optimizing Improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002)
Garcia, J.P., Peral, I.: Hardy inequalities and some critical elliptic and parabolic problems. J. Diff. Equations 144, 441–476 (1998)
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Laptev, A.: A geometrical version of Hardyapos; inequality. J. Funct. Anal. 189, 539–548 (2002)
Mazapos;a, V.: Sobolev spaces. Springer (1985)
Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N.J. (1951)
Peral, I., Vázquez, J.L.: On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term. Arch. Rational Mech. Anal. 129, 201–224 (1995)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970)
Tidblom, J.: A geometrical version of Hardyapos; inequality for W1,p0(Ω), Proc. Amer. Math. Soc. 132(8), 2265–2271 (2004)
Vázquez, J.L., Zuazua, E.: The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential, J. Funct. Anal. 173, 103–153 (2000)
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Filippas, S., Maz'ya, V. & Tertikas, A. On a question of Brezis and Marcus. Calc. Var. 25, 491–501 (2006). https://doi.org/10.1007/s00526-005-0353-6
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DOI: https://doi.org/10.1007/s00526-005-0353-6