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On best constants in Hardy inequalities with a remainder term. (English) Zbl 1220.35031

Summary: Let \(\Omega\) be a bounded open set of \(\mathbb R^N\) containing the origin. We compute the best value of the constant \(C(\alpha,|\Omega|)\) in
\[ \int_\Omega |\nabla u|^2\,dx- \frac{(N-2)^2}{4} \int_\Omega \frac{u^2}{|x|^2}\,dx\geq C(\alpha,|\Omega|)\|u\|_{L(\frac{2N}{N-\alpha},2)}^2, \]
with \(\alpha <2\) and \(u\in H_0^1(\Omega)\). Then we get the optimal value of \(C(|\Omega|)\) in
\[ \int_\Omega |\nabla u|^3\,dx- \bigg(\frac{N-3}{3}\bigg)^3 \int_\Omega \frac{u^3}{|x|^3}\,dx\geq C(|\Omega|)\|u\|_{L^3}^3, \]
where \(u\in W_0^{1,3}(\Omega)\).

MSC:

35J20 Variational methods for second-order elliptic equations
26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D30 Weak solutions to PDEs
49J40 Variational inequalities
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[1] Hardy, G. H., Notes on some points in the integral calculus, Messenger Math., 48, 107-112 (1919) · Zbl 0002.13004
[2] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1934), Cambridge University Press · Zbl 0010.10703
[3] Brezis, H.; Vazquez, J. L., Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 2, 10, 443-469 (1997) · Zbl 0894.35038
[4] Cabré, X.; Martel, Y., Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal., 156, 30-56 (1998) · Zbl 0908.35044
[5] Vazquez, J. L., Domain of existence and blow-up for the exponential reaction diffusion equation, Indiana Univ. Math. J., 48, 677-709 (1999) · Zbl 0928.35080
[6] Vazquez, 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) · Zbl 0953.35053
[7] Chaudhuri, N.; Ramaswamy, M., Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proc. Roy. Soc. Edinburgh Sect. A, 6, 131, 1275-1295 (2001) · Zbl 0997.35020
[8] Ghoussoub, N.; Moradifam, A., On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA, 105, 13746-13751 (2008) · Zbl 1205.26033
[9] Alvino, A.; Volpicelli, R.; Volzone, B., On Hardy inequalities with a remainder term, Ric. Mat., 59, 265-280 (2010) · Zbl 1204.35079
[10] Gazzola, F.; Granau, H. C.; Mitidieri, E., Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356, 6, 2149-2168 (2004) · Zbl 1079.46021
[11] Adimurthi; Chaudhuri, N.; Ramaswamy, M., An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130, 485-505 (2002) · Zbl 0987.35049
[12] Adimurthi; Filippas, S.; Tertikas, A., On the best constant of Hardy-Sobolev inequalities, Nonlinear Anal., 70, 2826-2833 (2009) · Zbl 1165.35484
[13] Barbatis, G.; Filippas, S.; Tertikas, A., A unified approach to improved \(L^p\) Hardy inequalities with best constants, Trans. Amer. Math. Soc., 356, 2169-2196 (2004) · Zbl 1129.26019
[14] Brezis, H.; Marcus, M., Hardy’s inequalities revisited, Ann. Sc. Norm. Super Pisa Cl. Sci., 25, 4, 217-237 (1997) · Zbl 1011.46027
[15] Filippas, S.; Maz’ja, V. G.; Tertikas, A., Sharp Hardy-Sobolev inequalities, C. R. Math. Acad. Sci. Paris, 339, 483-486 (2004) · Zbl 1058.26007
[16] Filippas, S.; Tertikas, A., Optimizing improved Hardy inequalities, J. Funct. Anal., 192, 186-233 (2002) · Zbl 1030.26018
[17] Talenti, G., Best constant in Sobolev inequalities, Ann. Mat. Pura Appl., 110, 353-372 (1976) · Zbl 0353.46018
[18] Chong, K. M.; Rice, N. M., Equimeasurable rearrangements of functions, (Queen’s Papers in Pure and Applied Mathematics, vol. 28 (1971), Queen’s University: Queen’s University Kingston, Ontario) · Zbl 0275.46024
[19] Kawhol, B., Rearrangements and convexity of level sets in P.D.E., (Lecture Notes in Mathematics, vol. 1150 (1985), Springer: Springer Berlin)
[20] Maz’ya, V. G., On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc., 20, 135-172 (1969) · Zbl 0226.35027
[21] Talenti, G., Linear elliptic P.D.E.’s: level sets, rearrangements and a priori estimates of solutions, Boll. Unione Mat. Ital., 4-B, 917-949 (1985) · Zbl 0602.35025
[22] Sagan, H., Introduction to the Calculus of Variations (1969), Dover Publications, Inc: Dover Publications, Inc New York
[23] Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994), Chapman and Hall: Chapman and Hall New York · Zbl 0826.65082
[24] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE Suite, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040
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