Abstract
We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian −Δ A u = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset \({\Omega \subseteq \mathbb{R}^n}\). Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form
where \({\bar{A}(t)}\) is a Young function related to A and satisfying Δ′-condition, while \({F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}\). Examples involving \({\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}\) are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality −Δ p u ≥ Φ, leading to Hardy and Hardy–Poincaré inequalities with the best constants.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adimurthi C.N., Ramaswamy M.: An improved Hardy–Sobolev inequality and its application. Proc. Am. Math. Soc. 130(2), 489–505 (2002)
Alber, H.-D.: Materials with Memory: Initial-boundary value problems for constitutive equations with internal variable. In: Lecture Notes in Mathematics, vol. 1682, Springer, Berlin (1998)
Ancona A.: On strong barriers and an inequality of Hardy for domains in \({{\mathbb{R}^n}}\). J. London Math. Soc. 34(2), 274–290 (1986)
Ando H., Horiuchi T.: Missing terms in the weighted Hardy–Sobolev inequalities and its application. Kyoto J. Math. 52(4), 759–796 (2012)
Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63, 337–403 (1997)
Baras P., Goldstein J.A.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284(1), 121–139 (1984)
Barbatis G., Filippas S., Tertikas A.: A unified approach to improved L p Hardy inequalities with best constants. Trans. Am. Math. Soc. 356(6), 2169–2196 (2004)
Barbatis, G., Filippas, S., Tertikas, A.: Series expansion for L p Hardy inequalities. Indiana Univ. Math. J. 52(1),171–190
Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.-.L: Hardy–Poincaré inequalities and application to nonlinear diffusions. C. R. Acad. Sci. Paris Ser. I 344(2007), 431–436 (2007)
Bhattacharya T., Leonetti F.: A new Poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth. Nonlinear Anal. TMA 17(9), 833–839 (1991)
Boyd D.W.: Indices for Orlicz spaces. Pac. J. Math. 38(2), 315–323 (1971)
Brézis H., Vazquez J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Complut. 10(2), 443–469 (1997)
Brown, R.C., Hinton, D.B.: Weighted interpolation and Hardy inequalities with some spectral theoretic applications. In: Bainov, D. et al. (ed.) Proceedings of the fourth international colloquium on differential equations, Plovdiv, Bulgaria, August 18–22, 1993, pp 55–70. VSP, Utrecht (1994)
Buckley S.M., Hurri-Syrjänen R.: Iterated log-scale Orlicz–Hardy inequalities. Ann. Acad. Sci. Fenn. Math. 38(2), 757–770 (2013)
Buckley S.M., Koskela P.: Orlicz–Hardy inequalities. Ill. J. Math. 48(3), 787–802 (2004)
Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-dimensional variational problems. An introduction. Oxford Lecture Series in Mathematics and its Applications, vol. 15. The Clarendon Press, Oxford University Press, New York (1998)
Butzer, P.L., Fehér. F.: Generalized Hardy and Hardy–Littlewood inequalities in rearrangement-invariant spaces. Comment. Math. Prace Mat. Tomus Specialis in Honorum Ladislai Orlicz 1, 41–64 (1978)
Caccioppoli, R.: Limitazioni integrali per le soluzioni di un’equazione lineare ellitica a derivate parziali. Giorn. Mat. Battaglini (4) 4(80), 186–212 (1951)
Cianchi A.: Hardy inequalities in Orlicz spaces. Trans. Am. Math. Soc. 351(6), 2459–2478 (1999)
Cianchi, A.: Some results in the theory of Orlicz spaces and applications to variational problems, Nonlinear analysis, function spaces and applications, 6 (Prague, 1998). Acad. Sci. Czech Rep. Prague. 6, 50–92 (1999)
Chua S.K.: Sharp conditions for weighted Sobolev interpolation inequalities. Forum Math. 17(3), 461–478 (2005)
Chua S.K.: On weighted Sobolev interpolation inequalities. Proc. Am. Math. Soc 121, 441–449 (1994)
D’Ambrosio, L: Hardy type inequalities related to degenerate elliptic differential operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 5 IV, 451–486 (2005)
D’Ambrosio, L.: Some Hardy inequalities on the Heisenberg group (Russian) Differ. Uravn. 40(4) (2004) 509–521, 575; translation in Differ. Equ. 40(4), 552–564 (2004)
D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc. 132(3), 725–734 (2004)
D’Ambrosio, L., Dipierro, S.: Hardy inequalities on Riemannian manifolds and applications. Ann. I.H. Poincare-AN (2013). doi:10.1016/j.anihpc.2013.04.004
Donaldson T.K., Trudinger N.S.: Orlicz–Sobolev spaces and imbedding theorems. J. Funct. Anal 8, 52–75 (1971)
Detalla A., Horiuchi T., Ando H.: Missing terms in Hardy–Sobolev inequalities and its application. Far East J. Math. Sci. (FJMS) 14(3), 333–359 (2004)
Fernandez-Martinez P., Singes T.: Real interpolation with symmetric spaces and slowly varying functions. Q. J. Math. 63(1), 133–164 (2012)
Fiorenza A., Krbec M.: Indices of Orlicz spaces and some applications. Comment. Math. Univ. Carolin 38(3), 433–451 (1997)
Fuchs, M., Seregin, G.: A regularity theory for variational intgrals with L ln L-Growth. Calc. Var. 6, 171–187 (1998)
Fuchs M., Mingione G.: Full C 1-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth. Manuscripta Math. 102, 227–250 (2000)
Garcia Azorero J.P., Peral Alonso I.: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Eq. 144, 441–476 (1998)
Goldstein J.A., Zhang Q.S.: On a degenerate heat equation with a singular potential. J. Funct. Anal. 186, 342–359 (2001)
Gossez J.-P., Mustonen V.: Variational inequalities in Orlicz–Sobolev spaces. Nonlinear Anal. 11(3), 379–392 (1987)
Grillo, G.: On the equivalence between p-Poincaré inequalities and L r − L q regularization and decay estimates of certain nonlinear evolutions. J. Differ. Eq. 249(10), 2561–2576 (2010)
Gustavsson J., Peetre J.: Interpolation of Orlicz spaces. Studia Math. 60, 33–59 (1977)
Gutierrez C.E., Wheeden R.L.: Sobolev interpolation inequalities with weights. Trans. Am. Math. Soc 323, 263–281 (1991)
Hajłasz P., Koskela P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Ser. I Math. 320(10), 1211–1215 (1995)
Haroske D., Skrzypczak L.: Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases. J. Funct. Spaces Appl. 9(2), 129–178 (2011)
Iwaniec, T., Sbordone, C.: Caccioppoli estimates and very weak solutions of elliptic equations. Renato Caccioppoli and modern analysis. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14(2003), No. 3 (2004), 189–205
Kał amajska A., Krbec M.: Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions. J. Math. Anal. Appl. 362, 460–470 (2010)
Kał amajska A., Pietruska-Pał uba K.: Gagliardo–Nirenberg inequalities in weighted Orlicz spaces. Studia Math. 173, 49–71 (2006)
Kał amajska A., Pietruska-Pał uba K.: On a variant of the Hardy inequality between weighted Orlicz spaces. Studia Math. 193, 1–28 (2009)
Kał amajska A., Pietruska-Pał uba K.: On a variant of Gagliardo–Nirenberg inequality deduced from Hardy. Bull. Pol. Acad. Sci. Math. 59(2), 133–149 (2011)
Kał amajska A., Pietruska-Pał uba K.: New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights. Cent. Eur. J. Math. 10(6), 2033–2050 (2012)
Kał amajska, A., Pietruska-Pał uba, K., Skrzypczak, I.: Nonexistence results for differential inequalities involving A-Laplacian. Adv. Differ. Eq. 17(3–4), 307–336 (2012)
Kim D.: Elliptic equations with nonzero boundary conditions in weighted Sobolev spaces. J. Math. Anal. Appl 337, 1465–1479 (2008)
Kim, K.-H., Krylov, N.V.: On The Sobolev space theory of parabolic and elliptic equations in C 1-domains. SIAM J. Math. Anal. 36(2), 618–642 (electronic) (2004)
Kita H.: On Hardy–Littlewood maximal functions in Orlicz spaces. Math. Nachr 183, 135–155 (1997)
Kufner, A.: Weighted Sobolev spaces. A Wiley-Interscience Publication. Wiley, New York (1985) (translated from Czech)
Kufner, A., Maliganda, L., Persson, L.E. The Hardy Inequality. About its History and Some Related Results. Vydavatelský Servis, Plzeň (2007)
Kufner, A., Opic, B.: Hardy-type Inequalities. Longman Scientific and Technical, Harlow (1990)
Kufner, A., Persson, L.E.: Weighted Inequalities of Hardy Type. World Scientific, River Edge (2003)
Krasnoselskii, M.A., Rutickii, Ya.B. Convex Functions and Orlicz Spaces. P.~Noordhoff Ltd. Groningen (1961)
Luque T., Parissis I.: The endpoint Fefferman-Stein inequality for the strong maximal function. J. Funct. Anal. 266(1), 199–212 (2014)
Maz’ya, V.G. Sobolev Spaces. Springer, Berlin (1985)
Mingione G., Siepe F.: Full C 1 regularity for minimizers of integral functionals with L log L growth. Z. Anal. Anw. 18(4), 1083–1100 (1999)
Mitsis T.: The weighted weak type inequality for the strong maximal function. J. Fourier Anal. Appl. 12(6), 645–652 (2006)
Prez, C.: A remark on weighted inequalities for general maximal operators. Proc. Amer. Math. Soc 119(4) (1993)
Mitidieri E.: A simple approach to Hardy inequalities. Mat. Zametki 67, 563–572 (2000)
Moscariello G., Nania L.: Hölder continuity of minimizers of functionals with nonstandard growth conditions. Ricerche di Matematica 15(2), 259–273 (1991)
Nečas, J.: Direct methods in the theory of elliptic equations, Transl. from the French. Editorial coordination and preface by Sarka Necasova and a contribution by Christian G. Simader. In: Springer Monographs in Mathematics
Nikolskii, S.M.; Lizorkin, P.I.; Miroshin, N.V.: Weighted function spaces and their applications to the study of boundary value problems for elliptic equations in divergent form (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 8, 4–30 (1988); tanslation in Soviet Math. (Iz. VUZ) 32(8), 1–40 (1988)
Palmieri G.: An approach to the theory of some trace spaces related to the Orlicz–Sobolev spaces. (Italian). Boll. Un. Mat. Ital. 16, 100–119 (1979)
Rao, M.M., Ren, Z.D. Theory of Orlicz Spaces. M. Dekker, Inc. New York (1991)
Simonenko I.B.: Interpolation and extrapolation of linear operators in Orlicz spaces. (in Russian). Mat. Sb. (N. S.) 63(105), 536–553 (1964)
Skrzypczak I.: Hardy-type inequalities derived from p-harmonic problems. Nonlinear Anal. TMA 93, 30–50 (2013)
Skrzypczak, I.: Hardy-Poincaré type inequalities derived from p-harmonic problems. Banach Center Publications (to appear)
Stein E.: Note on the class L log L. Studia Math. 32, 305–310 (1969)
Stepanov, V.D.: Weighted norm inequalities and related topics, in Nonlinear analysis, function spaces and applications, vol. 5, In: Proceedings of the Spring School in Prague, Prometheus (1994)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was supported by NCN grant 2011/03/N/ST1/00111.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Skrzypczak, I. Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian. Nonlinear Differ. Equ. Appl. 21, 841–868 (2014). https://doi.org/10.1007/s00030-014-0269-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-014-0269-y