Optimal L p Hardy-type inequalities
Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 1, pp. 93-118.

Let Ω be a domain in Rn or a noncompact Riemannian manifold of dimension n2, and 1<p<. Consider the functional Q(φ):=Ω(|φ|p+V|φ|p)dν defined on C0(Ω), and assume that Q0. The aim of the paper is to generalize to the quasilinear case (p2) some of the results obtained in [6] for the linear case (p=2), and in particular, to obtain “as large as possible” nonnegative (optimal) Hardy-type weight W satisfying

Q(φ)ΩW|φ|pdνφC0(Ω).

Our main results deal with the case where V=0, and Ω is a general punctured domain (for V0 we obtain only some partial results). In the case 1<pn, an optimal Hardy-weight is given by

W:=(p1p)p|GG|p,
where G is the associated positive minimal Green function with a pole at 0. On the other hand, for p>n, several cases should be considered, depending on the behavior of G at infinity in Ω. The results are extended to annular and exterior domains.

DOI: 10.1016/j.anihpc.2014.08.005
Keywords: Hardy inequality, Optimal, p-Laplacian
@article{AIHPC_2016__33_1_93_0,
     author = {Devyver, Baptiste and Pinchover, Yehuda},
     title = {Optimal {\protect\emph{L}}         \protect\textsuperscript{            \protect\emph{p}         } {Hardy-type} inequalities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {93--118},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.08.005},
     mrnumber = {3436428},
     zbl = {1331.35013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.005/}
}
TY  - JOUR
AU  - Devyver, Baptiste
AU  - Pinchover, Yehuda
TI  - Optimal L                     p          Hardy-type inequalities
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 93
EP  - 118
VL  - 33
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.005/
DO  - 10.1016/j.anihpc.2014.08.005
LA  - en
ID  - AIHPC_2016__33_1_93_0
ER  - 
%0 Journal Article
%A Devyver, Baptiste
%A Pinchover, Yehuda
%T Optimal L                     p          Hardy-type inequalities
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 93-118
%V 33
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.005/
%R 10.1016/j.anihpc.2014.08.005
%G en
%F AIHPC_2016__33_1_93_0
Devyver, Baptiste; Pinchover, Yehuda. Optimal L                     p          Hardy-type inequalities. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 1, pp. 93-118. doi : 10.1016/j.anihpc.2014.08.005. http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.005/

[1] Adimurthi; Sekar, A. Role of the fundamental solution in Hardy–Sobolev-type inequalities, Proc. R. Soc. Edinb. A, Volume 136 (2006), pp. 1111–1130 | DOI | MR | Zbl

[2] Allegretto, W.; Huang, Y.X. A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., Volume 32 (1998), pp. 819–830 | DOI | MR | Zbl

[3] Bojarski, B.; Hajłasz, P.; Strzelecki, P. Sard's theorem for mappings in Hölder and Sobolev spaces, Manuscr. Math., Volume 118 (2005), pp. 383–397 | DOI | MR | Zbl

[4] Chen, G.-Q.; Torres, M.; Ziemer, W.P. Gauss–Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Commun. Pure Appl. Math., Volume 62 (2009), pp. 242–304 | MR | Zbl

[5] D'Ambrosio, L.; Dipierro, S. Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014), pp. 449–475 | DOI | Numdam | MR | Zbl

[6] Devyver, B.; Fraas, M.; Pinchover, Y. Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., Volume 266 (2014), pp. 4422–4489 | DOI | MR | Zbl

[7] Davies, E.B.; Hinz, A.M. Explicit constants for Rellich inequalities in Lp(Ω) , Math. Z., Volume 227 (1998), pp. 511–523 | DOI | MR | Zbl

[8] Fraas, M.; Pinchover, Y. Isolated singularities of positive solutions of p-Laplacian type equations in Rd , J. Differ. Equ., Volume 254 (2013), pp. 1097–1119 | DOI | MR | Zbl

[9] García-Melián, J.; Sabina de Lis, J. Maximum and comparison principles for operators involving the p-Laplacian, J. Math. Anal. Appl., Volume 218 (1998), pp. 49–65 | DOI | MR | Zbl

[10] Ladyženskaja, O.A.; Ural'ceva, N.N. A variational problem for quasilinear elliptic equations with many independent variables, Sov. Math. Dokl., Volume 1 (1960), pp. 1390–1394 | MR | Zbl

[11] Maz'ya, V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342, Springer, Heidelberg, 2011 | MR | Zbl

[12] Pinchover, Y. On criticality and ground states of second-order elliptic equations II, J. Differ. Equ., Volume 87 (1990), pp. 353–364 | DOI | MR | Zbl

[13] Pinchover, Y.; Tintarev, K. Ground state alternative for p-Laplacian with potential term, Calc. Var. Partial Differ. Equ., Volume 28 (2007), pp. 179–201 | MR | Zbl

[14] Pinchover, Y.; Tintarev, K.; Maz'ya, V. On the Hardy–Sobolev–Maz'ya inequality and its generalizations, Sobolev Spaces in Mathematics I: Sobolev Type Inequalities, International Mathematical Series, vol. 8, Springer, 2009, pp. 281–297 | DOI | MR | Zbl

[15] Pinchover, Y.; Tertikas, A.; Tintarev, K. A Liouville-type theorem for the p-Laplacian with potential term, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 25 (2008), pp. 357–368 | DOI | Numdam | MR | Zbl

[16] Poliakovsky, A.; Shafrir, I. Uniqueness of positive solutions for singular problems involving the p-Laplacian, Proc. Am. Math. Soc., Volume 133 (2005), pp. 2549–2557 | DOI | MR | Zbl

[17] Pucci, P.; Serrin, J. The Maximum Principle, Progress in Nonlinear Differential Equations and Their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007 | MR | Zbl

[18] Serrin, J. Local behavior of solutions of quasi-linear equations, Acta Math., Volume 111 (1964), pp. 247–302 | DOI | MR | Zbl

[19] Serrin, J. Isolated singularities of solutions of quasi-linear equations, Acta Math., Volume 113 (1965), pp. 219–240 | DOI | MR | Zbl

[20] Tolksdorf, P. Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., Volume 51 (1984), pp. 126–150 | DOI | MR | Zbl

Cited by Sources: