American Institute of Mathematical Sciences

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August  2018, 38(8): 4019-4040. doi: 10.3934/dcds.2018175

On fractional Hardy inequalities in convex sets

Lorenzo Brasco 1,2, and  Eleonora Cinti 3,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy
2. 

Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France
3. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received  October 2017 Revised  February 2018 Published  May 2018

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order $(s, p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<∞$ and zhongwenzy $0<s<1$, with a constant which is stable as $s$ goes to 1.

Keywords: Hardy inequality, nonlocal operators, fractional Sobolev spaces.
Mathematics Subject Classification: Primary: 39B72; Secondary: 35R11, 46E35.
Citation: Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
References:
[1]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr., 284 (2011), 629-638.  doi: 10.1002/mana.200810109.

[2]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[3]

L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[4]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.

[5]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[6]

Z.-Q. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482.

[7]

E. Cinti and F. Ferrari, Geometric inequalities for fractional Laplace operators and applications, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1699-1714.  doi: 10.1007/s00030-015-0340-3.

[8]

E. B. Davies, A review of Hardy inequalities, The Maz'ya anniversary collection, Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., Birkhäuser, Basel, 110 (1999), 55–67. doi: 10.1007/978-3-0348-8672-7_5.

[9]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[10]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. 

[11]

B. Dyda and A. V. Vähäkangas, A framework for fractional Hardy inequalities, Ann. Acad. Sci. Fenn. Math., 39 (2014), 675-689.  doi: 10.5186/aasfm.2014.3943.

[12]

R. L. Frank, R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the Research of Vladimir Maz'ya. I, Int. Math. Ser. (N. Y. ), Springer, New York, 11 (2010), 161–167. doi: 10.1007/978-1-4419-1341-8_6.

[13]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[14]

H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl., 1 (1997), 25-46.  doi: 10.1155/S1025583497000039.

[15]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[16]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6.

[17]

M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001.

[18]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[19]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.

[20]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990.

[21]

A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.

show all references

References:
[1]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr., 284 (2011), 629-638.  doi: 10.1002/mana.200810109.

[2]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[3]

L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[4]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.

[5]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[6]

Z.-Q. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482.

[7]

E. Cinti and F. Ferrari, Geometric inequalities for fractional Laplace operators and applications, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1699-1714.  doi: 10.1007/s00030-015-0340-3.

[8]

E. B. Davies, A review of Hardy inequalities, The Maz'ya anniversary collection, Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., Birkhäuser, Basel, 110 (1999), 55–67. doi: 10.1007/978-3-0348-8672-7_5.

[9]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[10]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. 

[11]

B. Dyda and A. V. Vähäkangas, A framework for fractional Hardy inequalities, Ann. Acad. Sci. Fenn. Math., 39 (2014), 675-689.  doi: 10.5186/aasfm.2014.3943.

[12]

R. L. Frank, R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the Research of Vladimir Maz'ya. I, Int. Math. Ser. (N. Y. ), Springer, New York, 11 (2010), 161–167. doi: 10.1007/978-1-4419-1341-8_6.

[13]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[14]

H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl., 1 (1997), 25-46.  doi: 10.1155/S1025583497000039.

[15]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[16]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6.

[17]

M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001.

[18]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[19]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.

[20]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990.

[21]

A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.

Figure 1.  The set $\Sigma_\sigma(x)$ and the supporting hyperplane $\Pi_{x'}$
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Figure 2.  The distance of $y$ from $\partial K$ is smaller than its distance from the hyperplane
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Figure 3.  The set $K_x$ in the second part of the proof of Theorem 1.1
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