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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

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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