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September  2018, 17(5): 2011-2037. doi: 10.3934/cpaa.2018096

Sharp Sobolev type embeddings on the entire Euclidean space

1. 

Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy

2. 

Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

3. 

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

4. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received  September 2017 Revised  January 2018 Published  April 2018

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space ${\mathbb R^n}$, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Citation: Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096
References:
[1]

E. Acerbi and R. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. 

[2]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1976/77), 337-403. 

[4]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston, 1988.

[5]

P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations, 53 (2015), 803-846. 

[6]

D. Breit and O. D. Schirra, Korn-type inequalities in Orlicz-Sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory, Z. Anal. Anwend., 31 (2012), 335-356. 

[7]

D. BreitB. Stroffolini and A. Verde, A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233. 

[8]

M. BulíčekL. Diening and S. Schwarzacher, Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems, Anal. PDE, 9 (2016), 1115-1151. 

[9]

M. BulíčekM. Majdoub and J. Málek, Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11 (2010), 3968-3983. 

[10]

M. CarroA. García del Amo and J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857. 

[11]

M. CarroL. PickJ. Soria and V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. 

[12]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. 

[13]

A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations, 22 (1997), 1629-1646. 

[14]

A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana, 20 (2004), 427-474. 

[15]

A. Cianchi, Higher-order Sobolev and Poincar´e inequalities in Orlicz spaces, Forum Math., 18 (2006), 745-767. 

[16]

A. Cianchi and L. Pick, Sobolev embeddings into BMO, VMO and L, Ark. Mat., 36 (1998), 317-340. 

[17]

A. Cianchi and L. Pick, Optimal Sobolev trace embeddings, Trans. Amer. Math. Soc., 368 (2016), 8349-8382. 

[18]

A. CianchiL. Pick and L. Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math., 273 (2015), 568-650. 

[19]

A. Cianchi and M. Randolfi, On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J., 60 (2011), 1939-1973. 

[20]

D. E. EdmundsR. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307-355. 

[21]

H. J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. 

[22]

R. Kerman and L. Pick, Optimal Sobolev imbeddings, Forum Math., 18 (2006), 535-570. 

[23]

A. G. Korolev, On the boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities, Mat. Sb., 180 (1989), 78-100. 

[24]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. 

[25]

G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math., 1 (1951), 411-429. 

[26]

P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333. 

[27]

L. Pick, A. Kufner, O. John and S. Fučík, Function Spaces, Vol. 1. Second revised and extended edition, Math. Inequal. Appl., de Gruyter & Co., Berlin, 2013.

[28]

S. I. Pohozaev, On the imbedding theorem by S.L.Sobolev in the case pl = n, Dokl. Conf., Sect. Math. Moscow Power Inst., (1965), 158-170. 

[29]

E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. 

[30]

J. Soria, Lorentz spaces of weak type, Quart. J. Math. Oxford Ser. (2), 49 (1998), 93-103. 

[31]

R. S. Strichartz, A note on Trudinger' s extension of Sobolev' s inequality, Indiana Univ. Math. J., 21 (1972), 841-842. 

[32]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184. 

[33]

G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279. 

[34]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. 

[35]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, in Russian, Soviet Math. Dokl., 2 (1961), 746-749. 

[36]

J. Vybíral, Optimal Sobolev imbeddings on ${\mathbb{R}^n}$, Publ. Mat., 51 (2007), 17-44. 

[37]

A. Wrióblewska, Steady flow of non-Newtonian fluids–monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147. 

show all references

References:
[1]

E. Acerbi and R. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. 

[2]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1976/77), 337-403. 

[4]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston, 1988.

[5]

P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations, 53 (2015), 803-846. 

[6]

D. Breit and O. D. Schirra, Korn-type inequalities in Orlicz-Sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory, Z. Anal. Anwend., 31 (2012), 335-356. 

[7]

D. BreitB. Stroffolini and A. Verde, A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233. 

[8]

M. BulíčekL. Diening and S. Schwarzacher, Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems, Anal. PDE, 9 (2016), 1115-1151. 

[9]

M. BulíčekM. Majdoub and J. Málek, Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11 (2010), 3968-3983. 

[10]

M. CarroA. García del Amo and J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857. 

[11]

M. CarroL. PickJ. Soria and V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. 

[12]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. 

[13]

A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations, 22 (1997), 1629-1646. 

[14]

A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana, 20 (2004), 427-474. 

[15]

A. Cianchi, Higher-order Sobolev and Poincar´e inequalities in Orlicz spaces, Forum Math., 18 (2006), 745-767. 

[16]

A. Cianchi and L. Pick, Sobolev embeddings into BMO, VMO and L, Ark. Mat., 36 (1998), 317-340. 

[17]

A. Cianchi and L. Pick, Optimal Sobolev trace embeddings, Trans. Amer. Math. Soc., 368 (2016), 8349-8382. 

[18]

A. CianchiL. Pick and L. Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math., 273 (2015), 568-650. 

[19]

A. Cianchi and M. Randolfi, On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J., 60 (2011), 1939-1973. 

[20]

D. E. EdmundsR. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307-355. 

[21]

H. J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. 

[22]

R. Kerman and L. Pick, Optimal Sobolev imbeddings, Forum Math., 18 (2006), 535-570. 

[23]

A. G. Korolev, On the boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities, Mat. Sb., 180 (1989), 78-100. 

[24]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. 

[25]

G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math., 1 (1951), 411-429. 

[26]

P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333. 

[27]

L. Pick, A. Kufner, O. John and S. Fučík, Function Spaces, Vol. 1. Second revised and extended edition, Math. Inequal. Appl., de Gruyter & Co., Berlin, 2013.

[28]

S. I. Pohozaev, On the imbedding theorem by S.L.Sobolev in the case pl = n, Dokl. Conf., Sect. Math. Moscow Power Inst., (1965), 158-170. 

[29]

E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. 

[30]

J. Soria, Lorentz spaces of weak type, Quart. J. Math. Oxford Ser. (2), 49 (1998), 93-103. 

[31]

R. S. Strichartz, A note on Trudinger' s extension of Sobolev' s inequality, Indiana Univ. Math. J., 21 (1972), 841-842. 

[32]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184. 

[33]

G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279. 

[34]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. 

[35]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, in Russian, Soviet Math. Dokl., 2 (1961), 746-749. 

[36]

J. Vybíral, Optimal Sobolev imbeddings on ${\mathbb{R}^n}$, Publ. Mat., 51 (2007), 17-44. 

[37]

A. Wrióblewska, Steady flow of non-Newtonian fluids–monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147. 

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