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The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials
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 |
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.
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. Breit, B. Stroffolini and A. Verde,
A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233.
|
| [8] |
M. Bulíček, L. 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. Bulíček, M. 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. Carro, A. García del Amo and J. Soria,
Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857.
|
| [11] |
M. Carro, L. Pick, J. 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. Cianchi, L. 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. Edmunds, R. 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. Breit, B. Stroffolini and A. Verde,
A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233.
|
| [8] |
M. Bulíček, L. 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. Bulíček, M. 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. Carro, A. García del Amo and J. Soria,
Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857.
|
| [11] |
M. Carro, L. Pick, J. 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. Cianchi, L. 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. Edmunds, R. 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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