We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $ g $-Laplacian $ \mathcal{L}_g^s $, with $ 0 < s < 1 $. We prove the strict T-monotonicity of $ \mathcal{L}_g^s $ and we obtain the Lewy-Stampacchia inequalities $ F\leq\mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $ and $ F\wedge\mathcal{L}_g^s\varphi\leq \mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $, respectively, for the one obstacle solution $ u\geq\psi $ and for the two obstacles solution $ \psi\leq u\leq\varphi $, with given data $ F $. We consider the approximation of the solutions through semilinear problems, for which we prove a global $ L^\infty $-estimate, and we extend the local Hölder regularity to the solutions of the obstacle problems in the case of the fractional $ p(x, y) $-Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.
Citation: Catharine W. K. Lo, José Francisco Rodrigues. On the obstacle problem in fractional generalised Orlicz spaces[J]. Mathematics in Engineering, 2024, 6(5): 676-704. doi: 10.3934/mine.2024026
We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $ g $-Laplacian $ \mathcal{L}_g^s $, with $ 0 < s < 1 $. We prove the strict T-monotonicity of $ \mathcal{L}_g^s $ and we obtain the Lewy-Stampacchia inequalities $ F\leq\mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $ and $ F\wedge\mathcal{L}_g^s\varphi\leq \mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $, respectively, for the one obstacle solution $ u\geq\psi $ and for the two obstacles solution $ \psi\leq u\leq\varphi $, with given data $ F $. We consider the approximation of the solutions through semilinear problems, for which we prove a global $ L^\infty $-estimate, and we extend the local Hölder regularity to the solutions of the obstacle problems in the case of the fractional $ p(x, y) $-Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.
| [1] |
D. R. Adams, Capacity and the obstacle problem, Appl. Math. Optim., 8 (1982), 39–57. https://doi.org/10.1007/BF01447750 doi: 10.1007/BF01447750
|
| [2] | D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Vol. 314, Springer, 1996. https://doi.org/10.1007/978-3-662-03282-4 |
| [3] | R. A. Adams, Sobolev spaces, Vol. 65, Elsevier, 1975. |
| [4] | H. Attouch, C. Picard, Problèmes variationnels et théorie du potentiel non linéaire, Ann. Fac. Sci. Toulouse Math., 1 (1979), 89–136. |
| [5] |
H. Attouch, C. Picard, Inéquations variationnelles avec obstacles et espaces fonctionnels en théorie du potentiel, Appl. Anal., 12 (1981), 287–306. https://doi.org/10.1080/00036818108839369 doi: 10.1080/00036818108839369
|
| [6] |
E. Azroul, A. Benkirane, M. Shimi, M. Srati, On a class of nonlocal problems in new fractional Musielak-Sobolev spaces, Appl. Anal., 101 (2022), 1933–1952. https://doi.org/10.1080/00036811.2020.1789601 doi: 10.1080/00036811.2020.1789601
|
| [7] |
E. Azroul, A. Benkirane, M. Shimi, M. Srati, Embedding and extension results in fractional Musielak-Sobolev spaces, Appl. Anal., 102 (2023), 195–219. https://doi.org/10.1080/00036811.2021.1948019 doi: 10.1080/00036811.2021.1948019
|
| [8] |
A. Bahrouni, Comparison and sub-supersolution principles for the fractional $p(x)$-Laplacian, J. Math. Anal. Appl., 458 (2018), 1363–1372. https://doi.org/10.1016/j.jmaa.2017.10.025 doi: 10.1016/j.jmaa.2017.10.025
|
| [9] |
D. Baruah, P. Harjulehto, P. Hästö, Capacities in generalized Orlicz spaces, J. Funct. Spaces, 2018 (2018), 8459874. https://doi.org/10.1155/2018/8459874 doi: 10.1155/2018/8459874
|
| [10] |
L. Brasco, E. Parini, M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813–1845. https://doi.org/10.3934/dcds.2016.36.1813 doi: 10.3934/dcds.2016.36.1813
|
| [11] |
S. S. Byun, H. Kim, J. Ok, Local Hölder continuity for fractional nonlocal equations with general growth, Math. Ann., 387 (2023), 807–846. https://doi.org/10.1007/s00208-022-02472-y doi: 10.1007/s00208-022-02472-y
|
| [12] |
M. L. M. Carvalho, E. D. Silva, J. C. de Albuquerque, S. Bahrouni, On the $L^\infty$-regularity for fractional Orlicz problems via Moser's iteration, Math. Meth. Appl. Sci., 46 (2023), 4688–4704. https://doi.org/10.1002/mma.8795 doi: 10.1002/mma.8795
|
| [13] |
J. Chaker, M. Kim, M. Weidner, Harnack inequality for nonlocal problems with non-standard growth, Math. Ann., 386 (2023), 533–550. https://doi.org/10.1007/s00208-022-02405-9 doi: 10.1007/s00208-022-02405-9
|
| [14] |
J. Chaker, M. Kim, M. Weidner, Regularity for nonlocal problems with non-standard growth, Calc. Var. Partial Differ. Equ., 61 (2022), 227. https://doi.org/10.1007/s00526-022-02364-8 doi: 10.1007/s00526-022-02364-8
|
| [15] |
S. Challal, A. Lyaghfouri, Hölder continuity of solutions to the $A$-Laplace equation involving measures, Commun. Pure Appl. Anal., 8 (2009), 1577–1583. https://doi.org/10.3934/cpaa.2009.8.1577 doi: 10.3934/cpaa.2009.8.1577
|
| [16] | I. Chlebicka, P. Gwiazda, A. Świerczewska Gwiazda, A. Wróblewska-Kamińska, Partial differential equations in anisotropic Musielak-Orlicz spaces, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-88856-5 |
| [17] |
J. C. de Albuquerque, L. R. S. de Assis, M. L. M. Carvalho, A. Salort, On fractional Musielak-Sobolev spaces and applications to nonlocal problems, J. Geom. Anal., 33 (2023), 130. https://doi.org/10.1007/s12220-023-01211-2 doi: 10.1007/s12220-023-01211-2
|
| [18] |
L. M. Del Pezzo, J. D. Rossi, Traces for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory, 2 (2017), 435–446. https://doi.org/10.22034/aot.1704-1152 doi: 10.22034/aot.1704-1152
|
| [19] |
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
|
| [20] |
J. Fernández Bonder, A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333–367. https://doi.org/10.1016/j.jfa.2019.04.003 doi: 10.1016/j.jfa.2019.04.003
|
| [21] |
J. Fernández Bonder, A. Salort, H. Vivas, Interior and up to the boundary regularity for the fractional $g$-Laplacian: the convex case, Nonlinear Anal., 223 (2022), 113060. https://doi.org/10.1016/j.na.2022.113060 doi: 10.1016/j.na.2022.113060
|
| [22] |
J. Fernández Bonder, A. Salort, H. Vivas, Global Hölder regularity for eigenfunctions of the fractional $g$-Laplacian, J. Math. Anal. Appl., 526 (2023), 127332. https://doi.org/10.1016/j.jmaa.2023.127332 doi: 10.1016/j.jmaa.2023.127332
|
| [23] |
N. Gigli, S. Mosconi, The abstract Lewy-Stampacchia inequality and applications, J. Math. Pures Appl., 104 (2015), 258–275. https://doi.org/10.1016/j.matpur.2015.02.007 doi: 10.1016/j.matpur.2015.02.007
|
| [24] | P. Harjulehto, P. Hästö, Orlicz spaces and generalized Orlicz spaces, Vol. 2236, Cham: Springer, 2019. https://doi.org/10.1007/978-3-030-15100-3 |
| [25] |
P. Harjulehto, P. Hästö, R. Klén, Generalized Orlicz spaces and related PDE, Nonlinear Anal., 143 (2016), 155–173. https://doi.org/10.1016/j.na.2016.05.002 doi: 10.1016/j.na.2016.05.002
|
| [26] |
P. Harjulehto, A. Karppinen, Stability of solutions to obstacle problems with generalized Orlicz growth, Forum Math., 36 (2024), 285–304. https://doi.org/10.1515/forum-2022-0099 doi: 10.1515/forum-2022-0099
|
| [27] | J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, 1993. |
| [28] |
U. Kaufmann, J. D. Rossi, R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Electron. J. Qual. Theory Differ. Equ., 76 (2017), 1–10. https://doi.org/10.14232/ejqtde.2017.1.76 doi: 10.14232/ejqtde.2017.1.76
|
| [29] | D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Vol. 31, Society for Industrial and Applied Mathematics, 2000. https://doi.org/10.1137/1.9780898719451 |
| [30] |
J. Korvenpää, T. Kuusi, G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differ. Equ., 55 (2016), 63. https://doi.org/10.1007/s00526-016-0999-2 doi: 10.1007/s00526-016-0999-2
|
| [31] | M. A. Krasnosel'ski$\breve{\rm{i}}$, J. B. Ruticki$\breve{\rm{i}}$, Convex functions and Orlicz spaces, P. Noordhoff Ltd., 1961. |
| [32] |
T. Kuusi, G. Mingione, Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57–114. https://doi.org/10.2140/apde.2015.8.57 doi: 10.2140/apde.2015.8.57
|
| [33] |
T. Leonori, I. Peral, A. Primo, F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031–6068. https://doi.org/10.3934/dcds.2015.35.6031 doi: 10.3934/dcds.2015.35.6031
|
| [34] |
E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differ. Equ., 49 (2014), 795–826. https://doi.org/10.1007/s00526-013-0600-1 doi: 10.1007/s00526-013-0600-1
|
| [35] | C. Lo, Nonlocal anisotropic problems with fractional type derivatives, Faculdade de Ciências da Universidade de Lisboa, Ph.D. Thesis, 2022. |
| [36] |
C. W. K. Lo, J. F. Rodrigues, On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative, Port. Math., 80 (2023), 157–205. https://doi.org/10.4171/pm/2100 doi: 10.4171/pm/2100
|
| [37] | C. W. K. Lo, J. F. Rodrigues, On the stability of the $s$-nonlocal $p$-obstacle problem and their coincidence sets and free boundaries, arXiv, 2024. https://doi.org/10.48550/arXiv.2402.18106 |
| [38] | O. Méndez, J. Lang, Analysis on function spaces of Musielak-Orlicz type, CRC Press, 2018. https://doi.org/10.1201/9781498762618 |
| [39] |
S. Molina, A. Salort, H. Vivas, Maximum principles, Liouville theorem and symmetry results for the fractional $g$-Laplacian, Nonlinear Anal., 212 (2021), 112465. https://doi.org/10.1016/j.na.2021.112465 doi: 10.1016/j.na.2021.112465
|
| [40] | U. Mosco, Implicit variational problems and quasi variational inequalities, In: J. P. Gossez, E. J. Lami Dozo, J. Mawhin, L. Waelbroeck, Nonlinear operators and the calculus of variations, Lecture Notes in Mathematics, Springer, 543 (1976), 83–156. https://doi.org/10.1007/BFb0079943 |
| [41] | J. Musielak, Orlicz spaces and modular spaces, Vol. 1034, Springer, 1983. https://doi.org/10.1007/BFb0072210 |
| [42] |
J. Ok, Local Hölder regularity for nonlocal equations with variable powers, Calc. Var. Partial Differ. Equ., 62 (2023), 32. https://doi.org/10.1007/s00526-022-02353-x doi: 10.1007/s00526-022-02353-x
|
| [43] | E. H. Ouali, A. Baalal, M. Berghout, Density properties for fractional Musielak-Sobolev spaces, arXiv, 2024. https://doi.org/10.48550/arXiv.2403.12305 |
| [44] |
G. Palatucci, The Dirichlet problem for the $p$-fractional Laplace equation, Nonlinear Anal., 177 (2018), 699–732. https://doi.org/10.1016/j.na.2018.05.004 doi: 10.1016/j.na.2018.05.004
|
| [45] |
M. Piccinini, The obstacle problem and the Perron method for nonlinear fractional equations in the Heisenberg group, Nonlinear Anal., 222 (2022), 112966. https://doi.org/10.1016/j.na.2022.112966 doi: 10.1016/j.na.2022.112966
|
| [46] | J. F. Rodrigues, Obstacle problems in mathematical physics, Vol. 134, Amsterdam: North-Holland Publishing Co., 1987. |
| [47] |
J. F. Rodrigues, R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and applications, Complex Var. Elliptic Equ., 56 (2011), 769–787. https://doi.org/10.1080/17476933.2010.505016 doi: 10.1080/17476933.2010.505016
|
| [48] |
X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3–26. https://doi.org/10.5565/PUBLMAT_60116_01 doi: 10.5565/PUBLMAT_60116_01
|
| [49] |
R. Servadei, E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091–1126. https://doi.org/10.4171/rmi/750 doi: 10.4171/rmi/750
|
| [50] |
S. Shi, J. Xiao, Fractional capacities relative to bounded open Lipschitz sets complemented, Calc. Var. Partial Differ. Equ., 56 (2017), 3. https://doi.org/10.1007/s00526-016-1105-5 doi: 10.1007/s00526-016-1105-5
|
| [51] |
S. Shi, J. Xiao, On fractional capacities relative to bounded open Lipschitz sets, Potential Anal., 45 (2016), 261–298. https://doi.org/10.1007/s11118-016-9545-2 doi: 10.1007/s11118-016-9545-2
|
| [52] |
S. Shi, L. Zhang, Dual characterization of fractional capacity via solution of fractional $p$-Laplace equation, Math. Nachr., 293 (2020), 2233–2247. https://doi.org/10.1002/mana.201800438 doi: 10.1002/mana.201800438
|
| [53] | G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189–258. |
| [54] |
M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499–547. https://doi.org/10.1007/s11118-014-9443-4 doi: 10.1007/s11118-014-9443-4
|
Catharine W. K. Lo, José Francisco Rodrigues. On the obstacle problem in fractional generalised Orlicz spaces[J]. Mathematics in Engineering, 2024, 6(5): 676-704. doi: 10.3934/mine.2024026
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