On the obstacle problem in fractional generalised Orlicz spaces
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...
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AIMS Press
2024-09-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mine.2024026 |
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| author | Catharine W. K. Lo José Francisco Rodrigues |
| author_facet | Catharine W. K. Lo José Francisco Rodrigues |
| author_sort | Catharine W. K. Lo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-3d15ed0c9eb44235a0a44a1a0ac93ef7 |
| institution | Kabale University |
| issn | 2640-3501 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj-art-3d15ed0c9eb44235a0a44a1a0ac93ef72025-01-24T01:08:20ZengAIMS PressMathematics in Engineering2640-35012024-09-016567670410.3934/mine.2024026On the obstacle problem in fractional generalised Orlicz spacesCatharine W. K. Lo0José Francisco Rodrigues1Department of Mathematics, City University of Hong Kong, Hong Kong, ChinaCMAFcIO – Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa P-1749-016 Lisboa, PortugalWe 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.https://www.aimspress.com/article/doi/10.3934/mine.2024026fractional generalised orlicz spacesnonlocal nonlinear anisotropic operatorsone and two obstacles problems |
| spellingShingle | Catharine W. K. Lo José Francisco Rodrigues On the obstacle problem in fractional generalised Orlicz spaces Mathematics in Engineering fractional generalised orlicz spaces nonlocal nonlinear anisotropic operators one and two obstacles problems |
| title | On the obstacle problem in fractional generalised Orlicz spaces |
| title_full | On the obstacle problem in fractional generalised Orlicz spaces |
| title_fullStr | On the obstacle problem in fractional generalised Orlicz spaces |
| title_full_unstemmed | On the obstacle problem in fractional generalised Orlicz spaces |
| title_short | On the obstacle problem in fractional generalised Orlicz spaces |
| title_sort | on the obstacle problem in fractional generalised orlicz spaces |
| topic | fractional generalised orlicz spaces nonlocal nonlinear anisotropic operators one and two obstacles problems |
| url | https://www.aimspress.com/article/doi/10.3934/mine.2024026 |
| work_keys_str_mv | AT catharinewklo ontheobstacleprobleminfractionalgeneralisedorliczspaces AT josefranciscorodrigues ontheobstacleprobleminfractionalgeneralisedorliczspaces |