Elliptic problems on the space of weighted with the distance to the boundary integrable functions revisited
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.contributor.author | Rakotoson, Jean-Michel | |
| dc.date.accessioned | 2023-06-19T13:32:55Z | |
| dc.date.available | 2023-06-19T13:32:55Z | |
| dc.date.issued | 2014 | |
| dc.description | Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). Electronic Journal of Differential Equations, Conference 21 (2014), | |
| dc.description.abstract | We revisit the regularity of very weak solution to second-order elliptic equations Lu = f in Ω with u = 0 on ∂Ω for f ∈ L1 (Ω, δ), δ(x) the distance to the boundary ∂Ω. While doing this, we extend our previous results(and many others in the literature)by allowing the presence of distributions f+g which are more general than Radon measures (more precisely with g in the dual of suitable Lorentz-Sobolev spaces) and by making weaker assumptions on the coefficients of L. One of the new tools is a Hardy type inequality developed recently by the second author. Applications to the study of the gradient of solutions of some singular semilinear equations are also given. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Unión Europea. FP7 | |
| dc.description.sponsorship | DGISPI, Spain | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/29595 | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.officialurl | http://ejde.math.txstate.edu/Volumes/2014/90/begout.pdf | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/33990 | |
| dc.journal.title | Electronic Journal of Differential Equations, Conference | |
| dc.language.iso | eng | |
| dc.page.final | 59 | |
| dc.page.initial | 45 | |
| dc.publisher | Department of Mathematics Texas State University | |
| dc.relation.projectID | FIRST (238702) | |
| dc.relation.projectID | MTM 2011-26119 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 517.9 | |
| dc.subject.keyword | Very weak solutions | |
| dc.subject.keyword | semilinear elliptic equations | |
| dc.subject.keyword | distance to the boundary | |
| dc.subject.keyword | weighted spaces measure | |
| dc.subject.keyword | Hardy inequalities | |
| dc.subject.keyword | Hardy spaces | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | Elliptic problems on the space of weighted with the distance to the boundary integrable functions revisited | |
| dc.type | journal article | |
| dc.volume.number | 21 | |
| dcterms.references | F. Abergel, J. M. Rakotoson; Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete and Continuous Dynamical Systems 33, 5 (2013) 1809-1818. C. Bennett, R. Sharpley; Interpolation of Operators, Academic Press, London, (1983). H. Brezis; Personal communication with J.I. DÍaz. H. Brezis, W. A. Strauss; Semilinear second order elliptic equations in L1, J.Math. Soc .Japan, 25 (1973) 565-590. H. Brezis, M. Marcus; Hardy’s inequality revisited, Ann. Scuola Norm. Sup. Pisa, cl. sci. 4(1997) 217-237. Sun-Sig Byun; Parabolic equations with BMO coefficients in Lipschitz domains, Journal of differential Equations, 209 (2005) 229-295. M. G. Crandall, P. H. Rabinowitz, L. Tartar; On a Dirichlet problem with a singular nonlinearity, Comm. Part. Diff. Eq. 2, (1977) 193-222. S. Campanato; Equazioni ellittiche dell IIe ordine e spacazi L2,λ, Annali di Matematica,(1965) 41-370. J. I. Díaz; Nonlinear partial differential equations and free boundaries, (1985) Pitman, London. M. Del Pino; A global estimate for the gradient in a singular elliptic boundary value problem. Proc. Roy. Soc. Edinburgh Sect A, 122, (1992), 341-352 J. I. Díaz, J. M. Rakotoson; On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary. J. Functional Analysis 257, (2009), 807-831. J. I. Díaz, J. M. Rakotoson; On very weak solution of semi-linear elliptic equation in the framework of weighted spaces with respect to the distance to the boundary.Discrete and Continuous Dynamical Systems , 27 (3), (2010), 1037-1085. J. I. Díaz, J. Hernández, J. M. Rakotoson; On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spacial dependence terms, Milan J. Math,79 (2011) 233-245. P. Hajlasz; Pointwise Hardy inequalities, Proc. Amer. Math. Soc., 127 2, (1999), 417-423. J. Hernández, F. Mancebo; Singular elliptic and parabolic equations in Handbook of differential equations Stationary partial differential equations, vol 3 317-400. M. Ghergu; Lane-Emden systems with negative exponents, J. Functional Analysis. 258(2010) 3295-3318. C. Gui, F. Hua Lin; Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy.Soc. Edimburg Sect. A 123 (1993) 1021-1029. F. John, L. Niremberg; On functions of bounded mean oscillation, Comm. Pure Appl. Math.,14,(1961) 415-426. A. Kufner; Weighted Sobolev spaces, John Wiley & Sons, New-York, 1985. A. C. Lazer, P. J. Mc Kenna; On a singular nonlinear elliptic boundary value problem, Proc.Amer. Math. Soc. 111 (1991), 721-730. J. Merker, J. M. Rakotoson; Very weak solutions of Poisson’s equation with singular data under Neumann boundary conditions. to appear. J. Mossino, R. Temam; Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Make J. 48 (1981) 475-495. J. M. Rakotoson; Regularity of a very weak solution for parabolic equations and applications,Adv. Diff. Equa. 16 9-10 (2011), 867-894. J. M. Rakotoson; New Hardy inequalities and behaviour of linear elliptic equations, J. Funct.Ann. 263 (2012), 2893-2920. J. M. Rakotoson; A few natural extension of the regularity of a very weak solution, Diff. Int. Eq. 24 (11-12) (2011), 1125-1140. J. M. Rakotoson; R´earrangement Relatif: un instrument d’estimation dans les problèmes aux limites, (2008), Springer Verlag, Berlin. G. Stampacchia; Some limit case of Lp-estimates for solutions of second order elliptic equations,Comm. Pure Appli. Math. 16 (1963) 505-510. [28] A. Torchinsky; Real-Variable Methods in Harmonic Analysis, Academic Press, Orlando, 1986. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
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