Artículo
Abstract:
If Ω ⊂ Rn is a bounded domain, the existence of solutions u ∈ W01, p (Ω) of div u = f for f ∈ Lp (Ω) with vanishing mean value and 1 < p < ∞, is a basic result in the analysis of the Stokes equations. It is known that the result holds when Ω is a Lipschitz domain and that it is not valid for domains with external cusps. In this paper we prove that the result holds for John domains. Our proof is constructive: the solution u is given by an explicit integral operator acting on f. To prove that u ∈ W01, p (Ω) we make use of the Calderón-Zygmund singular integral operator theory and the Hardy-Littlewood maximal function. For domains satisfying the separation property introduced in [S. Buckley, P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (5) (1995) 577-593], and 1 < p < n, we also prove a converse result, thus characterizing in this case the domains for which a continuous right inverse of the divergence exists. In particular, our result applies to simply connected planar domains because they satisfy the separation property. © 2005 Elsevier Inc. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | Solutions of the divergence operator on John domains |
| Autor: | Acosta, G.; Durán, R.G.; Muschietti, M.A. |
| Filiación: | Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, B1613GSX Provincia de Buenos Aires, Argentina Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Casilla de Correo 172, 1900 La Plata, Pr. de Buenos Aires, Argentina |
| Palabras clave: | Divergence operator; John domains; Singular integrals |
| Año: | 2006 |
| Volumen: | 206 |
| Número: | 2 |
| Página de inicio: | 373 |
| Página de fin: | 401 |
| DOI: | http://dx.doi.org/10.1016/j.aim.2005.09.004 |
| Título revista: | Advances in Mathematics |
| Título revista abreviado: | Adv. Math. |
| ISSN: | 00018708 |
| PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00018708_v206_n2_p373_Acosta.pdf |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v206_n2_p373_Acosta |
Referencias:
- Bogovskii, M.E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium (1979) Soviet Math. Dokl., 20, pp. 1094-1098
- Buckley, S., Koskela, P., Sobolev-Poincaré implies John (1995) Math. Res. Lett., 2 (5), pp. 577-593
- Calderón, A.P., Zygmund, A., On singular integrals (1956) Amer. J. Math., 78, pp. 289-309
- David, G., Semmes, S., Quasiminimal surfaces of codimension one and John domains (1998) Pacific J. Math., 183 (2), pp. 213-277
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- Martio, O., Sarvas, J., Injectivity theorems in plane and space (1979) Ann. Acad. Sci. Fenn. Ser. A I. Math., 4, pp. 383-401
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- Swanson, D., Ziemer, W.P., Sobolev functions whose inner trace at the boundary is zero (1999) Ark. Mat., 37 (2), pp. 373-380
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Citas:
---------- APA ----------
Acosta, G., Durán, R.G. & Muschietti, M.A. (2006) . Solutions of the divergence operator on John domains. Advances in Mathematics, 206(2), 373-401.http://dx.doi.org/10.1016/j.aim.2005.09.004
---------- CHICAGO ----------
Acosta, G., Durán, R.G., Muschietti, M.A. "Solutions of the divergence operator on John domains" . Advances in Mathematics 206, no. 2 (2006) : 373-401.http://dx.doi.org/10.1016/j.aim.2005.09.004
---------- MLA ----------
Acosta, G., Durán, R.G., Muschietti, M.A. "Solutions of the divergence operator on John domains" . Advances in Mathematics, vol. 206, no. 2, 2006, pp. 373-401.http://dx.doi.org/10.1016/j.aim.2005.09.004
---------- VANCOUVER ----------
Acosta, G., Durán, R.G., Muschietti, M.A. Solutions of the divergence operator on John domains. Adv. Math. 2006;206(2):373-401.http://dx.doi.org/10.1016/j.aim.2005.09.004
