Interior Estimates for Elliptic Partial Differential Equations in the L(q,λ) Spaces of Strong Type

Ono, Akira

Canadian journal of mathematics, Tome 36 (1984) no. 3, pp. 385-404

Résumé

Recently the L (q,λ) spaces have been investigated by many authors and the theory of these spaces has proved to be particularly important for research in partial differential equations (see for example [15], [16] and [18]).The equations of elliptic type in these spaces were first studied by C. B. Morrey [8], [9], who applied his well-known imbedding theorems, and afterwards by S. Campanato [3], [4] with the aid of isomorphism theorems and the so-called fundamental inequalities due to him.On the other hand, G. Stampacchia introduced the L (q,λ) spaces of strong type [17], the structures of which are more general and complicated than those of L (q,λ) Spaces in the usual sense, and greater part of them were characterized by him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [11]-[14], [16] and [17]).

Détail

DOI : 10.4153/CJM-1984-024-8

Bibliographie

[1] 1. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math 12 (1959), 623–727. Google Scholar

[2] 2. Campanato, S., Proprietà di Hölderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup Pisa 17 (1963), 177–188. Google Scholar

[3] 3. Campanato, S., Equazioni ellittiche del II° ordine e spazi ℒ2, λ Ann. Mat. Pura Appl. 69 (1965), 321–381. Google Scholar

[4] 4. Campanato, S., Equazioni ellittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. 86 (1970), 125–154. Google Scholar

[5] 5. Furusho, Y., On inclusion property for certain ℒ(p, λ) spaces of strong type, Funkcial. Ekvac. 23 (1980), 197–205. Google Scholar

[6] 6. John, F. and Nirenberg, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math 74 (1961), 415–426. Google Scholar

[7] 7. Meyers, G. N., Mean oscillation over cubes and Holder continuity, Proc. Amer. Math. Soc. 75 (1964), 717–721. Google Scholar

[8] 8. Morrey, C. B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer Math. Soc. 43 (1938), 126–166. Google Scholar

[9] 9. Morrey, C. B., Second order elliptic equations in several variables and Holder continuity, Math. Z. 72 (1959), 146–164. Google Scholar

[10] 10. Nakamura, M., Interior estimates for the solutions of elliptic partial differential equations in the strong ℒ(q, μ) spaces, Rep. Fac. Sci. Engi. Saga Univ. 3 (1975), 65–72. Google Scholar

[11] 11. Ono, A., On isomorphism between certain strong ℒ(p, λ) spaces and the Sobolev spaces Hl, p, Boll. Unione Mat. Itali. 5 (1972), 230–233. Google Scholar

[12] 12. Ono, A., On isomorphism between certain strong ℒ(p, λ) spaces and the Lipschitz spaces and its applications, Funkcial. Ekvac. 27 (1978), 261–270. Google Scholar

[13] 13. Ono, A., Remarks on the Morrey-Sobolev type imbedding theorems in the strong ℒ(p,λ) spaces, Math. Rep. Coll. Gene. Educ. Kyushu Univ. 77 (1978), 149–155. Google Scholar

[14] 14. Ono, A. and Furusho, Y., On isomorphism and inclusion property for certain ℒ(p,λ) spaces of strong type, Ann. Mat. Pura Appl. 114 (1977), 289–304. Google Scholar

[15] 15. Peetre, J., On the theory of ℒ spaces, J. Func. Anal. 4 (1969), 71–87. Google Scholar

[16] 16. Piccinini, L. C., Proprietá di inclusione e di interpolazione tra apazi di Morrey e loro generalizzazioni (Thesis, Scuola Norm. Sup. Pisa, 1969). Google Scholar

[17] 17. Stampacchia, G., The spaces ℒ(p, λ) N(p, λ) and interpolation, Ann. Scuola Norm. Sup. Pisa 79 (1965), 443–462. Google Scholar

[18] 18. Stampacchia, G., The ℒ(p, λ) spaces and applications to the theory of partial differential equations, Proc. Conf. Diff. Equ. Appl, Bratislava (1969), 129–141. Google Scholar