Characterization of the interior regularity for parabolic systems with discontinuous coefficients

Palagachev, Dian K. ; Softova, Lubomira G.

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 125-132 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We deal in this Note with linear parabolic (in sense of Petrovskij) systems of order $2b$ with discontinuous principal coefficients belonging to $VMO \bigcap L^{\infty}$. By means of a priori estimates in Sobolev-Morrey spaces we give a precise characterization of the Morrey, BMO and Hölder regularity of the solutions and their derivatives up to order $2b-1$.

Résumé

In questa Nota si studiano dei sistemi lineari parabolici (in senso di Petrovskij) di ordine $2b$ con coefficienti principali appartenenti a $VMO \bigcap L^{\infty}$. Per mezzo di stime a priori negli spazi di Sobolev-Morrey si propone una caratterizzazione precisa della regolaritaÁ (Morrey, BMO e Hölder) all'interno delle soluzioni e le loro derivate fino all'ordine $2b-1$.

Détail

Publié le : 2005-06-01

MR 2225506 | Zbl 1225.35116

Bibliographie

[1] Agmon, S. - Douglis, A. - Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math., vol. 12, 1959, 623-727. | MR 125307 | Zbl 0093.10401

[2] Agmon, S. - Douglis, A. - Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math., vol. 17, 1964, 35-92. | MR 162050 | Zbl 0123.28706

[3] Campanato, S., Proprietà di Hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa, vol. 17, 1963, 175-188. | MR 156188 | Zbl 0121.29201

[4] Chiarenza, F. - Franciosi, M. - Frasca, M., $L^{p}$-Estimates for linear elliptic systems with discontinuous coefficients. Rend. Mat. Acc. Lincei, s. 9, vol. 5, 1994, 27-32. | MR 1273890 | Zbl 0803.35016

[5] Da Prato, G., Spazi $\mathcal L^{p,\theta}(\Omega,\delta)$ e loro proprietà. Ann. Mat. Pura Appl., vol. 69, 1965, 383-392. | MR 192330 | Zbl 0145.16207

[6] Douglis, A. - Nirenberg, L., Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math., vol. 8, 1955, 503-538. | MR 75417 | Zbl 0066.08002

[7] Ejdel'Man, S.D., Parabolic Equations. In: Yu.V. Egorov - M.A. Shubin (eds.), Partial Differential Equations VI: Elliptic and Parabolic Operators. Encycl. Math. Sci., vol. 63, Springer-Verlag, Berlin 1994, 203-316. | MR 1313734 | Zbl 0802.35061

[8] Fabes, E.B. - Rivière, N., Singular integrals with mixed homogeneity. Studia Math., vol. 27, 1966, 19-38. | MR 209787 | Zbl 0161.32403

[9] Friedman, A., Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J., 1964. | MR 181836 | Zbl 0144.34903

[10] John, F. - Nirenberg, L., On functions of bounded mean oscillation. Comm. Pure Appl. Math., vol. 14, 1961, 415-426. | MR 131498 | Zbl 0102.04302

[11] Kronz, M., Some function spaces on spaces of homogeneous type. Manuscr. Math., vol. 106, 2001, 219-248. | MR 1865566 | Zbl 0989.43004

[12] Maugeri, A. - Palagachev, D.K. - Softova, L.G., Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley-VCH, Berlin 2000. | MR 2260015 | Zbl 0958.35002

[13] Palagachev, D.K. - Softova, L.G., Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's. Potential Anal., vol. 20, 2004, 237-263. | MR 2032497 | Zbl 1036.35045

[14] Sarason, D., Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., vol. 207, 1975, 391-405. | MR 377518 | Zbl 0319.42006

[15] Solonnikov, V.A., On the boundary value problems for linear parabolic systems of differential equations of general form. Proc. Steklov Inst. Math., vol. 83, 1965 (in Russian); English translation: O.A. Ladyzhenskaya (ed.), Boundary Value Problems of Mathematical Physics III. Amer. Math. Soc., Providence, R.I., 1967. | Zbl 0164.12502