Parabolic De Giorgi classes of order p and the Harnack inequality
Tóm tắt
We define suitable homogeneous parabolic De Giorgi classes of order p and show that in this context a Harnack inequality holds with a time gap of order ρ
p
.
Tài liệu tham khảo
Bombieri, E.: Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimensions. Mimeographed Notes of Lectures held at Courant Institute, New York University (1970)
Bombieri, E., Giusti, E.: Harnack's inequality for elliptic differential equations on minimal surfaces. Inventiones Math. 15, 26–46 (1972)
DeGiorgi, E.: Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. Mem. Acc. Sci. Torino, Cl. Sc. Fis. Mat. Nat. 3, 25–43 (1957)
DiBenedetto, E.: Harnack estimates in certain function classes. Atti. Sem. Mat. Fis. Univ. Modena 37, 173–182 (1989)
DiBenedetto, E.: Degenerate Parabolic Equations. Springer Verlag, Series Universitext, New York (1993)
DiBenedetto, E.: Real Analysis. Birkhäuser Verlag (2002)
DiBenedetto, E., Gianazza, U., Vespri, V.: Local Clustering of the Non-Zero Set of Functions in W 1,1 (Kρ). preprint (2005)
DiBenedetto, E., Trudinger, N.S.: Harnack inequalities for quasi-minima of variational integrals. Ann. Inst. Henri Poincaré 1, 295–308 (1984)
DiBenedetto, E., Urbano, J.M., Vespri, V.: Current issues on singular and degenerate evolution equations. vol. I, pp. 169–286. North-Holland, Amsterdam (2004)
DiBenedetto, E., Vespri, V.: On the singular equation β (u) t = Δu. Arch. Rat. Mech. Anal. 132(3), 247–309 (1995)
Fabes, E.B., Garofalo, N.: Parabolic B.M.O. and Harnack's Inequality. Proc. A.M.S. 95, 63–69 (1985)
Fabes, E.B., Stroock, D.W.: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 327–338 (1986)
Giaquinta, M., Giusti, E.L Quasi-Minima. Ann. Inst. Henri Poincaré, Analyse Non Linéaire 1, 79–107 (1984)
Giusti, E.: Metodi diretti nel Calcolo delle Variazioni. Unione Matematica Italiana, Bologna (1994)
Ladyzhenskaja, O.A., Solonnikov, V.A., Ural'tzeva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS Transl. Math. Mono, vol. 23. Providence RI, USA (1968)
Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge, NJ (1996)
Moser, J.: A harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)
Moser, J.: On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24, 727–740 (1971)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)
Saloff-Coste, L.: Aspects of Sobolev—Type Inequalities. London Mathematical Society Lecture Notes Series, #289 (2002)
Wang, G.: Harnack inequalities for functions in the De Giorgi parabolic classes. Lect. Notes Math. 1306, 182–201 (1989)
Wang, G.L., Sun, A.X.: Weak Harnack inequality for functions from class B 2 and application to parabolic Q-minima. Chinese J. Contemp. Math. 14, 75–84 (1993)
Wieser, W.: Parabolic Q-minima and minimal solutions to variational flows. Manuscripta Math. 59, 63–107 (1987)
Zhou, S.L.: On the local behaviour of parabolic Q-minima. J. Partial Diff. Eq. 6, 255–272 (1993)
Zhou, S.L.: Parabolic Q-minima and their application. J. Partial Diff. Eq. 7, 289–322 (1994)