On the regularity theory of fully nonlinear parabolic equations. I. (English) Zbl 0832.35025
Recently, L. Caffarelli studied interior regularity of viscosity solutions to fully nonlinear elliptic equations \(F(D^2 u, x)= g(x)\). Caffarelli showed the maximum principle for viscosity solutions based on which he proved a Harnack inequality, \(C^{1, \alpha}\), \(C^{2, \alpha}\) and all the way through \(W^{2, p}\) theory for such solutions.
In this paper, we extend the results of Caffarelli to general parabolic equations. The main difficulties lie in the proof of the Harnack inequality and \(W^{2, p}\) estimates. Unlike the elliptic equations, the estimates for parabolic equations always have a shift in time which might produce errors in the process of iteration. This prevents us from using the argument of Caffarelli directly.
To overcome this defect, we use a form of Calderon-Zygmund decomposition particularly for parabolic equations, and show that the error is under control. To get the \(W^{2,p}\) theory, we show that there is no error if we properly extend the definition of the solution to later time. \(C^{1, \alpha}\) and \(C^{2, \alpha}\) theory for parabolic equations is very close to that of elliptic equations. The boundary regularities are new, even for linear equations. Since we use only the maximum principle and iteration, the theorems are very precise and, in our opinion, transparent and geometrical.
In this paper, we extend the results of Caffarelli to general parabolic equations. The main difficulties lie in the proof of the Harnack inequality and \(W^{2, p}\) estimates. Unlike the elliptic equations, the estimates for parabolic equations always have a shift in time which might produce errors in the process of iteration. This prevents us from using the argument of Caffarelli directly.
To overcome this defect, we use a form of Calderon-Zygmund decomposition particularly for parabolic equations, and show that the error is under control. To get the \(W^{2,p}\) theory, we show that there is no error if we properly extend the definition of the solution to later time. \(C^{1, \alpha}\) and \(C^{2, \alpha}\) theory for parabolic equations is very close to that of elliptic equations. The boundary regularities are new, even for linear equations. Since we use only the maximum principle and iteration, the theorems are very precise and, in our opinion, transparent and geometrical.
MSC:
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
| 35K55 | Nonlinear parabolic equations |
| 35B45 | A priori estimates in context of PDEs |
| 35B50 | Maximum principles in context of PDEs |
References:
| [1] | Caffarelli, Ann. of Math. 130 pp 189– (1989) |
| [2] | Caffarelli, Comm. Pure Appl. Math. 38 pp 209– (1985) |
| [3] | Campanato, Ann. Mat. Pura Appl. 69 pp 321– (1965) |
| [4] | Campanato, Ann. Sc. Norm. Sup. Pisa 18 pp 137– (1964) |
| [5] | Crandall, Trans. Amer. Math. Soc. 277 pp 1– (1983) |
| [6] | Evans, Comm. Pure Appl. Math. 25 pp 333– (1982) |
| [7] | Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983. |
| [8] | and , Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 |
| [9] | Ishii, Comm. Pure Appl. Math. 42 pp 15– (1989) |
| [10] | Ishii, Duke Math. J. 55 pp 369– (1987) |
| [11] | Jensen, Arch. Rational Mech. Anal. 101 pp 1– (1988) |
| [12] | Krylov, Izv. Akad. Nauk SSSR, Ser. Mat. 42 pp 1050– (1978) |
| [13] | Nonlinear Elliptic and Parabolic Equations of Second Order, Mathematics and its Applications, Reidel, Dordrecht-Boston-London, 1985. |
| [14] | Krylov, Izv. Akad. Nauk SSSR 44 pp 161– (1980) |
| [15] | Krylov, Dokl. Akad. Nauk SSSR 274 pp 21– (1983) |
| [16] | , and , Linear and Quasilinear Parabolic Equations, Nauka, Moscow, 1967. |
| [17] | Moser, Comm. Pure Appl. Math. 17 pp 101– (1964) |
| [18] | Pucci, Ann. Mat. Pura Appl. 72 pp 141– (1966) |
| [19] | Singular Integrals and the Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. |
| [20] | Tso, Comm. PDE 10 pp 543– (1985) |
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