By Brezis H., Browder F.

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Schauder, C. Morrey, L. Bers, L. Nirenberg, O. Ladyzhenskaya, N. Uraltseva, and J. Serrin. A complete theory, developed by O. Ladyzhenskaya N. Uraltseva [L-U], provides interior estimates of solutions of (17), as well as estimates up to the boundary for the same equation with a boundary condition. For a detailed account see [L-U], [G-T] and [Ser1]. For a broad survey of the theory of singularities for quasilinear equations we refer to the book of L. g. the early work of L. Bers and J. Serrin. More recently, corresponding results have been established for a broad class of fully nonlinear equations (18), notably including the Monge AmpeÁre equation det(D 2u)= f (x, u, Du) (19) of great importance in geometrical problems, and the Hamilton Jacobi Bellman equation Sup[A i u& f i ]=0 (20) i#I where (A i ) i # I are a family of linear second order elliptic operator.

Estimates for such problems can be perturbed to yield local estimates for variable coefficient problems under suitable hypotheses on the coefficients (C : for C : estimates and uniformly continuous for L p estimates). The estimates are of the following type &u& C 2m, : (0) \ C & f & C 0, : (0 ) +&u& C 0, : (0 ) +: &g j & C 2m&mj, : ( j 0) + and &u& W 2m, p (0) \ C & f & L p (0) +&u& L p (0) +: & g j & W 2m&mj&(1Â p), p( j 0) +. Here, the boundary term involves a fractional Sobolev norm. When 0~~
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A fairly complete generalization of the Hille Yosida theory in Hilbert spaces has been developed by many authors File: DISTL2 171350 . By:CV . Date:23:03:98 . Time:07:52 LOP8M. B. Page 01:01 Codes: 3145 Signs: 2334 . Length: 45 pic 0 pts, 190 mm 126 BREZIS AND BROWDER including F. Browder, T. Kato, Y. Komura, M. Crandall, A. Pazy and H. Brezis. The principal result asserts that there is a one-to-one correspondence between continuous semi-groups of contractions and maximal monotone operators. We refer to the books of H.

### Partial differential equations in the 20th century by Brezis H., Browder F.

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