Local Gradient Estimates of Solutions to Some Conformally Invariant Fully Nonlinear Equations
نویسندگان
چکیده
The Laplacian operator is invariant under rigid motions: For any function u on R and for any rigid motion T W R ! R, .u ı T / D . u/ ı T: T is called a rigid motion if T x Ox C b for some n n orthogonal matrix O and some vector b 2 R. It is clear that a linear second-order partial differential operator Lu WD aij .x/uij C bi .x/ui C c.x/u is invariant under rigid motion, i.e., L.u ı T / D .Lu/ ı T for any function u and any rigid motion T if and only if L D a C c for some constants a and c. Instead of rigid motions, we look at Möbius transformations of R [ f1g and nonlinear operators that are invariant under Möbius transformations. A map ' W R [ f1g ! R [ f1g is called a Möbius transformation if it is a composition of finitely many of the following three types of transformations: a translation W x ! x C N x where N x is a given point in R;
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2 00 6 Local gradient estimates of solutions to some conformally invariant fully nonlinear equations
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