We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.Mathematical Subject Classification (2010): 35J15, 58J05, 35B45. Keywords: Fully nonlinear elliptic equations on Riemannian manifolds, Dirichlet problem, a priori estimates, concavity, subsolutions.
We derive a priori estimates for second order derivatives of solutions to a wide class of fully nonlinear elliptic equations on Riemannian manifolds. There had been significant work in this direction, especially in connection with important geometric problems and other applications, but one had to make use of the special structures or needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper we are able to remove most of the technical assumptions and derive the estimates under conditions which are close to optimal. These estimates enable one to prove existence results which are new even for bounded domains in Euclidean space.
Mathematics Subject Classification
Abstract-The authors have derived a receiver model that provides an explicit relationship between the factor and the optical signal-to-noise ratio (OSNR) in optical fiber communication systems for arbitrary pulse shapes, realistic receiver filters, and arbitrarily polarized noise. It is shown how the system performance depends on both the degree of polarization of the noise and the angle between the Stokes' vectors of the signal and the noise. The results demonstrate that the relationship between the OSNR and the factor is not unique when the noise is partially polarized. This paper defines the enhancement factor and three other parameters that explicitly quantify the relative performance of different modulation formats in a receiver. The theoretical and experimental results show that the performance of the return-to-zero format is less sensitive to variations in the receiver characteristics than is the performance of the nonreturn-to-zero format. Finally, a validation of the formula is presented for computing the factor from the OSNR and the Stokes vectors of the signal and the noise by comparison with both experiments and Monte Carlo simulations.
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