17,924 research outputs found
A priori estimates for the complex Hessian equations
We prove some a priori estimates as well as existence and
stability theorems for the weak solutions of the complex Hessian equations in
domains of and on compact K\"ahler manifolds. We also show optimal
integrability for m-subharmonic functions with compact singularities, thus
partially confirming a conjecture of Blocki. Finally we obtain a local
regularity result for solutions of the real and complex Hessian
equations under suitable regularity assumptions on the right hand side. In the
real case the method of this proof improves a result of Urbas.Comment: 18 pages, preliminary versio
A priori estimates for the Hill and Dirac operators
Consider the Hill operator in , where is a 1-periodic real potential. The spectrum of is is absolutely
continuous and consists of bands separated by gaps \g_n,n\ge 1 with length
|\g_n|\ge 0. We obtain a priori estimates of the gap lengths, effective
masses, action variables for the KDV. For example, if \m_n^\pm are the
effective masses associated with the gap \g_n=(\l_n^-,\l_n^+), then
|\m_n^-+\m_n^+|\le C|\g_n|^2n^{-4} for some constant and any . In order prove these results we use the analysis of a conformal mapping
corresponding to quasimomentum of the Hill operator. That makes possible to
reformulate the problems for the differential operator as the problems of the
conformal mapping theory. Then the proof is based on the analysis of the
conformal mapping and the identities. Moreover, we obtain the similar estimates
for the Dirac operator
A priori estimates for Donaldson's equation over compact Hermitian manifolds
In this paper we prove a priori estimates for Donaldson's equation
over a compact Hermitian manifold X of complex dimension n, where and
are arbitrary Hermitian metrics. Our estimates answer a question of
Tosatti-Weinkove.Comment: 16 pages. Calculus of Variations and Partial Differential Equations,
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