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Register a new userDirichlet Problem of Quaternionic Monge-Amp`ere Equations
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Authors
Jingyong Zhu
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In this paper, the author studies quaternionic Monge-Amp`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to the open problem proposed by Semyon Alesker in [3], but also extends relevant results in [7] to the quaternionic vector space.
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We prove the existence of entire solutions of the Monge-Amp`ere equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Amp`ere equation $det abla^2 u=1$ with $kge 2$ singular points, which was established by Galvez-Martinez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Amp`ere equation, and under degenerations to a big class in the case of Hessian equations.
We obtain a genuine local $C^2$ estimate for the Monge-Amp`ere equation in dimension two, by using the partial Legendre transform.
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.
We develop a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kahler geometry. In cite{GL21b} we have studied the behavior of Monge-Amp`ere volumes on hermitian manifolds. We extend here the techniques of cite{GL21a} to the hermitian setting and use the bounds established in cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp`ere equations on compact hermitian manifolds.
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