Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties [PDF]
, 2012 We show, using a direct variational approach, that the second boundary value problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P.
Berman, Robert J., Berndtsson, Bo
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On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows
, 2020 Let $X$ be a compact K\"ahler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson ...
Xia, Mingchen
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Numerical solution of the simple Monge–Ampère equation with nonconvex dirichlet data on non-convex domains [PDF]
, 2017 The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Ampere equation is known independently of the convexity of the domain or Dirichlet boundary data - when the Monge-Ampere equation is posed as a Bellman ...
Jensen, Max
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Functions dividing their Hessian determinants and affine spheres [PDF]
, 2016 The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function.
Fox, Daniel J. F.
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A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows [PDF]
, 2016 In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses.
Ambrosio L. +7 more
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Attractors and Finite‐Dimensional Behaviour in the 2D Navier‐Stokes Equations
International Scholarly Research Notices, Volume 2013, Issue 1, 2013., 2013 The purpose of this review is to give a broad outline of the dynamical systems approach to the two‐dimensional Navier‐Stokes equations. This example has led to much of the theory of infinite‐dimensional dynamical systems, which is now well developed.
James C. Robinson +3 more
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Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model
, 2015 This paper deals with several qualitative properties of solutions of some parabolic equations associated to the Monge--Ampere operator arising in suitable formulations of the Gauss curvature flow and the worn stone problems.
G. Díaz, J. I. Díaz
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The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge‐Ampère Equation
Abstract and Applied Analysis, Volume 2013, Issue 1, 2013., 2013 We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge‐Ampère type. We show that such solution exists for all times and is unique.
Juan Wang +3 more
wiley +1 more source
Viscosity solutions of fully nonlinear functional parabolic PDE
International Journal of Mathematics and Mathematical Sciences, Volume 2005, Issue 22, Page 3539-3550, 2005., 2005 By the technique of coupled solutions, the notion of viscosity solutions is extended to fully nonlinear retarded parabolic equations. Such equations involve many models arising from optimal control theory, economy and finance, biology, and so forth. The comparison principle is shown.
Liu Wei-an, Lu Gang
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Convergence of the weak K\"ahler-Ricci Flow on manifolds of general type
, 2019 We study the K\"ahler-Ricci flow on compact K\"ahler manifolds whose canonical bundle is big. We show that the normalized K\"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique
Tô, Tat Dat
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