Results 11 to 20 of about 56,245 (97)
A Liouville theorem of parabolic Monge-AmpÈre equations in half-space
Discrete and Continuous Dynamical Systems. Series A, 2021 In this paper, we establish the gradient and second derivative estimates for solutions to two kinds of parabolic Monge-Ampere equations in half-space under certain boundary data and growth condition.
Ziwei Zhou, J. Bao, Bo Wang
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Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients
Journal of the London Mathematical Society, Volume 104, Issue 2, Page 688-746, September 2021., 2021 Abstract This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the ...
Dohyun Kwon, Alpár Richárd Mészáros
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Three ways to solve partial differential equations with neural networks — A review
GAMM-Mitteilungen, Volume 44, Issue 2, June 2021., 2021 Abstract Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability for high‐dimensional problems: physics‐informed neural networks, methods based on the ...
Jan Blechschmidt, Oliver G. Ernst
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Error estimation for second‐order partial differential equations in nonvariational form
Numerical Methods for Partial Differential Equations, Volume 37, Issue 3, Page 2190-2221, May 2021., 2021 Abstract Second‐order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton–Jacobi–Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second‐order PDEs. The nondivergence form in these problems
Jan Blechschmidt +2 more
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A parabolic approach to the Calabi–Yau problem in HKT geometry [PDF]
, 2022 We consider the natural generalization of the parabolic Monge–Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex structure is locally flat and admits a compatible ...
Bedulli L., Gentili G., Vezzoni L.
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Canonical complex extensions of Kähler manifolds
Journal of the London Mathematical Society, Volume 101, Issue 2, Page 786-827, April 2020., 2020 Abstract Given a complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent ...
Daniel Greb, Michael Lennox Wong
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Simplified Geometric Approach to Freeform Beam Shaper Design
International Journal of Optics, Volume 2020, Issue 1, 2020., 2020 Beam of light shaping process can be considered ultimate, if both irradiance and wavefront spatial distributions are under control and both can be shaped arbitrarily. In order to keep these two quantities determined simultaneously, it is required to apply at least two powered refractive or reflective surfaces.
Jacek Wojtanowski +2 more
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Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams [PDF]
, 2015 We introduce a monotone (degenerate elliptic) discretization of the Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded.
Mirebeau, Jean-Marie
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Symmetry of solutions to parabolic Monge-Ampère equations
Boundary Value Problems, 2013 In this paper, we study the parabolic Monge-Ampère equation −utdet(D2u)=f(t,u)in Ω×(0,T]. Using the method of moving planes, we show that any parabolically convex solution is symmetric with respect to some hyperplane.
Limei Dai
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Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem [PDF]
, 2018 We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set.
Benamou, Jean-David, Duval, Vincent
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