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Hyperbolic Monge-Ampère Equation
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In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
Title: Hyperbolic Monge-Ampère Equation
Description:
In this paper we use the Sobolev steepest descent method introduced by John W.
Neuberger to solve the hyperbolic Monge-Ampère equation.
First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions.
Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation.
Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
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