
TL;DR
This paper studies the evolution of compact subsets in complex space under Levi form-driven dynamics, proving existence, uniqueness, and long-term convergence to Levi flat hypersurfaces.
Contribution
It establishes the existence and uniqueness of solutions to a Levi form evolution problem and describes the asymptotic behavior of evolving graphs.
Findings
Existence and uniqueness of the evolution solution for all time.
Evolution of smooth graphs remains a graph over time.
Long-term convergence to Levi flat hypersurfaces when boundary conditions are met.
Abstract
Let be a compact subset of , a closed subset. In this paper we are dealing with evolution of with fixed part by Levi form. This amounts to solve a parabolic problem for an elliptic operator. We prove existence and unicity for such a problem and the solution exists for any time .If is a smooth graph and the the evolution is still a graph. In particular, if bounds a Levi flat hypersurface then as .
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Taxonomy
TopicsNonlinear Partial Differential Equations · Geometric Analysis and Curvature Flows · Advanced Mathematical Modeling in Engineering
