Regularity of the Convex Solutions of the Monge-Ampère Equation

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Mahmoudian, Sahand | 2024

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 57432 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Fotouhi. Morteza
Abstract:
Partial Differential Equations (PDEs) play a crucial role in modeling various physical, biological, and engineering phenomena. One of the most important and complex of these equations is the Monge-Ampère equation, which appears in various fields including differential geometry and optimization theory. In this thesis, after defining and examining the preliminary properties of the Monge-Ampère measure and defining the Alexandrov weak solution, the existence and uniqueness of this weak solution for the Dirichlet problem are addressed, and finally, the regularity problem is studied
Keywords:
Partial Differential Equations ; Elliptic Differential Equations ; Monge-Ampère Equation ; Regularity Theory ; Fully Nonlinear Partial Differential Equations (PDE) ; Monge-Ampère Measure

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