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Singular Points of Free Boundary in Bernoulli Type Problems

Hajian, Matin | 2025

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58390 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Fotouhi, Morteza
  7. Abstract:
  8. Regularity theory of weak solutions to PDEs is a central question in the analysis of PDEs and has strong connections with geometric measure theory and geometric analysis. The classical problem in this area asks about the optimal regularity of a weak solution to a PDE on a smooth domain of real space with suitable boundary conditions. A free boundary problem is a type of PDE in which the domain of definition of the solution is not known a priori. The topological boundary of this domain is usually called the free boundary. Understanding and investigating the geometric and analytic properties of this set is the main goal of the regularity theory of free boundaries. These types of problems appear in physical situations such as the flow of fluid jets in two dimensions or symmetric fluids in three dimensions, the melting of ice in water, and tumor growth. One important class of free boundary problems is the Bernoulli-type free boundary problem. These problems arise as Euler–Lagrange equations of discontinuous energy functionals. The rigorous analysis of these problems began with the pioneering works of Alt, Caffarelli, and Friedman in the 1980s. They first introduced a notion of weak solutions to the one-phase Bernoulli-type problem via variational tools, and they laid the foundations of blow-up analysis to address the regularity of the free boundary. The blow-up analysis, as well as many later results that we will cover in this thesis, have their origins in the regularity theory of minimal surfaces. As we will see throughout this thesis, there is a strong analogy between the regularity theory of minimal surfaces and that of free boundaries. In this thesis, we investigate the regularity theory of one-phase Bernoulli-type free boundary problems and later introduce their generalisations to two-phase problems and systems of PDEs
  9. Keywords:
  10. Geometric Measure Theory ; Regularity Theory ; Partial Differential Equations ; Free Boundaries ; Laplacian Eigen Values

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