Connectedness of a zero-level set as a geometric estimate for parabolic PDEs

김용정 교수 (KAIST)
2013. 10. 17.

Studies on PDEs are mostly focused on ?nding properties of PDEs within a speci?c discipline and on developing a technique specialized to them. However, ?nding a common structure over di?erent disciplines and unifying theories from di?erent subjects into a generalized theory is the direction that mathematics should go in. The purpose of this talk is to introduce a geometric argument that combines Oleinik or Aronson-Benilan type one-sided estimates that arise from various disciplines from hyperbolic to parabolic problems. It is clear that algebraic or analytic formulas and estimates that depend on the speci?c PDE wouldn’t provide such a unified theory and hence we need a di?erent approach. In this talk we will see that a geometric structure of solutions will provide an excellent alternative in doing such a uni?cation. Ultimate goal of this project is to encourage people to make unified approach developing geometric view points.