My research subjects are the viscosity solution theory and its applications.
In 1983, M. G. Crandall and P.-L. Lions introduced the new notion for weak solutions of nonlinear partial differential equations, which is called ``viscosity solutions''. Afterwards, it has turned out that the theory can be applied to a large number of applications such as control theory, game theory, mean curvature motion, phase transition, image process, mathematical finance etc.
On the other hand, after a pioneering work by L. A. Caffarelli in 1989, the notion of Lp-viscosity solutions for fully nonlinear uniformly elliptic PDEs was introduced in 1996 by Caffarelli-Crandall-Kocan-Święch to study Lp regularity theory for PDEs of non-divergent type.
Key words of my recent (and/or future) interests are as follows:
1 : Lp viscosity solutions
2 : Comparison principle for unbounded viscosity solutions in unbounded domains with superlinear terms in Du
3 : Regularity of viscosity solutions arising in mathematical finance
4 : ``Potential theory'' for viscosity solutions
5 : Loo Laplacian problems
6 : fractional Laplacian

List of papers

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