Research
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 L^{p}viscosity solutions
for fully nonlinear uniformly elliptic PDEs was introduced in 1996 by CaffarelliCrandallKocanŚwięch
to study L^{p} regularity theory for PDEs of nondivergent type.
Key words of my recent (and/or future) interests are as follows:

1 : L^{p} 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 : L^{oo} Laplacian problems


6 : fractional Laplacian


List of papers


Vitae

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