zoom : 회의 ID 891 0030 7121


초록 : The analysis of fractals has been studied extensively in both analysis and probability approaches. In this thesis, we construct the non-linear elliptic equation involving second order terms on fractal spaces, and our main objective is to exhibit the regularity of their solutions by using an analytic argument. Since a calculus on fractals is not available, our approach is based on the graph approximation argument to construct Dirichlet forms. The central concept is in finding suitable cut-off functions and weighted inequalities, which can be obtained by using the special geometric properties of the fractal domain.

Another topic in this thesis is the homogenization theory for fully non-linear parabolic equations. In particular, we treat the case with different scales of the oscillating variables. 

The interesting point is that the homogenization occurs separately for time and space due to a mismatch in the scale of time and space fast variables. In addition, this phenomenon causes different order of convergence rates.