Special Session 10: Recent Developments in Regularity Theory for PDEs

Dirichlet problem and regular boundary points for elliptic equations in non-divergence and double divergence form
Dong-ha Kim
Research Institute of Mathematics, Seoul National University.
Korea
Co-Author(s):    Hongjie Dong, Seick Kim
Abstract:
We consider the Dirichlet problem for second-order elliptic equations in non-divergence form $L$ and double divergence form $L^*$. We introduce a potential theory framework for these operators, including Perron`s method, capacity theory, and Wiener`s criterion. We establish the equivalence between regular boundary points for these operators $L$ and $L^*$ with those for the Laplace operator, assuming that the principal coefficients satisfy the Dini mean oscillation condition.