We study the Dirichlet problem associated to the equation for self-similar surfaces under mean curvature flow for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries.
"Construction of complete embedded self-similar surfaces under mean curvature flow. Part II." Adv. Differential Equations 15 (5/6) 503 - 530, May/June 2010.