Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions

Abstract

It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.

Classical and Stochastic Laplacian Growth

“…Even if Ω(0) has a piecewise smooth boundary, but has a corner with the interior angle smaller than π/2, then it turns out [311,312,497,503,202] that ∂Ω(t) stays at the corner for some positive amount of time. Then Ω(s) ⊂ Ω(t) for all 0 ≤ s < t, but it is not always true that Ω(0) ⊂ Ω(t).…”

“…Compare also [100], [154]. Corner development in higher dimensions has been studied by S. Gardiner and T. Sjödin [202]. Under certain assumptions, like strong starlikeness of the initial domain, it is possible even to prove the existence globally in time in the well-posed time direction (injection) [248].…”