Boundary regularity for the exponential transform in higher dimension

Problem's Description

The exponential transform of a domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 2$) is defined by $$ E(x)= \exp [-\frac{2}{|S^{n-1}|}\int_\Omega \frac{dy}{|x-y|^{n}}], \quad x\in\mathbb{R}^n\setminus\overline{\Omega}, $$ where $|S^{n-1}|$ denotes the area of the unit sphere. The exponential transform is a positive function in the exterior of $\Omega$, with zero boundary values at all reasonably regular points of $\partial\Omega$. The problem is to study in more detail the relationship between the regularity of $\partial\Omega$ and the boundary behavior of $E(x)$. More specifically we conjecture that if $\partial\Omega$ is smooth real analytic then $E(x)$ has a real analytic continuation across $\partial\Omega$.

This has been shown to be the case in two dimensions. In fact, in this case the exponential transform can be polarized into a function of two complex variables, for which a rich theory has been developed. In particular this theory gives a complete equivalence between analyticity of the boundary and analytic continuation properties of the exponential transform. And this can be used in the other direction, namely to prove analyticity of the boundary provided it is known in advance that the exponential transform has an analytic continuation, which occurs in the context of certain obstacle problems.

In dimension $n\geq 3$ there are some limited progress, ensuring analytic continuation from convex cavities. The proof of this uses rather special tools, and in general the continuation problem remains open in dimension $n\geq 3$.

1. Article The exponential transform: a renormalized Riesz potential at critical exponent

   • Björn Gustafsson,
   • Mihai Putinar

   Indiana University Mathematics Journal, 52 (2003), 527-568, 2003

2. Article Analytic continuation of the exponential transform from convex cavities

   • Björn Gustafsson,
   • Mihai Putinar

   Journal of Mathematical Analysis and Applications 328 (2007), 995-1006., 2007