The exponential transform: A renormalized Riesz potential at critical exponent
2003 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 52, no 3, 527-568 p.Article in journal (Refereed) Published
For an arbitrary domain in R-n we consider the exponential of a suitably normalized Riesz potential of first nonintegrable index. This gives a positive function, with certain monotonicity properties, defined in the complement of the domain, and vanishing on its boundary. In two dimensions this function, polarized into two complex variables, has previously been studied and proved to be useful within operator theory, moment problems and other problems of domain identification, and for proving regularity of free boundaries. Even in the absence of a natural polarization there are noteworthy properties of the higher dimensional exponential transform. For instance we show that this function is superharmonic and that it tends to zero at smooth points of the boundary, with slope bounded from above and below by constants depending solely on the local curvature. The analytic continuation configuration of the exponential transform of a convex polyhedron shows that it is indeed a natural defining function of its boundary. In addition, computations of this transform are carried out in full detail for some domains bounded by lower degree algebraic surfaces. There is also an inner exponential transform, defined inside the domain. If the domain is convex then the ordinary Newtonian potential of it can, for points inside the domain, be interpreted is an arithmetic meanvalue with respect to the solid angle of the squared distance to the boundary. The interior exponential transform is then the corresponding geometric meanvalue.
Place, publisher, year, edition, pages
2003. Vol. 52, no 3, 527-568 p.
Riesz potential, exponential transform, defining function, subharmonic function, analytic continuation, quadrature domain, 2-dimensional l-problem, extremal solutions, moments, domains
IdentifiersURN: urn:nbn:se:kth:diva-22679ISI: 000184218700001OAI: oai:DiVA.org:kth-22679DiVA: diva2:341377
QC 201005252010-08-102010-08-10Bibliographically approved