A SINGULAR INTEGRAL APPROACH TO A TWO PHASE FREE BOUNDARY PROBLEM

作者：Bortz Simon^*; Hofmann Steve

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2016, 144(9): 3959-3973.

DOI：10.1090/proc/13035

• Singular integrals
• layer potentials
• free boundary problems
• Poisson kernels
• VMO

We present an alternative proof of a result of Kenig and Toro (2006), which states that if Omega subset of Rn+1 is a 2-sided NTA domain, with Ahlfors-David regular boundary, and the log of the Poisson kernel associated to Omega as well as the log of the Poisson kernel associated to Omega(ext) are in VMO, then the outer unit normal v is in VMO. Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that partial derivative Omega is uniformly rectifiable, and that partial derivative Omega coincides with the measure theoretic boundary of Omega a.e. with respect to Hausdorff H-n measure.

• 出版日期2016-9