Henrik Shahgholian, Georg S. Weiss
For the two-phase membrane problemwhere and we prove in two dimensions that the free boundary is in a neighborhood of each ?branch point? the union of two -graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection.
In higher dimensions we show that the free boundary has finite -dimensional Hausdorff measure.
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