José M. Urbano - On free boundary problems with varying singularities

Abstract: We consider free boundary models arising from minimising energy functionals with varying singularities. For measurable and bounded singular exponents, we establish the $C^{1,\alpha}$-regularity of local minima. Sharp regularity estimates along the free boundary are also obtained under a mild continuity assumption on the exponent. These estimates vary point-by-point along the interface, leading to multiple free boundary geometries.