Geometric Approach to Nonvariational Singular Elliptic Equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms, as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ - 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases.We establish existence as well as sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and we obtain fine geometric-measure properties of the free boundary ${mathfrak{F} = partial{u > 0}}$ . In particular, we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and the ${mathcal{H}^{n-1}}$ almost-everywhere weak differentiability property of ${mathfrak{F}}$ .