Some comparison results and a partial bang-bang property for two-phases problems in balls

In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $a$ under $L^{\mathrm{\infty }}$ and $L^1$ constraints, this function $a$ appearing in a diffusive term of the form $-\mathrm{\nabla }\cdot (a\mathrm{\nabla })$ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $L^{\mathrm{\infty }}$ constraints defining the admissible class. This is done using an oscillatory method.