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^\infty $ and $ L^1 $ constraints, this function $ a $ appearing in a diffusive term of the form $ -{{\nabla}} \cdot(a{{\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^\infty $ constraints defining the admissible class. This is done using an oscillatory method.