In a celebrated three-pages long paper in 1979, Aronson and Bénilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since its publication, the theory of porous medium flow has expanded relentlessly with applications including modeling of tissue growth and cancer development. In this talk, based on [1], we first motivate the use of the porous media equation to study the phenomenon of the tumor growth and then we present an extension of the Aronson-Bénilan estimate in 

L∞ holding for any fields of pressure.

[1] G. Bevilacqua, B. Perthame, M. Schmidtchen; The Aronson-Bénilan estimate in Lebesgue spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire (2022), published online first, arxiv.org/pdf/2007.15267.pdf.