# PDEs and applications seminar: Concave power solutions of the Dominative p-Laplace equation

- Date: –11:15
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
- Lecturer: Fredrik Hoeg
- Organiser: Matematiska institutionen
- Contact person: Kaj Nyström
- Seminarium

Abstract: In this talk, I will discuss a concavity property of the Dominative p-Laplace operator,

D_p u =∆u + (p − 2)λ_max(D^2u)

where λ_max(D^2u) is the largest eigenvalue of the Hessian matrix D^2u. If u is a viscosity solution of the following problem

− D_p u = 1 in Ω

u = 0 on ∂Ω

then it turns out that the square root of u is concave. Similar types of problems havebeen studied for both the Laplace equation and the p-Laplace equation.I will go through what has been done for these similar equations and tryto explain why the square root is concave. The main technical problem is solved by the *Theorem of sums* or *Ishii’s Lemma*. I will explain thistheorem and show how one can apply it in this setting.