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.