2024

Kurkinen, Tapio

This thesis studies a nonlinear parabolic equation that generalizes both the usual ρ-parabolic equation and the normalized ρ-parabolic equation arising from stochastic game theory. Apart from special cases, the equation is in non-divergence form and we use the concept of viscosity solutions. The articles [A] and [B] focus on Harnack’s inequalities. We prove that all non-negative viscosity solutions satisfy a parabolic Harnack’s inequality with intrinsic scaling. Intrinsic scaling here means that the needed waiting time between time slices depends on the value of the solution. We also show that for a singular range, this waiting time is not needed and a so-called elliptic Harnack’s inequality, where we get the estimate on both sides without the waiting time, holds. Exponent ranges for both inequalities are optimal as shown by counterexamples. We also show that for very singular exponents, all solutions vanish in finite time. The article [C] examines boundary regularity for this equation. We prove that there exists a barrier family at a boundary point if and only if that point is regular. We use this characterization to prove geometric conditions that also guarantee regularity. These include an exterior ball condition and a result that shows that all locally time-wise earliest points are regular.