arXiv subjects
Publications and source records attributed to Fredrik Arbo Høeg.
We study Hölder continuity for solutions of the $p$-Laplace equation. This is established through a method involving an ordinary differential inequality, in contrast to the classical proof of the De Giorgi-Nash-Moser Theorem which uses iteration of an inequality through concentric balls.
In this paper, we study properties of solutions of the Dominative $p$-Laplace equation with homogeneous Dirichlet boundary conditions in a bounded convex domain $Ω$. For the equation $-\mathcal{D}_p u= 1$, we show that $\sqrt{u}$ is concave, and for the eigenvalue problem $\mathcal{D}_p u + λu=0$, we show that $\log {u}$ is concave.
We show that value functions of a certain time-dependent control problem in $Ω\times (0,T)$, with a continuous payoff $F$ on the parabolic boundary, converge uniformly to the viscosity solution of the parabolic dominative $p$-Laplace equation $$2(n+p)u_t=Δu+(p-2)λ_n(D^2 u),$$ with the boundary data $F$. Here $2\leq p< \infty$, and $λ_n(D^2 u)$ is the largest eigenvalue of the Hessian $D^2 u$.
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.