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Eero Ruosteenoja

Publications and source records attributed to Eero Ruosteenoja.

8 recordsLinked to original sources
arXiv2023

Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian

We prove a local Hölder estimate with an exponent $0<δ<\frac 12$ for solutions of the dynamic programming principle $$u^\varepsilon (x) =\sum_{j=1}^n α_j\inf_{\dim(S)=j}\sup_{\substack{v\in S\\ |v|=1}}\frac{ u^\varepsilon (x + \varepsilon v) + u^\varepsilon (x - \varepsilon v)}{2}.$$ The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $$\sum_{i=1}^n α_iλ_i(D^2u)=0,$$ where $λ_1(D^2 u)\leq\cdots\leq λ_n(D^2 u)$ are the eigenvalues of the Hessian.

math.AP↗
arXiv2020

A control problem related to the parabolic dominative $p$-Laplace equation

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 $D^2 u$.

math.AP↗
arXiv2019

Gradient regularity for a singular parabolic equation in non-divergence form

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $$\partial_t u-|Du|^γΔ_p^N u=f,$$ where $-1<γ<0$, $1<p<\infty$, and $f$ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

math.AP↗
arXiv2018

Remarks on regularity for $p$-Laplacian type equations in non-divergence form

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^γ\left(Δu+(p-2)Δ_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ Ω.$$ We investigate local $C^{1,α}$ regularity of viscosity solutions in the full range $γ>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $γ$ is close to 0.

math.AP↗
arXiv2018

Local regularity for time-dependent tug-of-war games with varying probabilities

We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain Hölder and Harnack estimates. The games have a connection to the normalized $p(x,t)$-parabolic equation $(n+p(x,t))u_t=Δu+(p(x,t)-2) Δ_{\infty}^N u$.

math.AP↗
arXiv2016

$C^{1,α}$ regularity for the normalized $p$-Poisson problem

We consider the normalized $p$-Poisson problem $$-Δ^N_p u=f \qquad \text{in}\quad Ω.$$ The normalized $p$-Laplacian $Δ_p^{N}u:=|D u|^{2-p}Δ_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,α}_{loc}$ regularity with nearly optimal $α$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory.

math.AP↗