Small diffusion and short-time asymptotics for Pucci operators

Abstract

This paper presents asymptotic formulas in the case of the following two problems for the {\it Pucci's extremal operators} ${\mathcal{M}}^{\pm }$. It is considered the solution $u^{\epsilon }(x)$ of $-{\epsilon }^2{\mathcal{M}}^{\pm }\left({\nabla }^2{u}^{\epsilon }\right)+u^{\epsilon }=0$ in $\Omega$ such that $u^{\epsilon }=1$ on $\Gamma$. Here, $\Omega \subset {\mathbb{R}}^N$ is a domain (not necessarily bounded) and $\Gamma$ is its boundary. It is also considered $v(x,t)$ the solution of $v_t-{\mathcal{M}}^{\pm }\left({\nabla }^{2}v\right)=0$ in $\Omega \times (0,\infty )$, $v=1$ on $\Gamma \times (0,\infty )$ and $v=0$ on $\Omega \times \{0\}$. In the spirit of their previous works, the authors establish the profiles as $\epsilon$ or $t\to 0^{+}$ of the values of $u^{\epsilon }(x)$ and $v(x,t)$ as well as of those of their $q$-means on balls touching $\Gamma$. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime.

15 pages. Minor typos fixed. The article, which is dedicated to the 65th birthday of Sergio Vessella, has been accepted by Applicable Analysis: Special Issue on Partial Differential Equations, Inverse and Ill-Posed Problems, Unique Continuation and Applications. The paper is already available in its online version