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 Pucci's extremal operators ${\mathcal{M}}^{\pm }$. It is considered the solution $u^{\epsilon }\left(x\right)$ of $-{\epsilon }^2{\mathcal{M}}^{\pm }\left({\mathrm{\nabla }}^2{u}^{\epsilon }\right)+u^{\epsilon }=0$ in Ω such that $u^{\epsilon }=1$ on Γ. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a domain (not necessarily bounded) and Γ is its boundary. It is also considered $v(x,t)$ the solution of $v_t-{\mathcal{M}}^{\pm }\left({\mathrm{\nabla }}^{2}v\right)=0$ in $\mathrm{\Omega }\times (0,\mathrm{\infty })$, v = 1 on $\mathrm{\Gamma }\times (0,\mathrm{\infty })$ and v = 0 on $\mathrm{\Omega }\times \left\{0\right\}$. In the spirit of their previous works [Berti D, Magnanini R. Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian. Appl Anal. 2019;98(10):1827–1842.; Berti D, Magnanini R. Short-time behavior for game-theoretic p-caloric functions. J Math Pures Appl (9). 2019;(126):249–272.], the authors establish the profiles as ϵ or $t\to 0^{+}$ of the values of $u^{\epsilon }\left(x\right)$ and $v(x,t)$ as well as of those of their q-means on balls touching Γ. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime.

Keywords:

• Pucci operators
• asymptotic analysis
• q-means

AMS Classification 2010:

• Primary 35K55
• 35J60
• Secondary 35K20
• 35J25
• 35B40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper was partially supported by the Gruppo Nazionale di Analisi Matematica, Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). D. B. was supported by the Programa de Excelencia Severo Ochoa SEV-2015-0554 of the Ministerio de Economia y Competitividad.