Small diffusion and short-time asymptotics for Pucci operators. Issue 10 (3rd July 2022)

• Record Type: Journal Article
• Authors: Berti, Diego; Magnanini, Rolando
• Is Part Of: Applicable analysis. Volume 101:Issue 10(2022)
• Page Start: 3716
• Page End: 3732
• Publication Date: 2022-07-03
• Subjects: Pucci operators -- asymptotic analysis -- q-means; Primary 35K55 -- 35J60 -- Secondary 35K20 -- 35J25 -- 35B40; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1750602
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M ± . It is considered the solution u ε ( x ) of − ε 2 M ± ∇ 2 u ε + u ε = 0 in Ω such that u ε = 1 on Γ. Here, Ω ⊂ 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 − M ± ∇ 2 v = 0 in Ω × ( 0, ∞ ), v = 1 on Γ × ( 0, ∞ ) and v = 0 on Ω × { 0 } . 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 → 0 + of the values of u ε ( x ) 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.