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While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by: $$M_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{negative eigenvalues of }N\right),\text{ and}$$ $$M_\Lambda^+(N):=-M_\Lambda^-(-N),$$ where $N\in\text{Sym}_{n\times n}$ and $\Lambda\geq 1$.

Then the author claims that the problem $$M_\Lambda^-(D^2u)\leq 0\leq M_\Lambda^+(D^2u)$$ in viscosity sense includes all $C^2$ solutions to uniformly elliptic equations of the form $tr(A(x)D^2u)=0$ where $I\leq A(x)\leq\Lambda I$. Since there is no citation on this, it seems this is well-known result in the field, but I am new to this and I want to know about its motivation, history, etc. I did some article search but I could not find a good reference book or paper that introduces the concept of Pucci extremal operator (of course this may be due to my lacking of searching skill). I will appreciate if anyone would explain the concept or give me some good reference on it. Thank you in advance.

[1] Mooney, Connor, A proof of the Krylov-Safonov theorem without localization, Commun. Partial Differ. Equations 44, No. 8, 681-690 (2019). ZBL1426.35124.

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    $\begingroup$ Incidentally, Connor is active on MathOverflow. If you send him an email pointing out your question, he may drop by and give you an answer. $\endgroup$ May 6 at 19:54
  • $\begingroup$ @WillieWong That's nice, I may try, thank you for the advice! $\endgroup$ May 9 at 23:04
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The original references are the works of Pucci [2] and [3] which, however, are written in Italian. A perhaps more accessible introduction to these kind of operators is found in the monograph [1], §2.2, pp. 14-17, by Caffarelli and Cabré: in the latter reference, particularly relevant to your questions are lemma 2.12, §2.2 pp. 15-16 and the final part of remark 2, §2.2 p. 16.

References

[1] Luis A. Caffarelli, Xavier Cabré, Fully nonlinear elliptic equations, (English) Colloquium Publications. 43. Providence, RI: American Mathematical Society (AMS). v, 104 p. (1995), DOI: 10.1090/coll/043, MR1351007, Zbl 0834.35002.

[2] Carlo Pucci, "Operatori ellittici estremanti", (in Italian) Annali di Matematica Pura e Applicata (4) 72 (1966), 141--170, DOI: BF02414332, MR0208150, Zbl 0154.12402.

[3] Carlo Pucci, "Operatori massimanti" (in Italian) Monge-Ampère equations and related topics. Proceedings of a Seminar held in Firenze, September - October 1980, Roma: Istituto Nazionale di Alta Matematica ”Francesco Severi”, 169-176 (1982), Zbl 0519.35076.

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  • $\begingroup$ Thank you for your help. This was very useful. $\endgroup$ May 15 at 11:57
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    $\begingroup$ @Jin-geonAn-Lacroix you are welcome: I'm glad of having been of some help. $\endgroup$ May 15 at 12:35

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