ABP maximum principles for fully nonlinear integro-differential equations with unbounded inhomogeneous terms
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ORIGINAL PAPER
ABP maximum principles for fully nonlinear integrodifferential equations with unbounded inhomogeneous terms Shuhei Kitano1 Ó Springer Nature Switzerland AG 2020
Abstract Aleksandrov–Bakelman–Pucci maximum principles are studied for a class of fully nonlinear integro-differential equations of order r 2 ½2 e0 ; 2Þ, where e0 is a small constant depending only on given parameters. The goal of this paper is to improve an estimate of Guillen and Schwab (Arch Ration Mech Anal 206(1):111–157, 2012) in order to avoid the dependence on L1 norm of the inhomogeneous term. Mathematics Subject Classification 35R09 47G20
1 Introduction In this paper, we study the fully nonlinear nonlocal equation of the form: M uðxÞ f ðxÞ in B1 ;
ð1Þ
in Rn n B1 ;
uðxÞ 0 where by setting
dðu; x; yÞ :¼ uðx þ yÞ þ uðx yÞ 2uðxÞ; we define the minimal fractional Pucci operator: (
M uðxÞ :¼
inf
k TrðAÞ and O A KId
ð2 rÞ
Z dðu; x; yÞ Rn
yT Ay jyjnþrþ2
) dy :
This article is part of the topical collection ‘‘Viscosity solutions – Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize’’ edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa and Senjo Shimizu. & Shuhei Kitano [email protected] 1
Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan SN Partial Differential Equations and Applications
16 Page 2 of 11
SN Partial Differ. Equ. Appl. (2020)1:16
Throughout this paper, we suppose that n 2;
0\k nK;
n
and define Br :¼ fy 2 R : jyj\rg. Furthermore, Qr Rn denotes the open cube with its center at 0 and its side-length r. We also set Br ðxÞ :¼ x þ Br and Qr ðxÞ :¼ x þ Qr . We will also use the maximal Pucci operator defined by ( ) Z yT Ay þ M uðxÞ :¼ sup ð2 rÞ dðu; x; yÞ nþrþ2 dy : jyj Rn k TrðAÞ and O A KId The main purpose of this paper is to show the Aleksandrov–Bakelman–Pucci (ABP for short) maximum principle depending only on Ln norm of f þ :¼ maxff ; 0g. Theorem 1.1 There exist constants C^ [ 0 and e0 2 ð0; 1Þ depending only on n; k and K such that if u 2 L1 ðRn Þ \ LSCðRn Þ is a viscosity supersolution of (1) with r 2 ½2 e0 ; 2Þ and f 2 CðB1 Þ; then it follows that ^ þk n inf u Ckf L ðfu 0gÞ : B1
ð2Þ
The ABP maximum principle plays a fundamental role in the regularity theory of fully nonlinear equations (see [1, 2] for instance). The ABP maximum principle for the second order equation presents a bound for the infimum of supersolutions by Ln norm of inhomogeneous terms. In the case of nonlocal equations, the ABP maximum principle is investigated in [3, 5, 7] and references therein. More precisely, for viscosity solutions, Caffarelli and Silvestre prove that the maximum is estimated from above by Riemann sums of f instead of their Ln norms in [3]. A more quantitative version of the ABP maximum principle is established by Guillen and Schwab in [5]. For an application of results in [5], W r;e -estimate is obtained by Yu [8], which was inspired by earlier works by [4, 6]. The ABP
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