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Keywords = Monge–Ampère operator

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32 pages, 416 KB  
Article
Averaging Effects and Their Applications to Fractional Elliptic and Parabolic Equations
by Wenxiong Chen and Yahong Guo
Fractal Fract. 2026, 10(6), 360; https://doi.org/10.3390/fractalfract10060360 - 26 May 2026
Viewed by 365
Abstract
The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove [...] Read more.
The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove various forms of the averaging effects for both fractional elliptic and parabolic equations, from the simplest one to the one under very relaxed conditions, including versions for antisymmetric functions. We then present examples to illustrate how to apply these effects to obtain radial symmetry and monotonicity for solutions in a unit ball and in a half space. In addition, we derive averaging effects for fractional Monge–Ampère operators and for fractional p-Laplacians, which will be potentially applied to obtain qualitative properties for solutions to equations involving these operators. Compared with the traditional approaches, methods based on the averaging effect require substantially weaker regularity assumptions and can even accommodate unbounded solutions. Full article
(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)
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19 pages, 388 KB  
Article
The Maximal Regularity of Nonlocal Parabolic Monge–Ampère Equations and Its Monotonicity in the Whole Space
by Xingyu Liu
Axioms 2025, 14(7), 491; https://doi.org/10.3390/axioms14070491 - 24 Jun 2025
Cited by 2 | Viewed by 1265
Abstract
The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to [...] Read more.
The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to seek the maximal regularity of the parabolic Monge–Ampère equations by following an approach similar to that used for finding the maximal regularity of the parabolic fractional Laplacian operator. The maximal regularity of nonlocal parabolic Monge–Ampère equations guarantees the existence of solutions in the whole space. Based on these conditions, a modified sliding method, an enhancement of the moving planes method, is employed to establish the monotonicity property of the solutions for the nonlocal parabolic Monge–Ampère equations in the whole space. Full article
(This article belongs to the Special Issue Current Perspectives in Fractional Calculus: Theory, Methods, and Applications)
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14 pages, 370 KB  
Article
Plurisubharmonic Interpolation and Plurisubharmonic Geodesics
by Alexander Rashkovskii
Axioms 2023, 12(7), 671; https://doi.org/10.3390/axioms12070671 - 7 Jul 2023
Cited by 2 | Viewed by 2369
Abstract
We give a short survey on plurisubharmonic interpolation, with a focus on the possibility of connecting two given plurisubharmonic functions by plurisubharmonic geodesics. Full article
(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))
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