Weighted Orlicz regularity estimates for fully nonlinear elliptic equations with asymptotic convexity

2019 ◽  
Vol 21 (04) ◽  
pp. 1850024 ◽  
Author(s):  
Mikyoung Lee
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Operator ◽  
Orlicz Spaces ◽  
Nonlinear Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Regularity Estimates ◽  
Hessian Estimates ◽  
Asymptotic Convexity

We prove interior Hessian estimates in the setting of weighted Orlicz spaces for viscosity solutions of fully nonlinear, uniformly elliptic equations [Formula: see text] under asymptotic assumptions on the nonlinear operator [Formula: see text] The results are further extended to fully nonlinear, asymptotically elliptic equations.

scholarly journals Borderline regularity for fully nonlinear equations in Dini domains

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Karthik Adimurthi ◽  
Agnid Banerjee
Keyword(s):  
Nonlinear Equations ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Regularity Estimates ◽  
Compactness Arguments ◽  
Gradient Regularity

AbstractIn this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C^{1,\mathrm{Dini}}-domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention that, differently from the approach used there which is based on W^{1,q} estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Luis Caffarelli.

An asymptotic treatment for non-convex fully nonlinear elliptic equations: Global Sobolev and BMO type estimates

2019 ◽  
Vol 21 (07) ◽  
pp. 1850053 ◽  
Author(s):  
J. V. da Silva ◽  
G. C. Ricarte
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
Integrability Condition ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Borderline Case ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Text Type ◽  
Nonlinear Elliptic

In this paper, we establish global Sobolev a priori estimates for [Formula: see text]-viscosity solutions of fully nonlinear elliptic equations as follows: [Formula: see text] by considering minimal integrability condition on the data, i.e. [Formula: see text] for [Formula: see text] and a regular domain [Formula: see text], and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting “fine” regularity estimates from a limiting operator, the Recession profile, associated to [Formula: see text] to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when [Formula: see text]. In such a scenery, we show that solutions admit [Formula: see text] type estimates for their second derivatives.

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