Regularity for fully nonlinear elliptic equations and motion by mean curvature

1997 ◽  
pp. 98-133 ◽  
Author(s):  
Lawrence C. Evans
Keyword(s):  
Mean Curvature ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Motion By Mean Curvature

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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