scholarly journals On the existence of smooth solutions for fully nonlinear elliptic equations with measurable “coefficients” without convexity assumptions

2012 ◽  
Vol 19 (2) ◽  
pp. 119-146
Author(s):  
Nicolai V. Krylov
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Smooth Solutions ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Measurable Coefficients ◽  
Nonlinear Elliptic ◽  
Convexity Assumptions

scholarly journals On the existence of Wp2 solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

2017 ◽  
Vol 19 (06) ◽  
pp. 1750009 ◽  
Author(s):  
N. V. Krylov
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Equations ◽  
Second Order ◽  
Fully Nonlinear Elliptic Equations ◽  
Sobolev Classes ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Convexity Assumptions ◽  
Second Order Equations

We establish the existence of solutions of fully nonlinear elliptic second-order equations like [Formula: see text] in smooth domains without requiring [Formula: see text] to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of [Formula: see text] at points at which [Formula: see text], where [Formula: see text] is any given constant. For large [Formula: see text] some kind of relaxed convexity assumption with respect to [Formula: see text] mixed with a VMO condition with respect to [Formula: see text] are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on [Formula: see text], apart from ellipticity, but of a “cut-off” version of the equation [Formula: see text].

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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