scholarly journals Convex solutions of boundary value problem arising from Monge-Ampère equations

2006 ◽  
Vol 16 (3) ◽  
pp. 705-720 ◽  
Author(s):  
Shouchuan Hu ◽  
◽  
Haiyan Wang ◽  
2007 ◽  
Vol 76 (2) ◽  
pp. 215-218 ◽  
Author(s):  
John Urbas

We construct a counterexample to a theorem of Jon Wolfson concerning the existence of globally smooth solutions of the second boundary value problem for Monge-Ampère equations in two dimensions, or equivalently, on the existence of minimal Lagrangian diffeomorphisms between simply connected domains in R2.


2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Baoqiang Yan ◽  
Meng Zhang

This paper considers the following boundary value problem:((-u'(t))n)'=ntn-1f(u(t)),  0<t<1,  u'(0)=0,  u(1)=0, wheren>1is odd. We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.


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