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Quasiconvex functions and nonlinear PDEs

Authors: E. N. Barron, R. Goebel and R. R. Jensen
Journal: Trans. Amer. Math. Soc. 365 (2013), 4229-4255
MSC (2010): Primary 35D40, 35B51, 35J60, 52A41, 53A10
Published electronically: March 11, 2013
MathSciNet review: 3055695
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Abstract: A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator $ L_0(Du,D^2u)=\linebreak\operatorname {min}\{v\cdot D^2u\,v^T\;\vert\;\vert v\vert=1,\vert v\cdot Du\vert=0\}.$ In two dimensions this is the mean curvature operator, and in any dimension $ L_0(Du,D^2u)/\vert Du\vert$ is the first principal curvature of the surface $ S=u^{-1}(c).$ Our main results include a comparison principle for $ L_0(Du,D^2u)=g$ when $ g \geq C_g>0$ and a comparison principle for quasiconvex solutions of $ L_0(Du,D^2u)=0.$ A more regular version of $ L_0$ is introduced, namely $ L_\alpha (Du,D^2u)= \operatorname {min}\{v\cdot D^2u\,v^T\;\vert\;\vert v\vert=1,\vert v\cdot Du\vert \leq \alpha \}$, which characterizes functions which remain quasiconvex under small linear perturbations. A comparison principle is proved for $ L_\alpha $. A representation result using stochastic control is also given, and we consider the obstacle problems for $ L_0$ and $ L_\alpha $.

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Additional Information

E. N. Barron
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660

R. Goebel
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660

R. R. Jensen
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660

Keywords: Quasiconvex, robustly quasiconvex, principal curvature, comparison principles
Received by editor(s): February 16, 2011
Received by editor(s) in revised form: November 23, 2011
Published electronically: March 11, 2013
Additional Notes: The authors were supported by grant DMS-1008602 from the National Science Foundation
Article copyright: © Copyright 2013 American Mathematical Society

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