Quasiconvex functions and nonlinear PDEs
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- by E. N. Barron, R. Goebel and R. R. Jensen PDF
- Trans. Amer. Math. Soc. 365 (2013), 4229-4255 Request permission
Abstract:
A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator $L_0(Du,D^2u)= \operatorname {min}\{v\cdot D^2u v^T\;|\;|v|=1,|v\cdot Du|=0\}.$ In two dimensions this is the mean curvature operator, and in any dimension $L_0(Du,D^2u)/|Du|$ 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 $L_\alpha (Du,D^2u)= \operatorname {min}\{v\cdot D^2u v^T\;|\;|v|=1,|v\cdot Du| \leq \alpha \}$, 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 $L_\alpha$.References
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Additional Information
- E. N. Barron
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660
- MR Author ID: 31685
- Email: ebarron@luc.edu
- R. Goebel
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660
- Email: rgoebel1@luc.edu
- R. R. Jensen
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660
- MR Author ID: 205502
- Email: rjensen@luc.edu
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4229-4255
- MSC (2010): Primary 35D40, 35B51, 35J60, 52A41, 53A10
- DOI: https://doi.org/10.1090/S0002-9947-2013-05760-1
- MathSciNet review: 3055695