Quasiconvex functions and nonlinear PDEs

Barron, E.; Goebel, R.; Jensen, R.

doi:10.1090/S0002-9947-2013-05760-1

Quasiconvex functions and nonlinear PDEs
HTML articles powered by AMS MathViewer

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

O. Alvarez, J.-M. Lasry, and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9) 76 (1997), no. 3, 265–288 (English, with English and French summaries). MR 1441987, DOI 10.1016/S0021-7824(97)89952-7
Phan Thanh An, Stability of generalized monotone maps with respect to their characterizations, Optimization 55 (2006), no. 3, 289–299. MR 2238417, DOI 10.1080/02331930600705242
Martino Bardi and Federica Dragoni, Convexity and semiconvexity along vector fields, Calc. Var. Partial Differential Equations 42 (2011), no. 3-4, 405–427. MR 2846261, DOI 10.1007/s00526-011-0392-0
G. Barles and Jérôme Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations 26 (2001), no. 11-12, 2323–2337. MR 1876420, DOI 10.1081/PDE-100107824
E. N. Barron and W. Liu, Calculus of variations in $L^\infty$, Appl. Math. Optim. 35 (1997), no. 3, 237–263. MR 1431800, DOI 10.1007/s002459900047
Stephen Boyd and Lieven Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 2004. MR 2061575, DOI 10.1017/CBO9780511804441
R. Buckdahn, P. Cardaliaguet, and M. Quincampoix, A representation formula for the mean curvature motion, SIAM J. Math. Anal. 33 (2001), no. 4, 827–846. MR 1884724, DOI 10.1137/S0036141000380334
Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Wendell H. Fleming and H. Mete Soner, Controlled Markov processes and viscosity solutions, 2nd ed., Stochastic Modelling and Applied Probability, vol. 25, Springer, New York, 2006. MR 2179357
Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1829162, DOI 10.1007/978-1-4612-0195-3
Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 59 (2006), no. 3, 344–407. MR 2200259, DOI 10.1002/cpa.20101
Adam M. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689–1694. MR 2286077, DOI 10.1090/S0002-9939-07-08887-9
Adam M. Oberman, Computing the convex envelope using a nonlinear partial differential equation, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 759–780. MR 2413037, DOI 10.1142/S0218202508002851
Jean-Paul Penot, Glimpses upon quasiconvex analysis, RFMAO 05—Rencontres Franco-Marocaines en Approximation et Optimisation 2005, ESAIM Proc., vol. 20, EDP Sci., Les Ulis, 2007, pp. 170–194 (English, with English and French summaries). MR 2402770, DOI 10.1051/proc:072016
Jean-Paul Penot and Michel Volle, Duality methods for the study of Hamilton-Jacobi equations, CSVAA 2004—control set-valued analysis and applications, ESAIM Proc., vol. 17, EDP Sci., Les Ulis, 2007, pp. 96–142 (English, with English and French summaries). MR 2362694, DOI 10.1051/proc:071708
H. X. Phu and P. T. An, Stable generalization of convex functions, Optimization 38 (1996), no. 4, 309–318. MR 1434252, DOI 10.1080/02331939608844259
A. Popier, Contrôle optimal stochastique en essential supremum, Tech. report, Université de Bretagne Occidentale in Brest, available from http://perso.univ-lemans.fr/˜ apopier/Research.html, 2001.
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
H. Mete Soner, Stochastic representations for nonlinear parabolic PDEs, Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 477–526. MR 2549373, DOI 10.1016/S1874-5717(07)80009-0
H. Mete Soner and Nizar Touzi, The dynamic programming equation for second order stochastic target problems, SIAM J. Control Optim. 48 (2009), no. 4, 2344–2365. MR 2556347, DOI 10.1137/07071130X
H. Mete Soner and Nizar Touzi, Dynamic programming for stochastic target problems and geometric flows, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 201–236. MR 1924400, DOI 10.1007/s100970100039