Convex viscosity solutions and state constraints

• Published in: Journal de mathématiques pures et appliquées Vol. 76; no. 3; pp. 265 - 288
• Main Authors: Alvarez, O., Lasry, J.-M., Lions, P.-L.
• Format: Journal Article
• Language: English
• Published: Paris Elsevier SAS 01.03.1997; Elsevier
• Subjects: Calculus of variations and optimal control; conjugate function; convex envelope; Convexity; elliptic equation; Exact sciences and technology; fully nonlinear equation; Mathematical analysis; Mathematics; partial convexity; Partial differential equations; Sciences and techniques of general use; state constraints boundary condition; viscosity solution; 35 J 65; convex envelope; partial convexity; 26 B 25; 35 B 99; viscosity solution; conjugate function; Convexity; fully nonlinear equation; state constraints boundary condition; 49 L 25; 35 J 60; elliptic equation; Non linear equation; Convex hull; Elliptic equation; Viscosity solution; Coercive function; Comparison principle; Convex solution; Boundary condition; Supersolution; State constraint
• ISSN: 0021-7824
• DOI: 10.1016/S0021-7824(97)89952-7

We establish the convexity of a viscosity solution of some general second order fully nonlinear elliptic equation with state constraints boundary conditions. Our method combines a comparison principle with the observation that, under suitable assumptions, the convex envelope of the solution is a supersolution. This property relies on the characterization of the viscosity subject of the convex envelope of a lower semicontinuous coercive function. The equation solved by the conjugate of a convex solution as well as partial convexity are topics we also discuss. On établit la convexité de solutions de viscosité d'équations elliptiques du seconda ordre, complètement non linéaires, avec contraintes d'état au bord. Notre méthode repose sur le principe de comparaison et sur l'observation que, sous des hypothèses convenables, l'enveloppe convexe d'une solution est une sursolution. Cette dernière propriété utilise une caractérisation du sous différentiel d'ordre deux (au sens de la théorie des solutions de viscosité) de l'enveloppe convexe d'une fonction coercive, semicontinue inférieurement. Des questions connexes sont traitées, comme la convexité partielle de solutions ou l'équation que doit vérifier la conjuguée d'une fonction convexe.