Bousquet, Pierre; Clarke, Francis Local Lipschitz continuity of solutions to a problem in the calculus of variations. (English) Zbl 1141.49033 J. Differ. Equations 243, No. 2, 489-503 (2007). The article deals with the problem of regularity of minimizers of the variational functional \[ I(u)= \int_{\Omega} F(Du) + G(x,u) \,dx \] over the function \(u \in W^{1,1}\) that assumme a given boundary values \(\phi\) on \(\partial\Omega\). It is well known that the minimizers are Lipschitz if the boundary data satisfies the so-called boundary slope condition. In the case \(G=0\) Clark introduced the lower (and also the upper) bounded slope condition, which is less restrictive. Under this condition and by means a new interesting methods the authors show the Lipschitz continuous regularity when \(G \neq 0\) is locally Lipschitz. Moreover the continuity of the solution at the boundary is discussed. Reviewer: Elvira Mascolo (Firenze) Cited in 12 Documents MSC: 49N60 Regularity of solutions in optimal control Keywords:regularity; boundary slope condition; bounded slope condition PDFBibTeX XMLCite \textit{P. Bousquet} and \textit{F. Clarke}, J. Differ. Equations 243, No. 2, 489--503 (2007; Zbl 1141.49033) Full Text: DOI References: [1] Bousquet, P., On the lower bounded slope condition, J. Convex Anal., 14, 119-136 (2007) · Zbl 1132.49031 [2] Cellina, A., On the bounded slope condition and the validity of the Euler Lagrange equation, SIAM J. Control Optim., 40, 1270-1279 (2001) · Zbl 1013.49015 [3] Clarke, F., Continuity of solutions to a basic problem in the calculus of variations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4, 511-530 (2005) · Zbl 1127.49001 [4] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 [5] Giusti, E., Direct Methods in the Calculus of Variations (2003), World Scientific: World Scientific Singapore · Zbl 1028.49001 [6] Hartman, P., On the bounded slope condition, Pacific J. Math., 18, 495-511 (1966) · Zbl 0149.32001 [7] Hartman, P.; Stampacchia, G., On some nonlinear elliptic differential-functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102 [8] Mariconda, C.; Treu, G., Gradient maximum principle for minima, J. Optim. Theory Appl., 112, 167-186 (2002) · Zbl 1019.49029 [9] Miranda, M., Un teorema di esistenza e unicità per il problema dell’area minima in \(n\) variabili, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 19, 233-249 (1965) · Zbl 0137.08201 [10] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0142.38701 [11] Stampacchia, G., On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math., 16, 383-421 (1963) · Zbl 0138.36903 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.