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**Regularity properties of solutions to the basic problem in the calculus of variations.**
*(English)*
Zbl 0563.49009

This paper concerns the classical problem in the calculus of variations
\[
(1)\quad \min \{\int^{b}_{a}L(t,x(t),x'(t))dt:\quad x(a)=A,\quad x(b)=B\}
\]
where the minimum is taken over all absolutely continuous arcs \(x:[a,b]\to {\mathbb{R}}^ n\). It was L. Tonelli (see ”Opere Scelte”, Cremonese, Rome (1961) and ”Fondamenti di Calcolo delle Variazioni”, Zanichelli, Bologna (1921-1923)) who attacked successfully this problem by obtaining the first general existence and regularity results. In this paper, problem (1) is studied under hypotheses considerably weaker than the classical ones of Tonelli. In fact, the integrand \(L:[a,b]\times {\mathbb{R}}^ n\times {\mathbb{R}}^ n\to {\mathbb{R}}\) is assumed to satisfy the following \(hypotheses:\)

(H\({}_ 1)\) L(t,x,v) is locally bounded, measurable in t, and convex in \(v;\)

(H\({}_ 2)\) L(t,x,v) is locally Lipschitz in (x,y) uniformly in \(t;\)

(H\({}_ 3)\) L(t,x,v)\(\geq \theta (| v|)-\alpha | x|\) where \(\alpha\) is a constant and \(\theta\) is a convex function with \(\lim_{r\to +\infty}\theta (r)/r=+\infty.\)

Under these conditions, the authors prove that a solution of (1) exists and is locally Lipschitz on an open set \(\Omega\) with \(meas([a,b]- \Omega)=0\). Moreover, if L(t,x,v) is \(C^ 1\) in (x,v), then the Euler- Lagrange equation \(\frac{d}{dt}(L_ v(t,x,x'))=L_ x(t,x,x')\) holds in \(\Omega\). It is known (see F. H. Clarke and R. B. Vinter [Appl. Math. Optim. 12, 73-79 (1984)] and J. M. Ball and V. J. Mizel [Arch. Ration. Mech. Anal. (to appear)]) that in general the set \(\Omega\) does not coincide with the whole interval [a,b]. In Section 3, many cases in which \(\Omega =[a,b]\) are examined. An interesting case is when the integrand L does not depend on t. Methods and techniques of nonsmooth analysis (see F. H. Clarke [Wiley Interscience, New York (1983)]) play an essential role in the proofs.

(H\({}_ 1)\) L(t,x,v) is locally bounded, measurable in t, and convex in \(v;\)

(H\({}_ 2)\) L(t,x,v) is locally Lipschitz in (x,y) uniformly in \(t;\)

(H\({}_ 3)\) L(t,x,v)\(\geq \theta (| v|)-\alpha | x|\) where \(\alpha\) is a constant and \(\theta\) is a convex function with \(\lim_{r\to +\infty}\theta (r)/r=+\infty.\)

Under these conditions, the authors prove that a solution of (1) exists and is locally Lipschitz on an open set \(\Omega\) with \(meas([a,b]- \Omega)=0\). Moreover, if L(t,x,v) is \(C^ 1\) in (x,v), then the Euler- Lagrange equation \(\frac{d}{dt}(L_ v(t,x,x'))=L_ x(t,x,x')\) holds in \(\Omega\). It is known (see F. H. Clarke and R. B. Vinter [Appl. Math. Optim. 12, 73-79 (1984)] and J. M. Ball and V. J. Mizel [Arch. Ration. Mech. Anal. (to appear)]) that in general the set \(\Omega\) does not coincide with the whole interval [a,b]. In Section 3, many cases in which \(\Omega =[a,b]\) are examined. An interesting case is when the integrand L does not depend on t. Methods and techniques of nonsmooth analysis (see F. H. Clarke [Wiley Interscience, New York (1983)]) play an essential role in the proofs.

Reviewer: G.Buttazzo

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

49K05 | Optimality conditions for free problems in one independent variable |

49K15 | Optimality conditions for problems involving ordinary differential equations |

49J05 | Existence theories for free problems in one independent variable |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

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\textit{F. H. Clarke} and \textit{R. B. Vinter}, Trans. Am. Math. Soc. 289, 73--98 (1985; Zbl 0563.49009)

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### References:

[1] | Serge Bernstein, Sur les équations du calcul des variations, Ann. Sci. École Norm. Sup. (3) 29 (1912), 431 – 485 (French). · JFM 43.0460.01 |

[2] | Gilbert A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. · Zbl 0036.34401 |

[3] | Lamberto Cesari, Optimization — theory and applications, Applications of Mathematics (New York), vol. 17, Springer-Verlag, New York, 1983. Problems with ordinary differential equations. · Zbl 0506.49001 |

[4] | Frank H. Clarke, The Euler-Lagrange differential inclusion, J. Differential Equations 19 (1975), no. 1, 80 – 90. · Zbl 0323.49021 |

[5] | Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247 – 262. · Zbl 0307.26012 |

[6] | Frank H. Clarke, Generalized gradients of Lipschitz functionals, Adv. in Math. 40 (1981), no. 1, 52 – 67. · Zbl 0463.49017 |

[7] | Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. · Zbl 0582.49001 |

[8] | Frank H. Clarke and R. B. Vinter, Existence and regularity in the small in the calculus of variations, J. Differential Equations 59 (1985), no. 3, 336 – 354. · Zbl 0727.49003 |

[9] | Frank H. Clarke and R. B. Vinter, On the conditions under which the Euler equation or the maximum principle hold, Appl. Math. Optim. 12 (1984), no. 1, 73 – 79. · Zbl 0559.49012 |

[10] | A. D. Ioffe and V. M. Tihomirov, Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979 (German). Translated from the Russian by Bernd Luderer. A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. |

[11] | V. Lakshmikantham and S. Leela, Differential and integral inequalities, Academic Press, New York, 1969. · Zbl 0177.12403 |

[12] | Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0142.38701 |

[13] | R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. · Zbl 0932.90001 |

[14] | L. Tonelli, Sur une méthode directe du calcul des variations, Rend. Circ. Mat. Palermo 39 (1915), 233-264, also appears in Opere scelte, Vol. 2, Cremonese, Rome, 1961, pp. 289-333. · JFM 45.0615.02 |

[15] | -, Fondamenti di calcolo delle variazioni, Vols. 1, 2, Zanichelli, Bologna, 1921, 1923. |

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