Regularity properties of solutions to the basic problem in the calculus of variations.

*(English)*Zbl 0563.49009This 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:

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