On the minimum problem for a class of non-coercive functionals
The authors study problems of the calculus of variations of the form Minimize \(\{I(v) \mid v \in {\mathcal A}_C\}\), where \(C\) is a given rectifiable curve, \({\mathcal A}_C\) is the set of abolutely continuous representatives of \(C\), \(I(v) = \int^T_0 (g(v(t)) + f(v'(t)))dt\), \(f \in C^1 (\mathbb{R}^n)\) and \(g \in C (\mathbb{R}^n)\) are nonnegative functions and \(f\) is strictly convex. If \(U_C\) is the reduced arc length parametrization of the curve \(C\), then \({\mathcal A}_C = \{U_C\circ\varphi \mid \varphi \in \Phi\}\) with \(\Phi = \{\varphi \in W^{1,1} (0,T) |\varphi (0) = 0\), \( \varphi (T) = T\), \(\varphi' (t) \geq 0\}\). By setting \(J(u, \varphi) = \int^T_0 (g(u(t)) + f(u'(t)/ \varphi' (t)))\varphi' (t) dt\), it is clear that \(J(v,\text{id}) = I(v)\). It is proved that \[ \inf_{v \in {\mathcal A}_C} \inf_{ \varphi \in \Phi} J(v, \varphi)= \inf_{v \in {\mathcal A}_C} J(v,\text{id}) \] and that the infimum is achieved at some \((\widehat U_C, \widehat \varphi) \in {\mathcal A}_C \times \Phi\) satisfying a Dubois-Reymond equation. This result is next used to prove that if \(f(y) \geq - A + B |y |\), \(A > 0\), \(B > 0\), and if \(\lim_{|y |\to \infty} (f(y) - \langle y, \nabla f(y) \rangle) = - \infty\), then the classical non-coercive problem Minimize \(\{I(v) \mid v \in W^{1,1} (0,T; \mathbb{R}^n)\), \(v(0) = a\), \(v(T) = b\}\), admits solutions. This result is closely connected with a recent paper by \textit{F. H. Clarke} [Trans. Am. Math. Soc. 336, No. 2, 655-673 (1993; Zbl 0780.49016)].
scientific article; zbMATH DE number 881709