One-dimensional non-coercive problems of the calculus of variations
The author characterizes the existence of solutions for the variational problem \[ \min \int_a^b f(t,v'(t)) dt \] with \(f(t,\cdot)\) convex, but not necessarily coercive, over the class \[ W^p_d=\left\{v\in W^{1,p}([a,b]): v(a)=0,\;v(b)=d\right\}\qquad 1\leq p\leq +\infty \] by means of a boundedness condition on the slope \(d\). This result extends and generalizes that given by \textit{P. Brandi} [Atti Semin. Mat. Fis. Univ. Modena 28, 15-32 (1979; Zbl 0472.49004)] and the former one due to \textit{P. J. Kaiser} [Atti Semin. Mat. Fis. Univ. Modena 24, 236-245 (1976; Zbl 0356.49026)]. The case of integrands of the type \(f(t,z)=\phi(t) h(z)\) -- both in the quasi-coercive and non-coercive case -- is discussed in detail. As an application, the author extends the suffient condition due to \textit{B. Botteron} and \textit{P. Marcellini} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 2, 197-223 (1991; Zbl 0729.49002)]. Analogous results are obtained by the author [J. Convex Anal. 5, No. 1, 1-17 (1998; Zbl 0929.49005)] for optimization problems over the subclass \(W=\{v\in W^p_d: v'(t)\geq \alpha\}\).
scientific article; zbMATH DE number 1383230