Hölder continuity of solutions of one-dimensional Lagrange problems of calculus of variations
Let \(L: [a,b]\times \mathbb{R}^ n\times \mathbb{R}^ n\to \mathbb{R}\) be a Borel function. A couple \((A,B)\) of vectors in \(\mathbb{R}^ n\) is assumed to be given. We also denote by \(\mathcal U\) the set of all the absolutely continuous functions \(u: [a,b]\to \mathbb{R}^ n\) such that \((u(a)= A\) and \(u(b)= B\). Then consider a classical variational problem \([*]\): \(\underset{u\in{\mathcal U}}{\text{Minimize}} \int^ b_ a L\bigl(t,u(t), \dot u(t)\bigr) dt\). The existence of a solution for the problem \([*]\) is presupposed. The main result of this paper can now be described as follows. Theorem. Assume that (i) for some \(\alpha> 1\), the limit \[ \lim_{\| z\| \to+\infty} {L(t,u,z)\over \| z\|^ \alpha}\equiv g(t,u) \] exists uniformly in any compact subset of \(I\times \mathbb{R}^ n\), and (ii) the function \(g: I\times \mathbb{R}^ n\to (0,+ \infty)\) thus defined is continuous. Then every solution \(u\) of the problem \([*]\) satisfies \(\int^ b_ a \|\dot u(t)\|^ q dt< +\infty\) for any \(q<\infty\). The authors make recourse to Gurov and Reshetnyak's result for the proof of the theorem. (However the nondecreasing property of the function \(p\) in Theorem 1.1 does not seem to be compatible with (1.1).) Illuminating examples are supplied in order to clarify such technical details as: (a) the conditions assumed in the theorem do not imply the Lipschitz continuity of the solutions for \([*]\), and (b) the continuity of the function \(g\) is indispensable.
scientific article; zbMATH DE number 96526