On the question of regularity of the solutions of variational problems

M. A. Sychev

Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 535-556

Résumé

Under the assumptions that $L(t,u,v)\in C(\mathbf R^3)$, $L_{vv}>\mu>0$, and $L>\mu v^2$ a study is made of the problem of minimizing the functional $\mathcal F(u(t))=\int_a^bL(t,u(t),\dot u(t))\,dt$ in the class of absolutely continuous functions $u(t)$ with $u(a)=A$ and $u(b)=B$. A direct method is presented for investigating the regularity of solutions and their dependence on the parameters of the problem. An example is given of a problem in which $L$ is analytic, $L_{vv}>\mu>0$, $L>\mu v^2$, and all the sequences minimizing the functional in the class of admissible smooth functions converge to a nonsmooth function $u_0(t)$ that is not a generalized solution of the Euler equation. An analogous example is given for the two-dimensional problem in the disk.