Diagonal non-semicontinuous variational problems

Zagatti, Sandro

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1333-1343.

We study the minimum problem for non sequentially weakly lower semicontinuos fucntionals of the form$$\begin{array}{c}\hfill ℱ\left(u\right)={\int }_{I}f(x,u\left(x\right),u\text{'}\left(x\right))\mathrm{d}x,\end{array}$$ defined on Sobolev spaces, where the integrand $f:I\times {ℝ}^m\times {ℝ}^m\to ℝ$ is assumed to be non convex in the last variable. Denoting by $\overline{f}$ the lower convex envelope of $f$ with respect to the last variable, we prove the existence of minimum points of $ℱ$ assuming that the application $p\mapsto \overline{f}(·,p,·)$ is separately monotone with respect to each component $p_i$ of the vector $p$ and that the Hessian matrix of the application $\xi \mapsto \overline{f}(·,·,\xi )$ is diagonal. In the special case of functionals of sum type represented by integrands of the form $f(x,p,\xi )=g(x,\xi )+h(x,p)$, we assume that the separate monotonicity of the map $p\mapsto h(,p)$, holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.

[1] G. Buttazzo and G. Dal Maso Г-Limits of Integral Functionals. J. Analyse Math. 37 (1980) 145–185 | DOI | MR | Zbl

[2] A. Cellina, G. Colombo, On a classical problem of the calculus of variations without convexity conditions. Ann. Inst. Henri Poincaré Anal. Non Lin. 7 (1990) 97–106 | DOI | Numdam | MR | Zbl

[3] L. Cesari, An existence theorem without convexity conditions. SIAM J. Control 12 (1974) 319–331 | DOI | MR | Zbl

[4] L. Cesari, Optimization – Theory and Applications. Springer-Verlag, New York (1983) | DOI | MR | Zbl

[5] B. Dacorogna, Non convex problems of the calculus of variations and differential inclusions, in Handbook of Differ. Equ. 2 Elsevier/North Holland (2005) 57–126 | MR | Zbl

[6] G. Dal Maso An introduction to Г-convergence. Birkhäuser, Boston (1993) | DOI | MR | Zbl

[7] E. Giusti, Metodi diretti nel calcolo delle variazioni, Unione Matematica Italiana, Bologna (1993) | MR | Zbl

[8] P. Marcellini, Non convex integrals of the calculus of variations, Methods of nonconvex analysis, edited by A. Cellina. In Vol. 1446 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin (1990) 16–57 | DOI | MR | Zbl

[9] S. Zagatti, On the minimum problem for non convex scalar functionals. SIAM J. Math. Anal. 37 (2005) 982–995 | DOI | MR | Zbl

[10] S. Zagatti, Uniqueness and continuous dependence on boundary data for integro-extremal minimizer of the functional of the gradient. J. Convex Anal. 14 (2007) 705–727 | MR | Zbl

[11] S. Zagatti, Solutions of Hamilton-Jacobi equations and minimizers of non quasiconvex functionals. J. Math. Anal. Appl. 335 (2007) 1143–1160 | DOI | MR | Zbl

[12] S. Zagatti, Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations. Calc. Var. PDE’s 31 (2008) 511–519 | DOI | MR | Zbl

[13] S. Zagatti, Qualitative properties of integro-extremal minimizers of non-homogeneous scalar functionals. Int. J. Pure Appl. Math. 51(2009) 103–116 | MR | Zbl

[14] S. Zagatti, Minimization of non quasiconvex functionals by integro-extremization method. Discrete Contin. Dyn. Syst. , Ser. A 21 (2008) 625–641 | DOI | MR | Zbl