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Stable minimizers of functionals of the gradient

  • Mikhail A. Sychev (a1), Giulia Treu (a2) and Giovanni Colombo (a2)


Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let $L: {\mathbb R}^n\rightarrow \bar {\mathbb R}= {\mathbb R}\cup \{+\infty \}$ be a continuous function with superlinear growth at infinity, and consider the functional $\mathcal {I}(u)=\int \nolimits _\Omega L(Du)$ , u ∈ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ∈ W1,1(Ω) such that $\mathcal {I}(f) < +\infty $ , the problem of minimizing $\mathcal {I}(u)$ with the boundary condition u|∂Ω = f has a solution which is stable, or – alternatively – is such that all of its solutions are stable. By stability of $\mathcal {I}$ at u we mean that $u_k\rightharpoonup u$ weakly in W1,1(Ω) together with $\mathcal {I}(u_k)\to \mathcal {I}(u)$ imply uk → u strongly in W1,1(Ω). This extends to general boundary data some results obtained by Cellina and Cellina and Zagatti. Furthermore, with respect to the preceding literature on existence results for scalar variational problems, we drop the assumption that the relaxed functional admits a continuous minimizer.



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1Celada, P. and Perrotta, S.. Minimizing non-convex multiple integrals: a density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 721741.
2Cellina, A.. On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993), 337341.
3Cellina, A.. On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993), 343347.
4Cellina, A. and Zagatti, S.. A version of Olech's lemma in a problem of the calculus of variations. SIAM J. Control Optim. 32 (1994), 11141127.
5Clarke, F.. Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 511530.
6Ekeland, I. and Témam, R.. Convex analysis and variational problems. Classics in Applied Mathematics,vol. 28 (Philadelphia: SIAM, 1999).
7Friesecke, G.. A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 437471.
8Giaquinta, M.. Growth conditions and regularity, a counterexample. Manuscripta Math. 59 (1987), 245248.
9Giaquinta, M. and Giusti, E.. On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 3146.
10Marcellini, P.. Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Rational Mech. Anal. 105 (1989), 267284.
11Mariconda, C. and Treu, G.. Gradient maximum principle for minima. J. Optim. Theory Appl. 112 (2002), 167186.
12Mariconda, C. and Treu, G.. Local Lipschitz regularity of minima for a scalar problem of the calculus of variations. Commun. Contemp. Math. 10 (2008), 11291149.
13Rickman, S.. Quasiregular mappings (Berlin: Springer-Verlag, 1993).
14Saks, S.. Theory of the Integral, 2nd revised edn. With two additional notes by Stefan Banach (New York: Dover Publ., 1964).
15Sychev, M. A.. Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 611631.
16Sychev, M. A.. Attainment and relaxation results in special classes of deformations. Calc. Var. PDE's 19 (2004), 183210.
17Sychev, M. A. and Sycheva, N. N.. Young measure approach to the weak convergence theory in the calculus of variations and strong materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XV (2016), 561598.
18Sychev, M. A. and Sycheva, N. N.. On Legendre and Weierstrass conditions in one-dimensional variational problems. J. Convex Anal. 24 (2017), 123133.
19Valadier, M.. Young measures. In Methods of Nonconvex Analysis. Varenna 1989 (C.I.M.E.) (ed. Cellina, A.). Lecture Notes in Math.vol. 1446, pp. 152188 (Berlin: Springer, 1990).
20Zagatti, S.. Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000), 384399.
21Ziemer, W. P.. Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics,vol. 120 (New York: Springer, 1989).


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