Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T05:57:41.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable minimizers of functionals of the gradient

Part of: Miscellaneous topics in calculus of variations and optimal control Existence theories

Published online by Cambridge University Press:  05 July 2019

Mikhail A. Sychev ,
Giulia Treu  and
Giovanni Colombo
Mikhail A. Sychev
Affiliation:
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 4 Acad. Koptyug avenue, 630090 Novosibirsk State University, Russia (masychev@math.nsc.ru)
Giulia Treu
Affiliation:
Dipartimento di Matematica, Università di Padova, ‘Tullio Levi-Civita’, via Trieste 63, 35121Padova, Italy (treu@math.unipd.it; colombo@math.unipd.it)
Giovanni Colombo
Affiliation:
Dipartimento di Matematica, Università di Padova, ‘Tullio Levi-Civita’, via Trieste 63, 35121Padova, Italy (treu@math.unipd.it; colombo@math.unipd.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Relaxed functionalstrict convexityweak and strong convergence

MSC classification

Primary: 49N60: Regularity of solutions
Secondary: 49N05: Linear optimal control problems 49J52: Nonsmooth analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2642 - 2655
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Celada, P. and Perrotta, S.. Minimizing non-convex multiple integrals: a density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 721–741.CrossRefGoogle Scholar
2Cellina, A.. On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993), 337–341.CrossRefGoogle Scholar
3Cellina, A.. On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993), 343–347.CrossRefGoogle Scholar
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), 1114–1127.CrossRefGoogle Scholar
5Clarke, F.. Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 511–530.Google Scholar
6Ekeland, I. and Témam, R.. Convex analysis and variational problems. Classics in Applied Mathematics,vol. 28 (Philadelphia: SIAM, 1999).Google Scholar
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), 437–471.Google Scholar
8Giaquinta, M.. Growth conditions and regularity, a counterexample. Manuscripta Math. 59 (1987), 245–248.CrossRefGoogle Scholar
9Giaquinta, M. and Giusti, E.. On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31–46.CrossRefGoogle Scholar
10Marcellini, P.. Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Rational Mech. Anal. 105 (1989), 267–284.Google Scholar
11Mariconda, C. and Treu, G.. Gradient maximum principle for minima. J. Optim. Theory Appl. 112 (2002), 167–186.CrossRefGoogle Scholar
12Mariconda, C. and Treu, G.. Local Lipschitz regularity of minima for a scalar problem of the calculus of variations. Commun. Contemp. Math. 10 (2008), 1129–1149.CrossRefGoogle Scholar
13Rickman, S.. Quasiregular mappings (Berlin: Springer-Verlag, 1993).CrossRefGoogle Scholar
14Saks, S.. Theory of the Integral, 2nd revised edn. With two additional notes by Stefan Banach (New York: Dover Publ., 1964).Google Scholar
15Sychev, M. A.. Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 611–631.CrossRefGoogle Scholar
16Sychev, M. A.. Attainment and relaxation results in special classes of deformations. Calc. Var. PDE's 19 (2004), 183–210.CrossRefGoogle Scholar
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), 561–598.Google Scholar
18Sychev, M. A. and Sycheva, N. N.. On Legendre and Weierstrass conditions in one-dimensional variational problems. J. Convex Anal. 24 (2017), 123–133.Google Scholar
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. 152–188 (Berlin: Springer, 1990).CrossRefGoogle Scholar
20Zagatti, S.. Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000), 384–399.Google Scholar
21Ziemer, W. P.. Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics,vol. 120 (New York: Springer, 1989).Google Scholar