Stable minimizers of functionals of the gradient

Standard

Stable minimizers of functionals of the gradient. / Sychev, Mikhail A.; Treu, Giulia; Colombo, Giovanni.

In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 150, No. 5, 01.10.2020, p. 2642-2655.

Research output: Contribution to journal › Article › peer-review

Harvard

Sychev, MA, Treu, G & Colombo, G 2020, 'Stable minimizers of functionals of the gradient', Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 150, no. 5, pp. 2642-2655. https://doi.org/10.1017/prm.2019.38

APA

Sychev, M. A., Treu, G., & Colombo, G. (2020). Stable minimizers of functionals of the gradient. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150(5), 2642-2655. https://doi.org/10.1017/prm.2019.38

Vancouver

Sychev MA, Treu G, Colombo G. Stable minimizers of functionals of the gradient. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2020 Oct 1;150(5):2642-2655. doi: 10.1017/prm.2019.38

Author

Sychev, Mikhail A. ; Treu, Giulia ; Colombo, Giovanni. / Stable minimizers of functionals of the gradient. In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2020 ; Vol. 150, No. 5. pp. 2642-2655.

BibTeX

@article{7037ae3cfd7a47c69a64090d72b3ddbd,

title = "Stable minimizers of functionals of the gradient",

abstract = "Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let be a continuous function with superlinear growth at infinity, and consider the functional, u ϵ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ϵ W1,1(Ω) such that I(f) < +∞, the problem of minimizing 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 at u we mean that weakly in W1,1(Ω) together with 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 functional, strict convexity, weak and strong convergence, INTEGRALS, REGULARITY, MINIMA, CALCULUS",

author = "Sychev, {Mikhail A.} and Giulia Treu and Giovanni Colombo",

year = "2020",

month = oct,

day = "1",

doi = "10.1017/prm.2019.38",

language = "English",

volume = "150",

pages = "2642--2655",

journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",

issn = "0308-2105",

publisher = "Cambridge University Press",

number = "5",

}

RIS

TY - JOUR

T1 - Stable minimizers of functionals of the gradient

AU - Sychev, Mikhail A.

AU - Treu, Giulia

AU - Colombo, Giovanni

PY - 2020/10/1

Y1 - 2020/10/1

N2 - Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let be a continuous function with superlinear growth at infinity, and consider the functional, u ϵ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ϵ W1,1(Ω) such that I(f) < +∞, the problem of minimizing 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 at u we mean that weakly in W1,1(Ω) together with 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.

AB - Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let be a continuous function with superlinear growth at infinity, and consider the functional, u ϵ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ϵ W1,1(Ω) such that I(f) < +∞, the problem of minimizing 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 at u we mean that weakly in W1,1(Ω) together with 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.

KW - Relaxed functional

KW - strict convexity

KW - weak and strong convergence

KW - INTEGRALS

KW - REGULARITY

KW - MINIMA

KW - CALCULUS

UR - http://www.scopus.com/inward/record.url?scp=85068683671&partnerID=8YFLogxK

U2 - 10.1017/prm.2019.38

DO - 10.1017/prm.2019.38

M3 - Article

AN - SCOPUS:85068683671

VL - 150

SP - 2642

EP - 2655

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 5

ER -

ID: 20838958