Cylindrical optimal rearrangement problem leading to a new type obstacle problem

Mikayelyan, Hayk

doi:10.1051/cocv/2017047

Mikayelyan, Hayk ¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 859-872.

Résumé

An optimal rearrangement problem in a cylindrical domain $Ω = D \times (0, 1)$ is considered, under the constraint that the force function does not depend on the $x_{n}$ variable of the cylindrical axis. This leads to a new type of obstacle problem in the cylindrical domain

\begin{matrix} Δ_{u} (x', x_{n}) = χ_{{v > 0}} (x') + χ_{{v = 0}} (x') [\partial_{ν} u (x', 0) + \partial_{ν} u (x', 1)] \end{matrix} l

arising from minimization of the functional

\begin{matrix} \int_{Ω} \frac{1}{2} {|\nabla u (x)|}^{2} + χ_{{v > 0}} (x^{'}) u (x) d x, \end{matrix}

where $v (x') = \int_{0}^{1} u (x', t) d t, and$ $\partial_{ν} u$ is the exterior normal derivative of $u$ at the boundary. Several existence and regularity results are proven and it is shown that the comparison principle does not hold for minimizers.

MR Zbl

DOI : 10.1051/cocv/2017047

Classification : 35R35, 49J20
Mots-clés : Obstacle problem, rearrangements

Affiliations des auteurs :

Mikayelyan, Hayk ¹

@article{COCV_2018__24_2_859_0,
     author = {Mikayelyan, Hayk},
     title = {Cylindrical optimal rearrangement problem leading to a new type obstacle problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {859--872},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {2},
     year = {2018},
     doi = {10.1051/cocv/2017047},
     zbl = {1402.49007},
     mrnumber = {3816419},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017047/}
}

TY  - JOUR
AU  - Mikayelyan, Hayk
TI  - Cylindrical optimal rearrangement problem leading to a new type obstacle problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 859
EP  - 872
VL  - 24
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017047/
DO  - 10.1051/cocv/2017047
LA  - en
ID  - COCV_2018__24_2_859_0
ER  -

%0 Journal Article
%A Mikayelyan, Hayk
%T Cylindrical optimal rearrangement problem leading to a new type obstacle problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 859-872
%V 24
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017047/
%R 10.1051/cocv/2017047
%G en
%F COCV_2018__24_2_859_0

Mikayelyan, Hayk. Cylindrical optimal rearrangement problem leading to a new type obstacle problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 859-872. doi : 10.1051/cocv/2017047. http://www.numdam.org/articles/10.1051/cocv/2017047/

Bibliographie
Cité par

[1] I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50 (2001) 1077–1112. | DOI | MR | Zbl

[2] G.R. Burton, Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276 (1987) 225–253. | DOI | MR | Zbl

[3] G.R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré Anal. Non Linéaire 6 (1989) 295–319. | DOI | Numdam | MR | Zbl

[4] G.R. Burton and J.B. Mcleod, Maximisation and minimisation on classes of rearrangements. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) 287–300. | DOI | MR | Zbl

[5] G.R. Burton and E.P. Ryan, On reachable sets and extremal rearrangements of control functions. SIAM J. Control Optimiz. 26 (1988) 1481–1489. | DOI | MR | Zbl

[6] L.A. Caffarelli, The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998) 383–402. | DOI | MR | Zbl

[7] F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory, vol. 178 of Graduate Texts in Mathematics. Springer Verlag, New York (1998). | MR | Zbl

[8] B. Emamizadeh and Y. Liu, Constrained and unconstrained rearrangement minimization problems related to the p-Laplace operator. Israel J. Math. 206 (2015) 281–298 | DOI | MR | Zbl

[9] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001) | MR | Zbl

[10] B. Kawohl, Rearrangements and convexity of level sets in PDE, vol. 1150 of Lect. Notes Math. Springer Verlag, Berlin (1985) | DOI | MR | Zbl

[11] E.H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics. AMS, Providence, RI, 2nd edition (2001) | DOI | MR | Zbl

Cité par Sources :