In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider
Mots-clés : Torsional rigidity, first Dirichlet eigenvalue, shape optimization
@article{COCV_2018__24_4_1585_0, author = {Henrot, Antoine and Lucardesi, Ilaria and Philippin, G\'erard}, title = {On two functionals involving the maximum of the torsion function}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1585--1604}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017069}, mrnumber = {3922448}, zbl = {1442.35281}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017069/} }
TY - JOUR AU - Henrot, Antoine AU - Lucardesi, Ilaria AU - Philippin, Gérard TI - On two functionals involving the maximum of the torsion function JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1585 EP - 1604 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017069/ DO - 10.1051/cocv/2017069 LA - en ID - COCV_2018__24_4_1585_0 ER -
%0 Journal Article %A Henrot, Antoine %A Lucardesi, Ilaria %A Philippin, Gérard %T On two functionals involving the maximum of the torsion function %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1585-1604 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017069/ %R 10.1051/cocv/2017069 %G en %F COCV_2018__24_4_1585_0
Henrot, Antoine; Lucardesi, Ilaria; Philippin, Gérard. On two functionals involving the maximum of the torsion function. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1585-1604. doi : 10.1051/cocv/2017069. http://archive.numdam.org/articles/10.1051/cocv/2017069/
[1] Solutions of the Cheeger problem via torsion functions. J. Math. Anal. Appl. 381 (2011) 263–279 | DOI | MR | Zbl
and ,[2] Variational methods in shape optimization problems. Progress Nonlin. Differ. Equ. Appl. 65. Birkhäuser. Boston (2005) | MR | Zbl
and ,[3] A strange term coming from nowhere. Topics in the mathematical modelling of composite materials. Progr. Nonlin. Differ. Equ. Appl. 31 (1997) 45–93. Birkhuser, Boston. | MR | Zbl
and ,[4] An introduction to Γ-convergence, In Vol. 8 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1993) | MR | Zbl
,[5] On functionals involving the torsional rigidity related to some classes of nonlinear operators. Preprint (2017) | arXiv | MR | Zbl
, and[6] Approximation of Dirichlet eigenvalues on domains with small holes. J. Math. Anal. Appl. 193 (1995) 169–199 | DOI | MR | Zbl
,[7] Extremum problems for eigenvalues of elliptic operators. Birkhäuser, Basel (2006) | MR | Zbl
,[8] Variation et Optimisation de Formes. Une Analyse Géométrique. In Vol. 48 of Mathématiques et Applications. Springer, Berlin (2005) | MR | Zbl
and ,[9] Estimation de l’erreur dans des problèmes de Dirichlet où apparait un terme étrange, Partial differential equations and the calculus of variations. Vol. II. In Vol. 2 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1989) 661–696 | MR | Zbl
and ,[10] Power concavity and boundary value problems. Indiana Univ. Math. J. 34 (1985) 687–704 | DOI | MR | Zbl
,[11] Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Ration. Mech. Anal. 97 (1987) 19–32 | DOI | MR | Zbl
and ,[12] Sufficient conditions for the concavity of the solution of the Dirichlet problem for the equation Δu = −1. Mat. Zametki 42 (1987) 537–542 | MR | Zbl
,[13] Solution of the Dirichlet’s problem for the equation Δu = −1 in a convex region. Math. Notes Akademy Sci. USSR 9 (1971) 52–53 | MR | Zbl
,[14] Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. 1. Birkhäuser, Basel (2000) | MR | Zbl
, and ,[15] Bounds for the maximum Stress in the St-Venant problem, special issue in honor of B. Sen. Part 1. Indian J. Mech. Math. (1968) 51–59 | MR
,[16] Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 251–265 | DOI | MR | Zbl
,[17] Some special maximum principles with applications to isoperimetric inequalities, Maximum principles and eigenvalue problems in partial differential equations. In Vol 175 of Pitman Research Notes in Math. Edited by . Longman (1988). | MR | Zbl
,[18] Some remarks on the problems of elastic torsion and of torsional creep, Some Aspects of Mechanics of Continua. Part 1. Jadavpur University (1977) 32–40
and ,[19] Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains. SIAM J. Math. Anal. 14 (1983) 1154–1162 | DOI | MR | Zbl
and ,[20] Isoperimetric inequalities and overdetermined problems for the Saint–Venant equation. New Zealand J. Math. 25 (1996) 217–227 | MR | Zbl
and ,[21] On extending some maximum principles to convex domains with nonsmooth boundaries. Math. Methods Appl. Sci. 33 (2010) 1850–1855 | DOI | MR | Zbl
and ,[22] Estimates for the Torsion Function and Sobolev Constants. Potential Anal. 36 (2012) 607–616 | DOI | MR | Zbl
,[23] Spectral bounds for the torsion function. Preprint (2017) | arXiv | MR | Zbl
,[24] On the torsion function with Robin or Dirichlet boundary conditions. J. Funct. Anal. 266 (2014) 1647–1666 | DOI | MR | Zbl
and ,[25] Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity, New trends in shape optimization. In Vol. 166 of International Series of Numerical Mathematics. Birkh?user/ Springer, Cham (2015) 19–41 | DOI | MR | Zbl
and ,[26] Hardy inequality and Lp estimates for the torsion function. Bull. Lond. Math. Soc. 41 (2009) 980–986 | DOI | MR | Zbl
and ,[27] On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue. Integr. Equ. Oper. Theory 86 (2016) 579–600 | DOI | MR | Zbl
, , and ,[28] Existence and regularity results for some shape optimization problems. In Vol. 19 of Theses, Scuola Normale Superiore di Pisa (Nuova Serie). Edizioni della Normale, Pisa (2015) | MR | Zbl
,[29] L∞-estimates for the torsion function and L∞ -growth of semigroups satisfying Gaussian bounds. Preprint (2018) | arXiv | MR
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