References
- Henrot , A and Oudet , E . 2003 . Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions . Arch. Ration. Mech. Anal. , 169 ( 1 ) : 73 – 87 .
- Troesch , BA . 1973 . Elliptical membranes with smallest second eigenvalue . Math. Comput. , 27 : 767 – 772 .
- Oudet , E . 2004 . Numerical minimization of eigenmodes of a membrane with respect to the domain . ESAIM: COCV , 10 : 315 – 330 .
- Henrot , A . 2006 . Extremum Problems for Eigenvalues of Elliptic Operators , Basel : Birkhauser .
- Bucur , D . 2003 . Regularity of optimal convex shapes . J. Convex Anal. , 10 ( 2 ) : 501 – 516 .
- Kinderlehrer , D and Nirenberg , L . 1977 . Regularity in free boundary problems . Ann. Scuola Norm. Sup. Pisa , 11 : 373 – 391 .
- Vogel , AL . 1992 . Symmetry and regularity for general regions having solutions to certain overdetermined boundary value problems . Atti Sem. Mat. Fis. Univ. Modena , 40 : 443 – 484 .
- Grisvard , P . 1985 . Boundary Value Problems in Non-Smooth Domains , London : Pitman .
- Pommerenke , Ch . 1992 . Boundary Behaviour of Conformal Maps , Berlin : Springer-Verlag .
- Garnett , JB and Marshall , DE . 2005 . “ Harmonic Measures ” . In New Mathematical Monographs , Vol. 2 , Cambridge : Cambridge University Press .
- Mannino , A . 2004 . Some inequalities concerning starlike and convex functions . Gen. Math. , 12 : 5 – 12 .
- Warschawski , SE . 1961 . On differentiability at the boundary in conformal mapping . Proc. Amer. Math. Soc. , 12 : 614 – 620 .
- J. Lamboley, On a non-regularity result in shape optimization, preprint.
- Fragalà , I and Gazzola , F . 2008 . Partially overdetermined problems . J. Differ. Eqns , 245 : 1299 – 1322 .
- J. Lamboley, Variations autour de formes irrégulières et propriétés de formes optimales, PhD Thesis, ENS Cachan, 2008.
- T. Briançon, Regularity of optimal shapes for the Dirichlet's energy with volume constraint, ESAIM: COCV, 10 (2004), 99–122.
- Briançon , T and Lamboley , J . 2009 . Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints . Ann. l'IHP, Anal. Non Linéaire , 26 ( 4 ) : 1149 – 1163 .
- Fragalà , I , Gazzola , F , Lamboley , J and Pierre , M . 2009 . Counterexamples to symmetry for partially overdetermined elliptic problems . Analysis , 29 : 85 – 93 .
- Stredulinsky , E and Ziemer , WP . 1997 . Area minimizing sets subject to a volume constraint in a convex set . J. Geom. Anal. , 7 ( 4 ) : 653 – 677 .
- Shahgholian , H and Uraltseva , N . 2003 . Regularity properties of a free boundary near contact points with the fixed boundary . Duke Math. J. , 116 ( 1 ) : 1 – 34 .
- Dauge , M . 1988 . “ Elliptic Boundary Value Problems on Corner Domains ” . In Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics , Vol. 1341 , Berlin : Springer-Verlag .
- Shamir , E . 1968 . Regularization of mixed second-order elliptic problems . Israel J. Math. , 6 : 150 – 168 .