Sharp Estimates for Eigenfunctions of a Neumann Problem

References

• Alvino , A. , Ferone , V. , Trombetti , G. ( 1998 ). On the properties of some nonlinear eigenvalues . SIAM J. Math. Anal. 29 : 437 – 451 .
   Web of Science ®Google Scholar
• Alvino , A. , Lions , P.-L. , Trombetti , G. ( 1989 ). On optimization problems with prescribed rearrangements . Nonlinear Anal. 13 : 185 – 220 .
   Web of Science ®Google Scholar
• Alvino , A. , Matarasso , S. , Trombetti , G. ( 2002 ). Elliptic boundary value problems: Comparison results via symmetrization . Ricerche Mat. 51 : 341 – 355 .
   Google Scholar
• Ashbaugh , M. S. , Benguria , R. D. ( 1992 ). A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions . Ann. Math. (2) 135 : 601 – 628 .
   Web of Science ®Google Scholar
• Atar , R. , Burdzy , K. ( 2002 ). On nodal lines of Neumann eigenfunctions . Electron. Comm. Probab. 7 : 129 – 139 .
   Web of Science ®Google Scholar
• Atar , R. , Burdzy , K. ( 2004 ). On Neumann eigenfunctions in lip domains . J. Amer. Math. Soc. 17 : 243 – 265 .
   Web of Science ®Google Scholar
• Avinyo , A. , Mora , X. ( 1997 ). Geometric inequalities of Cheeger type for the first positive eigenvalue of the n-dimensional free membrane problem . Ital. J. Pure Appl. Math. 2 : 133 – 140 .
   Google Scholar
• Bañuelos , R. , Burdzy , K. ( 1999 ). On the “hot spots” conjecture of J. Rauch . J. Funct. Anal. 164 ( 1 ): 1 – 33 .
   Web of Science ®Google Scholar
• Bhattacharya , T. , Weitsman , A. ( 1998 ). Some estimates for the symmetrized first eigenfunction of the Laplacian . Potential Anal. 9 : 143 – 172 .
   Web of Science ®Google Scholar
• Brandolini , B. , Trombetti , C. ( 2007 ). A symmetrization result for Monge–Ampère type equations . Math. Nachr. 280 : 467 – 478 .
   Web of Science ®Google Scholar
• Caffarelli , L. A. , Friedman , A. ( 1985 ). Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations . J. Differential Equations 60 ( 3 ): 420 – 433 .
   Web of Science ®Google Scholar
• Chiacchio , F. ( 2009 ). Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization . Publ. Mat. 53 : 47 – 71 .
   Web of Science ®Google Scholar
• Chiti , G. ( 1981 ). Inequalities for the first three membrane eigenvalues . Boll. Un. Mat. Ital. A 18 : 144 – 148 .
   Google Scholar
• Chiti , G. ( 1982 ). A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators . Z. Angew. Math. Phys. 33 : 143 – 148 .
   Web of Science ®Google Scholar
• Cianchi , A. ( 1989 ). On relative isoperimetric inequalities in the plane . Boll. Un. Mat. Ital. B 3 : 289 – 325 .
   Google Scholar
• Chong , K. M. , Rice , N. M. ( 1971 ) . Equimeasurable Rearrangements of Functions. Queen's Papers in Pure and Applied Mathematics, No. 28 . Kingston , Canada : Queen's University .
   Google Scholar
• Courant , R. , Hilbert , D. ( 1953 ) . Methods of Mathematical Physics. Vol. II . New York : Wiley .
   Google Scholar
• Federer , H. ( 1969 ) . Geometric Measure Theory. Berlin : Springer .
   Google Scholar
• Freitas , P. ( 2007 ). Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi . J. Funct. Anal. 251 : 376 – 398 .
   Web of Science ®Google Scholar
• Gilbarg , D. , Trudinger , N. S. ( 2001 ) . Elliptic Partial Differential Equations of Second Order. Berlin : Springer-Verlag .
   Google Scholar
• Hardt , R. , Simon L. ( 1989 ). Nodal sets for solutions of elliptic equations . J. Differential Geom. 30 : 505 – 522 .
   Web of Science ®Google Scholar
• Jerison , D. , Nadirashvili , N. ( 2000 ). The “hot spots” conjecture for domains with two axes of symmetry . J. Amer. Math. Soc. 13 : 741 – 772 .
   Web of Science ®Google Scholar
• Kawohl , B. ( 1985 ) . Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics 1150 . New York : Springer Verlag .
   Google Scholar
• Lions , P.-L. , Pacella , F. , Tricarico , M. ( 1988 ). Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions . Indiana Univ. Math. J. 37 : 301 – 324 .
   Web of Science ®Google Scholar
• Maderna , C. , Salsa , S. ( 1979 ). Symmetrization in Neumann problems . Applicable Anal. 9 : 247 – 256 .
   Google Scholar
• Maz'ja , V.G. ( 1985 ). Sobolev Spaces. Berlin : Springer-Verlag.
   Google Scholar
• Nečas , J. ( 1967 ) . Les Méthodes Directes En théorie Des équations Elliptiques [Direct Methods in the Theory of Elliptic Equations] . Paris : Masson .
   Google Scholar
• Pacella , F. , Tricarico , M. ( 1985/86 ). Symmetrization for a class of elliptic equations with mixed boundary conditions . Atti Sem. Mat. Fis. Univ. Modena 34 : 75 – 93 .
   Google Scholar
• Payne , L. E. , Rayner , M. ( 1972 ). An isoperimetric inequality for the first eigenfunction in the fixed membrane problem . Z. Angew. Math. Phys. 23 : 13 – 15 .
   Web of Science ®Google Scholar
• Payne , L. E. , Weinberger , H. F. ( 1960 ). An optimal Poincaré inequality for convex domains . Arch. Rational Mech. Anal. 5 : 286 – 292 .
   Google Scholar
• Talenti , G. ( 1976 ). Elliptic equations and rearrangements . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 : 697 – 718 .
   Google Scholar