On the Geometry of Optimal Windows, with Special Focus on the Square
For the Laplace operator with mixed (Dirichlet and Neumann) boundary conditions, the dependence of the principal eigenvalue on the placement of the Dirichlet part is investigated. An optimal window is a Dirichlet part of the boundary that minimizes the principal eigenvalue among all competitors of t...
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| Vydáno v: | SIAM journal on mathematical analysis Ročník 37; číslo 6; s. 1800 - 1827 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2006
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| ISSN: | 0036-1410, 1095-7154 |
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| Abstract | For the Laplace operator with mixed (Dirichlet and Neumann) boundary conditions, the dependence of the principal eigenvalue on the placement of the Dirichlet part is investigated. An optimal window is a Dirichlet part of the boundary that minimizes the principal eigenvalue among all competitors of the same area. In the special case of a square, we provide both numerical evidence and rigorous partial results for the conjecture that optimal windows in a square are segments centered at either a corner or the midpoint of a side. In particular, we prove that the principal eigenvalue decreases as a window is shifted from a side-centered position towards the corner. An optimal window contained in two sides of the square is connected and contains a corner in its interior. Optimal windows whose length does not exceed the length of one side break the symmetry of the square. We also construct a star-shaped domain whose optimal window(s) must be disconnected. Finally we give, for general domains in $\R^d$, continuity results for the eigenvalue as a function of the window, and examples of discontinuity when crucial hypotheses are violated. We also give a variation formula that relates the eigenvalue to the singularities of the eigenfunction (stress intensity coefficient) near the boundary of the window. Methods are based on the variational problem and include rearrangement, Dirichlet--Neumann bracketing, capacity estimates, and deformation under a flow. |
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| AbstractList | For the Laplace operator with mixed (Dirichlet and Neumann) boundary conditions, the dependence of the principal eigenvalue on the placement of the Dirichlet part is investigated. An optimal window is a Dirichlet part of the boundary that minimizes the principal eigenvalue among all competitors of the same area. In the special case of a square, we provide both numerical evidence and rigorous partial results for the conjecture that optimal windows in a square are segments centered at either a corner or the midpoint of a side. In particular, we prove that the principal eigenvalue decreases as a window is shifted from a side-centered position towards the corner. An optimal window contained in two sides of the square is connected and contains a corner in its interior. Optimal windows whose length does not exceed the length of one side break the symmetry of the square. We also construct a star-shaped domain whose optimal window(s) must be disconnected. Finally we give, for general domains in $\R^d$, continuity results for the eigenvalue as a function of the window, and examples of discontinuity when crucial hypotheses are violated. We also give a variation formula that relates the eigenvalue to the singularities of the eigenfunction (stress intensity coefficient) near the boundary of the window. Methods are based on the variational problem and include rearrangement, Dirichlet--Neumann bracketing, capacity estimates, and deformation under a flow. |
| Author | Denzler, Jochen Burchard, Almut |
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| Cites_doi | 10.1007/s000390050027 10.1016/0031-8914(68)90059-1 10.1006/jmaa.1997.5716 10.1007/978-1-4612-9917-2 10.1137/S1052623497326083 10.1002/mana.19770760103 10.1137/S0036141099357574 10.1090/S0002-9947-99-02558-1 10.1090/S0002-9947-99-02207-2 10.1006/jmaa.1999.6558 10.1007/BFb0075060 10.1090/surv/051 10.1007/BF00249677 |
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| Keywords | Laplace operator Discontinuity Neumann problem Singularity Variational methods 31C40 shape optimization Eigenfunction Optimization method Optimal eigenvalue 35J05 Boundary condition mixed boundary conditions singular coefficient Capacity rearrangement; 49R50 Dirichlet problem 35R05 |
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| References | R2 R4 R5 Kawohl Bernhard (R15) 1985 R6 R8 R9 Dauge Monique (R7) 1987; 304 R21 Lieb Elliott (R18) 1997 R11 R14 R13 R16 R19 Evans Lawrence (R10) 1992 Brothers John (R3) 1988; 384 |
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| Title | On the Geometry of Optimal Windows, with Special Focus on the Square |
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