The Method of Fundamental Solutions applied to the calculation of eigensolutions for simply connected plates

Freitas

2010

Proc. R. Soc. A.

Self Cite

We consider the inverse spectral problem for the Laplace operator on triangles with Dirichlet boundary conditions, providing numerical evidence to the effect that the eigenvalue triplet (l 1 , l 2 , l 3 ) is sufficient to determine a triangle uniquely. On the other hand, we show that other combinations such as (l 1 , l 2 , l 4 ) will not be enough, and that there will exist at least two triangles with the same values on these triplets.

Section: (A) the Admissible Regionmentioning

confidence: 99%

“…In this case, the computation of the eigenvalues was done using a mesh-free numerical method known as method of fundamental solutions (MFSs), as studied in Alves & Antunes (2005). In some stiffer cases such as for thin triangles, we used an enriched version of the MFS as described in Antunes & Valtchev (2010) instead.…”

Section: Introductionmentioning

confidence: 99%

On the inverse spectral problem for Euclidean triangles

Freitas

2010

Proc. R. Soc. A.

Self Cite

“…In the MFS, these base functions are built translating the fundamental solution to some source points placed outside U and this method can produce highly accurate approximations when the domain is smooth (e.g. Alves & Antunes 2005;Barnett & Betcke 2008). However, for polygonal shapes it is convenient to enrich the space of functions with some particular solutions of the PDE, which also satisfy the null boundary conditions along a corner (cf.…”

Section: Introductionmentioning

confidence: 99%

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Henrot

2010

Proc. R. Soc. A.

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In this paper, we study the set of points in the plane defined by {(x, y) = (l 1 (U), l 2 (U)), |U| = 1}, where (l 1 (U), l 2 (U)) are either the first two eigenvalues of the Dirichlet-Laplacian, or the first two non-trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

“…The coefficients α j and β j will be calculated by fitting the boundary conditions. As in [2] or [3] we define m collocation points x 1 , . .…”

Section: Brief Description Of the Numerical Methodsmentioning

confidence: 99%

“…The variational formulation in (1.1) may be found in [13,17]. For hinged plates the natural boundary conditions lead to minimize J in the Sobolev space H 2 A formal integration by parts (see [15], Sect. 1.1.2 for the details) yields the strong Euler-Lagrange equation…”

Section: Introductionmentioning

confidence: 99%

Convex shape optimization for the least biharmonic Steklov eigenvalue

Gazzola

2013

ESAIM: COCV

Abstract. The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.