Eigenvalues of polyharmonic operators on variable domains

Self Cite

We consider the biharmonic operator subject to homogeneous intermediate boundary conditions of Steklov-type. We prove an analyticity result for the dependence of the eigenvalues upon domain perturbation and compute the appropriate Hadamard-type formulas for the shape derivatives. Finally, we prove that balls are critical domains for the symmetric functions of multiple eigenvalues subject to volume constraint

“…The proof of the analyticity of Λ F , s follows by the abstract results in and applied to the operator

{((), Δ_{φ}^{2})}^{(MathClass-bin- 1)} MathClass-bin\circ {scriptJ}_{φ}

. See also . We now prove Formula (2.8).…”

Section: An Analyticity Resultsmentioning

confidence: 86%

“…Let

u_{l} MathClass-rel= v_{l} MathClass-bin\circ \overset{φ}{true}

for all l ∈ F . By proceeding as in , we have that

normald {MathClass-rel|}_{φ MathClass-rel= \overset{φ}{true}} Λ_{F MathClass-punc, s} [ψ] MathClass-rel= MathClass-bin- λ_{F}^{s MathClass-bin+ 1} [\overset{φ}{true}] ((), \binom{| F | - 1}{s - 1}) {MathClass-op\sum}_{l MathClass-rel\in F} Δ_{\overset{φ}{true}}^{2} ([], normald {MathClass-rel|}_{φ MathClass-rel= \overset{φ}{true}} ((), {((), Δ_{φ}^{2})}^{(MathClass-bin- 1)} MathClass-bin\circ {scriptJ}_{φ}) [ψ] (u_{l})) [u_{l}] MathClass-punc.

The proof of (2.8) will follow by combining (2.9) with the following formula

\begin{matrix} aligned \\ right & left Δ_{\overset{MathClass-op̃}{φ}}^{2} [d |_{φ = \overset{MathClass-op̃}{φ}} ({(Δ_{φ}^{2})}^{MathClass-open(- 1 MathClass-close)} \circ J_{φ})] \end{matrix}

…”

Section: An Analyticity Resultsmentioning

confidence: 99%

“…Proof Assume that

\overset{φ}{true} (Ω)

{MathClass-op\sum}_{l MathClass-rel\in F} {((), \frac{\partial v_{l}}{∂r})}^{2}

is constant on ∂B , hence

{MathClass-op\sum}_{l MathClass-rel\in F} Δ_{\partial \overset{φ}{true} (Ω)} {((), \frac{\partial v_{l}}{∂ν})}^{2} MathClass-rel= 0 MathClass-punc, on ∂B MathClass-punc.

The function

\frac{\partial^{4}}{∂r^{4}} {MathClass-op\sum}_{l MathClass-rel\in F} v_{l}^{2} MathClass-rel= {MathClass-op\sum}_{l MathClass-rel\in F} ((), 6 {((), \frac{\partial^{2} v_{l}}{\partial r^{2}})}^{2} MathClass-bin+ 8 \frac{\partial v_{l}}{∂r} \frac{\partial^{3} v_{l}}{\partial r^{3}} MathClass-bin+ 2 v_{l} \frac{\partial^{4} v_{l}}{∂r^{4}})

is clearly radial; hence,

{MathClass-op\sum}_{l MathClass-rel\in F} \partial v_{l}

…”

Section: Criticality Of Balls In Isovolumetric Perturbationsmentioning

confidence: 99%

“…We recall that in the case of the first eigenvalue of the Dirichlet Laplacian, the corresponding condition is

\frac{∂u}{∂ν} MathClass-rel= C

\partial \overset{φ}{true} (Ω)

in which case it is a classical result that the existence of a positive solution implies that

\overset{φ}{true} (Ω)

is ball, see the celebrated paper . We also refer to for more references.…”

Section: Criticality Of Balls In Isovolumetric Perturbationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Shape deformation for vibrating hinged plates

Buoso

Lamberti

2013

Self Cite

On the optimization of Maxwell's eigenvalues as functions of the electric permittivity in a cavity

Lamberti,

Luzzini,

Zaccaron

2024

Boundary homogenization for a triharmonic intermediate problem

Arrieta

Ferraresso

Lamberti

2016

Self Cite

We consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N‐dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems that depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case that provides a typical homogenization problem leading to a strange boundary term in the limit problem. Copyright © 2016 John Wiley & Sons, Ltd.