Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems

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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.

“…Arguing as in , Section 8.3], we obtain that

\int_{normalΩ ∖ (K_{ε} \cup Q_{ε})} D^{3} v_{ε} : D^{3} T_{ε} φ normald x \to 0

as ε →0. Thus, it remains to study the limiting behaviour of the last summand in the right hand‐side of , and this is carried out by unfolding it.…”

Section: Proof Of Theoremmentioning

confidence: 68%

“…Then, it is easy to see that

\int_{{normalΩ}_{ε}} v_{ε} T_{ε} ψ_{ε} normald x, \int_{{normalΩ}_{ε}} f_{ε} T_{ε} ψ_{ε} normald x

\int_{{normalΩ}_{ε} ∖ normalΩ} D^{3} v_{ε} : D^{3} T_{ε} ψ_{ε} normald x \to 0

as ε →0. Moreover, a slight modification of , Lemma 8.47] combined with Lemma yields

\int_{normalΩ} D^{3} v_{ε} : D^{3} T_{ε} ψ_{ε} normald x \to \int_{W \times Y \times (- \infty, 0)} D_{y}^{3} \overset{v}{true} ((), \overset{x}{true}, y) : D_{y}^{3} ψ ((), \overset{x}{true}, y) normald \overset{x}{true} normald y

. Thus, passing to the limit in , we obtain .…”

Section: Proof Of Theoremmentioning

confidence: 73%

{normalΦ}_{ε} ((), \overset{x}{true}, x_{N}) = ((), \overset{x}{true}, x_{N} - h_{ε} ((), \overset{x}{true}, x_{N}))

for all

((), \overset{x}{true}, x_{N}) \in {normalΩ}_{ε}

where

h_{ε} (\bar{x}, x_{N}) = \{\begin{matrix} 0, & if - 1 \leq x_{N} \leq - ε, \\ g_{ε} ((), \overset{x}{true}) {((), \frac{x_{N} + ε}{g_{ε} ((), \overset{x}{true}) + ε})}^{4}, & if - ε \leq x_{N} \leq g_{ε} ((), \overset{x}{true}) . \end{matrix}

The map Φ ε is a diffeomorphism of class C 3 , even though the highest order derivatives may not be uniformly bounded as ε →0.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Because

T_{ε} φ \in W^{3, 2} ({normalΩ}_{ε}) \cap W_{0}^{2, 2} ({normalΩ}_{ε})

, by , we obtain

\int_{{normalΩ}_{ε}} D^{3} v_{ε} : D^{3} T_{ε} φ normald x + \int_{{normalΩ}_{ε}} v_{ε} T_{ε} φ normald x = \int_{{normalΩ}_{ε}} f_{ε} T_{ε} φ normald x,

and passing to the limit as ε →0 we have that

\int_{{normalΩ}_{ε}} v_{ε} T_{ε} φ normald x \to \int_{normalΩ} vφ normald x, \int_{{normalΩ}_{ε}} f_{ε} T_{ε} φ normald x \to \int_{normalΩ} fφ normald x .

\int_{K_{ε}} D^{3} v_{ε} : D^{3} T_{ε} φ normald x \to \int_{normalΩ} D^{3} v : D^{3} φ normald x, \int

…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In this paper, we continue the analysis addressed in for the case of the biharmonic operator, and we study the compact convergence of the resolvent operators

H_{{normalΩ}_{ε}}^{- 1}

{double-struckR}^{N - 1}, {normalΩ}_{ε} = ({}, ((), \overset{x}{true}, x_{N}) \in {double-struckR}^{N} : \overset{x}{true} \in W, - 1 < x_{N} < g_{ε} ((), \overset{x}{true}))

where

g_{ε} ((), \overset{x}{true}) = ε^{α} b ((), \overset{x}{true} / ε)

for all

\overset{x}{true} \in W

, and b is a Y ‐periodic smooth function with Y = (−1/2,1/2) N − 1 .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Boundary homogenization for a triharmonic intermediate problem

Arrieta

Ferraresso

Lamberti

2016

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A note on the Neumann eigenvalues of the biharmonic operator

Provenzano

2016

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio σ. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to σ∈[0,1[and that all the Neumann eigenvalues tend to zero as σ→1−. Moreover, we show that the Neumann problem defined by setting σ = 1 admits a sequence of positive eigenvalues of finite multiplicity that are not limiting points for the Neumann eigenvalues with σ∈[0,1[as σ→1− and that coincide with the Dirichlet eigenvalues of the biharmonic operator. Copyright © 2016 John Wiley & Sons, Ltd.

On the spectral instability for weak intermediate triharmonic problems

Ferraresso

2022

We define the weak intermediate boundary conditions for the triharmonic operator −Δ 3 . We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation (Ω 𝜖 ) 𝜖 > 0 of a smooth domain Ω of R N for which the weak intermediate boundary conditions on 𝜕Ω 𝜖 are not preserved in the limit on 𝜕Ω, analogously to the Babuška paradox for the hinged plate. Four different boundary conditions can be produced in the limit, depending on the convergence of 𝜕Ω 𝜖 to 𝜕Ω. In one particular case, we obtain a 'strange' boundary condition featuring a microscopic energy term related to the shape of the approaching domains. Many aspects of our analysis could be generalised to an arbitrary order elliptic differential operator of order 2m and to more general domain perturbations. MSC CLASSIFICATION35J40; 35P99; 35B25