Wyniki wyszukiwania

1

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

100%

Triggiani R.

Applicationes Mathematicae

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2008

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tom 35

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nr 4

481-512

EN

This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let $λ_i$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let ${φ_{ij}}^{ℓ_i}_{j=1}$ be the corresponding linearly independent (normalized) eigenfunctions in L₂(Ω), so that $ℓ_i$ is the geometric multiplicity of $λ_i$. We prove that the Dirichlet boundary traces ${φ_{ij}|_{Γ₁}}^{ℓ_i}_{j=1}$ are linearly independent in L₂(Γ₁). Here Γ₁ is an arbitrary open, connected portion of Γ, of positive surface measure. The same conclusion holds true if the setting {Neumann B.C., Dirichlet boundary traces} is replaced by the setting {Dirichlet B.C., Neumann boundary traces}. These results are motivated by boundary feedback stabilization problems for parabolic equations [L-T.2]. Part II: The same problem is posed for the Stokes operator with motivation coming from the boundary stabilization problems in [B-L-T.1]- [B-L-T.3] (with tangential boundary control), and [R] (with just boundary control), where we take Γ₁ = Γ. The aforementioned property of boundary traces of eigenfunctions critically hinges on a unique continuation result from the boundary of corresponding over-determined problems. This is well known in the case of second-order elliptic operators of Part I; but needs to be established in the case of Stokes operators. A few proofs are given here.

2

Sharp regularity of the second time derivative w_tt of solutions to Kirchhoff equations with clamped Boundary Conditions

63%

Lasiecka I., Triggiani R.

International Journal of Applied Mathematics and Computer Science

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2001

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tom 11

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nr 4

753-772

EN

We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. (“clamped control”). If w denotes elastic displacement and θ temperature, we establish optimal regularity of {w, w_t, w_tt} in the elastic case, and of {w, w_t, w_tt, θ} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped B.C., and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped B.C. are ‘exceptional’ within the set of canonical B.C. (hinged, clamped, free B.C.). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.

3

Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions

63%

Lebiedzik C., Triggiani R.

Control and Cybernetics

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2009

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tom 38

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nr 4B

1461-1499

4

Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation

51%

Lasiecka I., Triggiani R.

Control and Cybernetics

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2008

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tom 37

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nr 4

935-969

Ograniczanie wyników

2 Control and Cybernetics

1 Applicationes Mathematicae

1 International Journal of Applied Mathematics and Computer Science

4 Triggiani R.

2 Lasiecka I.

1 Lebiedzik C.

1 2009

2 2008

1 2001

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Sharp regularity of the second time derivative w_tt of solutions to Kirchhoff equations with clamped Boundary Conditions

Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions

Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation