On an Alternative Approach for Mixed Boundary Value Problems for the Lamé System

Natroshvili, David; Tsertsvadze, Tornike

doi:10.1007/s10659-023-10004-1

Cited by 2 publications

(3 citation statements)

References 12 publications

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“…Using the arguments employed in [30] with appropriate modifications, we can prove the following regularity results for solutions to the mixed BVP of steady state elastic oscillation equations.…”

Section: Regularity Resultsmentioning

confidence: 99%

“…It should be mentioned that similar approaches for interior and exterior mixed BVPs are used in [28], [29] and [30] for the Laplace equation, the Helmholtz equation, and the Lamé system. In the case of the steady state elastic oscillation equations, the arguments used in the above references are not directly applicable and we need appropriate modifications to justify our alternative approach.…”

Section: Introductionmentioning

confidence: 99%

“…Note that continuity of the operator (6.2) follows from Theorem 3.3. Due to Theorem 7(ii) in the reference [30], the operator…”

mentioning

confidence: 99%

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An alternative potential method for mixed steady state elastic oscillation problems

Natroshvili,

Mrevlishvili,

Tediashvili

2023

Preprint

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We consider an alternative approach to investigate three-dimensional exterior mixed boundary value problems (BVP) for the steady state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω − ⊂ R 3 , has a compact boundary surface S = ∂ Ω − , which is divided into two disjoint parts, the Dirichlet part S D and the Neumann part S N , where the displacement vector (the Dirichlet type condition) and the stress vector (the Neumann type condition) are prescribed respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov-Poincaré type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right-hand sides of the resulting boundary integral equations system are vector-functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector-functions belonging to the Sobolev space [ W 2 , loc 1 ( Ω − ) ] 3 and satisfying the Sommerfeld-Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the L p -based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses C α -Hölder continuity property in the closed domain Ω − ‾ with α = 1 2 − ε , where ε>0 is an arbitrarily small number.

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Section: Regularity Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An alternative potential method for mixed steady state elastic oscillation problems

Natroshvili,

Mrevlishvili,

Tediashvili

2023

Preprint

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An alternative potential method for mixed steady‐state elastic oscillation problems

Natroshvili,

Mrevlishvili,

Tediashvili

2024

Math Methods in App Sciences

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We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

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On an Alternative Approach for Mixed Boundary Value Problems for the Lamé System

Cited by 2 publications

References 12 publications

An alternative potential method for mixed steady state elastic oscillation problems

An alternative potential method for mixed steady state elastic oscillation problems

An alternative potential method for mixed steady‐state elastic oscillation problems

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