Boundary Integral Equations

Hsiao, George; Wendland, Wolfgang L.

doi:10.1007/978-3-030-71127-6_2

Cited by 67 publications

(122 citation statements)

References 3 publications

Supporting

Mentioning

121

Contrasting

Unclassified

Order By: Relevance

“…We shall work mainly with the spaces of Bessel potentials H s which coincide with the Sobolev spaces W s,p in the case s ∈ Z , p = 2 and the Sobolev-Slobodeckij spaces for s ∈ R , p = 2, all briefly referred to as Sobolev spaces [3,41,52,119,120,124].…”

Section: The Wh Equations In Sommerfeld Half-plane Problemsmentioning

confidence: 99%

“…The different analytical nature of the spaces H ±1/2 (R + ) and H ±1/2 + is crucial. It is well-known [46,52] that the spaces…”

Section: The Wh Equations In Sommerfeld Half-plane Problemsmentioning

confidence: 99%

“…Formulations of Sommerfeld half-plane problems as boundary value and transmission (or interface) problems in Sobolev spaces started in the 1970s with Giorgio Talenti [117], continued in the 1980s by Erhard Meister and his co-authors, touching the more general theory of partial and pseudo-differential equations, for instance by Eli Shamir [99,100], Mikhail Agranovich, Marko Vishik, and Gregory Eskin [3,120,41], Joseph Wloka [124], George Hsiao and Wolfgang Wendland [52], to mention some closely related work.…”

Section: Historical Remarksmentioning

confidence: 99%

“…Remark 5.5. In potential theory, the reasoning is often based on a proof that a BVP is uniquely solvable if and only if a system of boundary integral equations is uniquely solvable, see [37,52]. This would be a consequence of (5.52), however there are other important operator relations which also imply the equivalence of the unique solubility and much more, cf.…”

Section: Operator Factorisation In Boundary Value Problemsmentioning

confidence: 99%

“…As a prototype, we focus again the Sommerfeld half-plane problems (3.40) with jump conditions (3.42) on Σ . Let H 1 (Ω) denote the weak solutions [52] of the Helmholtz equation in the two-dimensional slit domain Ω = R 2 \ Σ where Σ = {x = (x 1 , x 2 ) ∈ R 2 : x 1 ≥ 0 , x 2 = 0} is the half-line identified with R + as in Section 3. The Sommerfeld half-plane problems with two transmission conditions of first and second kind, respectively, are briefly written as [108] u ∈ H 1 (Ω), (5.53)…”

Section: Operator Factorisation In Boundary Value Problemsmentioning

confidence: 99%

See 4 more Smart Citations

From Sommerfeld Diffraction Problems to Operator Factorisation

Speck¹

2019

Constructive Mathematical Analysis

View full text Add to dashboard Cite

This article presents a brief survey devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by Sommerfeld half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear in a natural way from applications and how they can help to find solutions rigorously in case of well-posed problems or how to normalise problems by an adequate change of function spaces.

show abstract

Section: The Wh Equations In Sommerfeld Half-plane Problemsmentioning

confidence: 99%

“…The different analytical nature of the spaces H ±1/2 (R + ) and H ±1/2 + is crucial. It is well-known [46,52] that the spaces…”

Section: The Wh Equations In Sommerfeld Half-plane Problemsmentioning

confidence: 99%

Section: Historical Remarksmentioning

confidence: 99%

Section: Operator Factorisation In Boundary Value Problemsmentioning

confidence: 99%

Section: Operator Factorisation In Boundary Value Problemsmentioning

confidence: 99%

See 3 more Smart Citations

From Sommerfeld Diffraction Problems to Operator Factorisation

Speck¹

2019

Constructive Mathematical Analysis

View full text Add to dashboard Cite

show abstract

Boundary‐domain integral equation systems to the Dirichlet and Neumann problems for compressible Stokes equations with variable viscosity in 2D

Ayele

Dagnaw

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, the Dirichlet and Neumann boundary value problems for the steady-state Stokes system of partial differential equations for a compressible viscous fluid with variable viscosity coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to a system of direct segregated boundary-domain integral equations (BDIEs). The BDIEs in the two-dimensional case have special properties in comparison with the three dimension because of the logarithmic term in the parametrix for the associated partial differential equations. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of corresponding parametrix-based hydrodynamic single layer and hypersingular potentials and hence the unique solvability of BDIEs. Equivalence of the BDIE systems to the Dirichlet and Neumann BVPs and the invertibility of the corresponding boundary-domain integral operators in appropriate Sobolev spaces are shown. KEYWORDS boundary-domain integral equations, equivalence, invertibility, parametrix, Stokes equations MSC CLASSIFICATION 76D07; 35J57; 31A10; 45A05 H 1,0 * (Ω;)≔ ||p|| 2We define also the space H 1  (Ω) = H 1 (Ω)∕ = {v ∈ H 1 (Ω) ∶ ∫ Ω v • wdx = 0, for all w ∈ }, where  = {a + b(−x 2 , x 1 ) T ; a, b are constant vector and scalar, respectively} is the space of rigid body motions and  =

show abstract

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H⁻¹(Ω) source term on Lipschitz domains

Fresneda-Portillo

Woldemicheal

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

The interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is reduced to a system of boundary-domain integral equations (BDIEs) employing the parametrix obtained in Portillo (2019). We further extend the results obtained in Portillo ( 2019) for the mixed problem in a smooth domain with L 2 (Ω) right-hand side to Lipschitz domains and partial differential equation (PDE) right-hand side in the Sobolev space H −1 (Ω), where neither the classical nor the canonical conormal derivatives are well-defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

show abstract

Boundary Integral Equations

Cited by 67 publications

References 3 publications

From Sommerfeld Diffraction Problems to Operator Factorisation

From Sommerfeld Diffraction Problems to Operator Factorisation

Boundary‐domain integral equation systems to the Dirichlet and Neumann problems for compressible Stokes equations with variable viscosity in 2D

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H⁻¹(Ω) source term on Lipschitz domains

Contact Info

Product

Resources

About

Boundary Integral Equations

Cited by 67 publications

References 3 publications

From Sommerfeld Diffraction Problems to Operator Factorisation

From Sommerfeld Diffraction Problems to Operator Factorisation

Boundary‐domain integral equation systems to the Dirichlet and Neumann problems for compressible Stokes equations with variable viscosity in 2D

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H−1(Ω) source term on Lipschitz domains

Contact Info

Product

Resources

About

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H⁻¹(Ω) source term on Lipschitz domains