Spectral properties for layer potentials associated to the Stokes equation in Lipschitz domains

Chang, Tongkeun; Pahk, Dae Hyeon

doi:10.1007/s00229-009-0292-1

Cited by 9 publications

(6 citation statements)

References 10 publications

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“…Since λI − K * C is invertible, K * #,C − K * C is a compact operator [3], λI − K * #,C is a Fredholm operator and it is enough to show that it is one-to-one. The proof goes exactly as in [4]. Let us assume that λI − K * #,C is not one-to-one.…”

Section: Appendix Spectrum Of Some Periodic Integral Operatorsmentioning

confidence: 89%

Spectroscopic conductivity imaging of a cell culture

Ammari

Giovangigli

Kwon

et al. 2016

ASY

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Abstract. In this paper, we present a simplified electrical model for tissue culture. We derive a mathematical structure for overall electrical properties of the culture and study their dependence on the frequency of the current. We introduce a method for recovering the microscopic properties of the cell culture from the spectral measurements of the effective conductivity. Numerical examples are provided to illustrate the performance of our approach.

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Section: Appendix Spectrum Of Some Periodic Integral Operatorsmentioning

confidence: 89%

Spectroscopic conductivity imaging of a cell culture

Ammari

Giovangigli

Kwon

et al. 2016

ASY

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“…As Γ(t) is unbounded we cannot rely on compactness arguments to show the solvability of this equation. Instead, we modify arguments from [7,10] to obtain the necessary information on the spectrum of the corresponding integral operator via a Rellich identity. Moreover, we also rely on a further Rellich identity used in [18] in the study of the Muskat problem.…”

Section: Introductionmentioning

confidence: 99%

Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

Matioc

Prokert

2022

Nonlinear Differ. Equ. Appl.

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We study the two-phase Stokes flow driven by surface tension for two fluids of different viscosities, separated by an asymptotically flat interface representable as graph of a differentiable function. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single and double layer potential, spectral results on the corresponding integral operators, and abstract results on nonlinear parabolic evolution equations.

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“…Many papers study the Brinkman transmission problem and the Stokes-Brinkman transmission problem by the integral equation method ( [21], [22], [23], [25], [20], [24]). The transmission problem for the Stokes system was studied by the integral equation method in [3], [36]. The following transmission problem was studied:…”

Section: Introductionmentioning

confidence: 99%

Integral equations method and the transmission problem for the stokes system

Medková¹

2015

Kragujevac J Math

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Abstract. The transmission problem for the Stokes system is studied:is a bounded open set with Lipschitz boundary and G − is the corresponding complementary open set. Using the integral equation method we study the problem in homogeneous Sobolev spaces. Under assumption that ∂G + is of class C 1 we study this problem also in Besov spaces and L q -solutions of the problem. We show the unique solvability of the problem. Moreover, we solve the corresponding boundary integral equations by the successive approximation method.

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Spectral properties for layer potentials associated to the Stokes equation in Lipschitz domains

Cited by 9 publications

References 10 publications

Spectroscopic conductivity imaging of a cell culture

Spectroscopic conductivity imaging of a cell culture

Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

Integral equations method and the transmission problem for the stokes system

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