On a Transmission Problem Related to Models of Electrocardiology

Shefer, Yulia L.

doi:10.17516/1997-1397-2020-13-5-596-607

Cited by 3 publications

(6 citation statements)

References 9 publications

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“…As we have mentioned above, in the very particular case where, additionally, the operators A (e) , A (i) , A (b) are proportional, similar theorems were proved in [24].…”

Section: ) With An Arbitrary Element Hsupporting

confidence: 62%

“…Problem 2.1 was considered in [24] in the Sobolev spaces over smooth domains under the following rather restrictive but partially natural assumptions:…”

Section: A More General Steady Problemmentioning

confidence: 99%

“…Now we are ready to describe the null-space of Problem 2.1. We slightly differ the approach of [11] and [24] in this more general situation.…”

Section: A More General Steady Problemmentioning

confidence: 99%

“…Consider the situation where the case A (e) , A (i) and A (b) satisfy (2.31) and the operators A (e) , A (i) are proportional (cf. [11], [24]). Then, in particular, the pair ( (2.32) ∆ (e) = γ∆ (i) with some γ > 0.…”

Section: ) With An Arbitrary Element Hmentioning

confidence: 99%

“…Recently, theoretical investigations of the steady part of the model led to interesting results about non-uniqueness and existence of its solutions in Hardy type spaces, see [11]. Paper [24] was devoted to a larger class of similar transmission problems in the Sobolev spaces in the framework of general theory of elliptic operators with constant coefficients. However the both results were obtained under the following very restrictive assumptions: all the elliptic operators involved in the model should be proportional.…”

Section: Introductionmentioning

confidence: 99%

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On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

Shlapunov¹,

Shefer²

2021

Preprint

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We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.

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“…As we have mentioned above, in the very particular case where, additionally, the operators A (e) , A (i) , A (b) are proportional, similar theorems were proved in [24].…”

Section: ) With An Arbitrary Element Hsupporting

confidence: 62%

“…Problem 2.1 was considered in [24] in the Sobolev spaces over smooth domains under the following rather restrictive but partially natural assumptions:…”

Section: A More General Steady Problemmentioning

confidence: 99%

“…Now we are ready to describe the null-space of Problem 2.1. We slightly differ the approach of [11] and [24] in this more general situation.…”

Section: A More General Steady Problemmentioning

confidence: 99%

Section: ) With An Arbitrary Element Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

Shlapunov¹,

Shefer²

2021

Preprint

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On uniqueness theorems for the inverse problem of electrocardiography in the Sobolev spaces

Kalinin¹,

Shlapunov

Ushenin³

2022

Z Angew Math Mech

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We consider a mathematical model related to reconstruction of cardiac electrical activity from ECG measurements on the body surface. An application of recent developments in solving boundary value problems for elliptic and parabolic equations in Sobolev type spaces allows us to obtain uniqueness theorems for the model. The obtained results can be used as a sound basis for creating numerical methods for non-invasive mapping of the heart.

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On the uniqueness theorems for transmission problems related to models of elasticity, diffusion and electrocardiography

Shlapunov,

Shefer

2023

Journal of Inverse and Ill-Posed Problems

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We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials, we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.

show abstract

On a Transmission Problem Related to Models of Electrocardiology

Abstract: We consider a generalization of a transmission problem for matrix elliptic operators related to mathematical models of cardiology. We find sufficient conditions when the approach developed for scalar elliptic operators is still valid in this much more general situation

Cited by 3 publications

References 9 publications

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

On uniqueness theorems for the inverse problem of electrocardiography in the Sobolev spaces

On the uniqueness theorems for transmission problems related to models of elasticity, diffusion and electrocardiography

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