On the Possibility of Holomorphic Extension Into a Domain of Functions Defined on a Connected Piece of Its Boundary

Aizenberg, Lev; Kytmanov, Alexander M.

doi:10.1070/sm1992v072n02abeh002148

Cited by 20 publications

(40 citation statements)

References 10 publications

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“…Moreover, its extension is given by (33). Clearly, We note that Theorem 5 is an analogue of Theorem by Aizenberg and Kytmanov [8]) describing solvability conditions of the Cauchy problem for the Cauchy-Riemann system (cf. also [9] in the Cauchy Problem for Laplace Equation or [15] in the Cauchy problem for general elliptic systems).…”

Section: And Only If There Is a Vector Functionmentioning

confidence: 99%

“…[8]). Moreover, Corollary 2 gives us a possibility to use Hilbert space methods for this purpose in the case where q = 2 (cf.…”

Section: And Only If There Is a Vector Functionmentioning

confidence: 99%

“…Green formula (8) and the Uniqueness Theorem 3 suggest us to consider two kind of problems for the parabolic Lamé type operator.…”

Section: The Boundary Problemsmentioning

confidence: 99%

“…Actually, the approach was invented for the investigation of the famous ill-posed Cauchy problem for elliptic equations (see, for instance, [8] for the Cauchy-Riemann operator, [9] for the Laplace equation, [10] for the elliptic Lamé operator and [11], [12], [15], for general systems with injective principal symbols).…”

Section: Introductionmentioning

confidence: 99%

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On a mixed problem for the parabolic Lamé type operator

Puzyrev

Shlapunov

2014

Journal of Inverse and Ill-Posed Problems

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Abstract. We consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier-Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvabilty of the problem.

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Section: And Only If There Is a Vector Functionmentioning

confidence: 99%

“…[8]). Moreover, Corollary 2 gives us a possibility to use Hilbert space methods for this purpose in the case where q = 2 (cf.…”

Section: And Only If There Is a Vector Functionmentioning

confidence: 99%

“…Green formula (8) and the Uniqueness Theorem 3 suggest us to consider two kind of problems for the parabolic Lamé type operator.…”

Section: The Boundary Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a mixed problem for the parabolic Lamé type operator

Puzyrev

Shlapunov

2014

Journal of Inverse and Ill-Posed Problems

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“…in order to get information about extendability of the potential T + i f (cf. [8], [2]). However in this paper we will use the theory of Fourier series with respect to the bases with the double orthogonality property (cf.…”

Section: A Homotopy Formulamentioning

confidence: 99%

On the Cauchy problem for the elliptic complexes in spaces of distributions

Fedchenko

Shlapunov

2013

Complex Variables and Elliptic Equations

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Let D be a bounded domain in R n with a smooth boundary ∂D. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A i } of first order operators. In particular, we describe traces on ∂D of tangential part τ i (u) and normal part ν i (u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue spaces L 2 (D) we construct the approximate and exact solutions to the Cauchy problem with maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H 1 (D) by its Cauchy data τ i (u) on a subset Γ ⊂ ∂D and the values of A i u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.Key words: Elliptic differential complexes, ill-posed Cauchy problem, Carleman's formula.It is well-known that the Cauchy problem for an elliptic system A is ill-posed (see, for instance, [1]). Apparently, the serious investigation of the problem was stimulated by practical needs. Namely, it naturally appears in applications: in hydrodynamics (as the Cauchy problem for holomorphic functions), in geophysics (as the Cauchy problem for the Laplace operator), in elasticity theory (as the Cauchy problem for the Lamé system) etc., see, for instance, the book [2] and its bibliography. The problem was actively studied through the XX century (see, for instance, [3] Differential complexes appeared as compatibility conditions for overdetermined operators (see, for instance, [13]). Thus, the Cauchy problem for them is of special interest. One of the first problems of this kind was the Cauchy problem for the Dolbeault complex (the compatibility complex for the multidimensional Cauchy-Riemann system), see [14]. The interest to it was great because of the famous example by H. Lewy of differential equation without solutions, constructed with the use of the tangential Cauchy-Riemann operator, see [15]. Recently new approaches to the problem were found in spaces of smooth functions (see [16], [17]).We consider the Cauchy problem in spaces of distributions with some restrictions on growth in order to correctly define its traces on boundaries of domains (see, for instance, [2], [18], [19], [20]). In this paper we develop the approach presented in [9] to study the homogeneous Cauchy problem for overdetermined elliptic partial differential operators. Instead we consider the nonhomogeneous Cauchy problem for elliptic complexes.

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Monogenic Functions in Complexified Infinite-Dimensional Spaces with Commutative Multiplication

Plaksa

Shpakivskyi

2023

Frontiers in Mathematics

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On the Possibility of Holomorphic Extension Into a Domain of Functions Defined on a Connected Piece of Its Boundary

Cited by 20 publications

References 10 publications

On a mixed problem for the parabolic Lamé type operator

On a mixed problem for the parabolic Lamé type operator

On the Cauchy problem for the elliptic complexes in spaces of distributions

Monogenic Functions in Complexified Infinite-Dimensional Spaces with Commutative Multiplication

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