Mixed Dirichlet-Transmission Problems in Non-smooth Domains

Vasylyeva, Nataliya

doi:10.1007/978-3-030-50302-4_9

Cited by 2 publications

(1 citation statement)

References 11 publications

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“…Here we focus on the solution of the problem which vanishes as r → +∞. First of all, we remark that the explicit solutions of classical Dirichlet-transmission problem to Poisson equations in domain G 1 ∪ g ∪ G 2 were searched in Section 9.4 [32]. In light of this fact, we are left to construct solutions in the case of homogenous equations (1.5) and homogenous Dirichlet and transmission conditions (1.8) and (1.9), i.e.…”

Section: Explicit Solutions Of Transmission Boundary Value Problems W...mentioning

confidence: 99%

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

Vasylyeva¹

2022

Preprint

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For ν ∈ [0, 1] and a complex parameter σ, Re σ > 0, we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane z ∈ C:where Ω(z) and F(z) are given complex functions, while a 1 and a 2 are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as |z| → +∞. Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed.

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Section: Explicit Solutions Of Transmission Boundary Value Problems W...mentioning

confidence: 99%

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

Vasylyeva¹

2022

Preprint

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Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector

Krasnoshchok

2023

Proc. IAMM NASU

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We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_\theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_\mu(s_\theta$ under some conditions on weight $\mu$, higher order of derivatives $k$ and the opening of the angle $\theta$.

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Mixed Dirichlet-Transmission Problems in Non-smooth Domains

Cited by 2 publications

References 11 publications

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector

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