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Potential Theory and Its Applications to Basic Problems of Mathematical Physics
... The double-and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. For example, the representation of the solution of the Dirichlet problem for the Laplace equation is sought as a double-layer potential with unknown density and an application of certain property leads to a Fredholm equation of the second kind for determining the density function (see [1] and [2]). ...
... in the domain, bounded in the half-plane x 1 > 0. An exposition of the results on the potential theory for the two-dimensional singular elliptic equation (1) together with references to the original literature are to be found in the monograph by Smirnov [7], which is the standard work on the subject. This work also contains an extensive bibliography of all relevant papers up to 1966; the list of references given in the present work is largely supplementary to Smirnov's bibliography. ...
... However, only two works are devoted to the study of the mixed problem for a singular elliptic equation by the potential method: in [13] was considered a two-dimensional singular elliptic equation (1), and the work [14] is devoted to solving the mixed problem for the equation ...
Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [26], Kupradze, Gegelia, Basheleishvili and Burchuladze [33], Mikhlin [44], Mikhlin and Prössdorf [45], Buchukuri, Chkadua, Duduchava, and Natroshvili [2].) Here we provide a summary of the continuity properties of the boundary operator W Ω [a, S a , ·] (the so-called Neumann-Poincaré operator in case P [a, D] is the Laplace operator) in the frame of Hölder and Schauder spaces. ...
... Here we must say that we only consider the boundary behaviour of the double layer potential. Instead for the regularity properties of the double layer potential in the Schauder space C 1,α outside of the boundary we refer to Günter [26], Kupradze, Gegelia, Basheleishvili and Burchuladze [33], Mikhlin [44], Mikhlin and Prössdorf [45], Miranda [46], [47], Wiegner [65], Dalla Riva [7], Dalla Riva, Morais and Musolino [9], Mitrea, Mitrea and Verdera [52] and references therein. ...
... We first mention some known results in the classical case of the boundary behaviour of the double layer potential in Schauder spaces with m = 2. Instead for the regularity properties of the double layer potential in Schauder spaces with m = 2 outside of the boundary we refer to Günter [26], Kupradze, Gegelia, Basheleishvili and Burchuladze [33], Mikhlin [44], Mikhlin and Prössdorf [45], Miranda [46], [47], Wiegner [65], Dalla Riva [7], Dalla Riva, Morais and Musolino [9], Mitrea, Mitrea and Verdera [52] and references therein. In case n = 3 and Ω is of class C 2 , α ∈]0, 1[ and if P [a, D] is the Helmholtz operator, Colton and Kress [6] have developed previous work of Günter [26] and Mikhlin [44] and proved that the operator W [∂Ω, a, S a , ·] is bounded from C 0,α (∂Ω) to C 1,α (∂Ω). ...
We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder and Schauder spaces on the boundary of a bounded open subset of . The purpose is two-fold. On one hand we try present in a single paper all the known continuity results on the topic with the best known exponents in a H\"{o}lder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garc\'{\i}a-Cuerva and Gatto in the frame of H\"{o}lder spaces and later by the author.
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [9], Kupradze, Gegelia, Basheleishvili and Burchuladze [15], Mikhlin [21], Mikhlin and Prössdorf [22], Buchukuri, Chkadua, Duduchava, and Natroshvili [1]. ) We now briefly summarize some known results in the classical case of the boundary behaviour double layer potential in Schauder spaces with m ≥ 2. Instead for the regularity properties of the double layer potential in Schauder spaces with m ≥ 2 outside of the boundary we refer to Günter [9], Kupradze, Gegelia, Basheleishvili and Burchuladze [15], Mikhlin [21], Mikhlin and Prössdorf [22], Miranda [23], [24], Wiegner [29], Dalla Riva [5], Dalla Riva, Morais and Musolino [7], Mitrea, Mitrea and Verdera [27] and references therein. ...
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [9], Kupradze, Gegelia, Basheleishvili and Burchuladze [15], Mikhlin [21], Mikhlin and Prössdorf [22], Buchukuri, Chkadua, Duduchava, and Natroshvili [1]. ) We now briefly summarize some known results in the classical case of the boundary behaviour double layer potential in Schauder spaces with m ≥ 2. Instead for the regularity properties of the double layer potential in Schauder spaces with m ≥ 2 outside of the boundary we refer to Günter [9], Kupradze, Gegelia, Basheleishvili and Burchuladze [15], Mikhlin [21], Mikhlin and Prössdorf [22], Miranda [23], [24], Wiegner [29], Dalla Riva [5], Dalla Riva, Morais and Musolino [7], Mitrea, Mitrea and Verdera [27] and references therein. ...
... 3, Theorems 3.26 and 3.28] and prove that if Ω is of class C m,α , then W [∂Ω, a, S a , ·] is bounded from C m−2,α ′ (∂Ω) to C m−1,α ′ (∂Ω) for α ′ ∈]0, α[. In case n = 3 and Ω is of class C 2 , α ∈]0, 1[ and if P [a, D] is the Helmholtz operator, Colton and Kress [4] have developed previous work of Günter [9] and Mikhlin [21] and proved that the operator W [∂Ω, a, S a , ·] is bounded from C 0,α (∂Ω) to C 1,α (∂Ω). ...
We prove the validity of a regularizing property on the boundary of the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M.~G\"{u}nter, S.~Mikhlin, V.D.~Kupradze, T.G.~Gegelia, M.O.~Basheleishvili and T.V.~Bur\-chuladze, U.~Heinemann and extends the work of A.~Kirsch who has considered the case of the Helmholtz operator.
... Numerous applications of simple-and double-layer potentials, as well as volumetric potentials, occur in fluid mechanics, elastodynamics, electromagnetizm, and acoustics [3]; therefore, the theory of potentials plays an important role in solving boundary value problems for elliptic equations. This, in particular, allows one to reduce the solution of boundary value problems to the solution of integral equations [1,2]. ...
... x u x + 2β y u y = 0, 0 < 2α, 2β < 1 (2) are devoted to relatively few works. In the works [15][16][17][18] the authors studied only the properties of the double-layer potentials for generalized biaxially symmetric elliptic equation (2). ...
... x u x + 2β y u y = 0, 0 < 2α, 2β < 1 (2) are devoted to relatively few works. In the works [15][16][17][18] the authors studied only the properties of the double-layer potentials for generalized biaxially symmetric elliptic equation (2). In this paper, for the equation (2), we construct the theory potential and apply it to the solution of the Dirichlet problem in the domain bounded in the first quarter R 2+ 2 := {(x, y) : x > 0, y > 0} of the xOy-plane. ...
Жалпыланған екi өске симметриялық эллиптикалық теңдеудiң iргелi шешiмдерi екi айныма- лысы бар Аппелдiң гипергеометриялық функциясы арқылы өрнектеледi, олардың қасиеттерi жоғарыда келтiрiлген теңдеу үшiн шектi есептердi зерттеу үшiн қажет. Бұл жұмыста Аппел- дiң гипергеометриялық функциясының кейбiр қасиеттерiн қолдана отырып, бiз қос қабатты және жай қабатты потенциалдардың тығыздығы үшiн шектi теоремаларды дәлелдеймiз және интегралдық теңдеулер аламыз. Құрылған потенциалдар теориясының нәтижелерiн жазы- қтықтың бiрiншi ширегiнде шектелген облыста екi сингулярлы коэффициентi бар екi өлшемдi эллиптикалық теңдеу үшiн Дирихле есебiн зерттеуге қолданамыз.
... in the domain, which is bounded in the half-plane x 1 > 0. An exposition of the results on the potential theory for the two-dimensional singular elliptic equation (1), together with references to the original literature on the subject, is to be found in the monograph by Smirnov,7 which is the standard work on the subject. This work also contains an extensive bibliography of all relevant papers up to 1966. ...
... where the density 1 ...
... we denote the double-layer potential (14) by w (1) 1 (x). We now investigate some properties of the double-layer potential w (1) 1 (x). Lemma 1. ...
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two‐dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double‐ and simple‐layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double‐ and simple‐layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball.
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [16], Kupradze et al. [22], Mikhlin [26].) ...
... In case n = 3 , α ∈]0, 1[, and Ω is of class C 2 and if P [a, D] is the Helmholtz operator, Colton and Kress [4] have developed previous work of Günter [16] and Mikhlin [26] and proved that the operator W Ω [a, S a , ·] is bounded from C 0,α (∂Ω) to C 1,α (∂Ω) and that accordingly it is compact in C 1,α (∂Ω). ...
We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Hölder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of kernels of integral operators and a generalization of a result for integral operators on differentiable manifolds.
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [13], Kupradze, Gegelia, Basheleishvili and Burchuladze [19], Mikhlin [23].) The analysis of the continuity and compactness properties of W Ω [a, S a , ·] is a classical topic and several results in the literature show that W Ω [a, S a , ·] improves the regularity of Hölder continuous functions on ∂Ω. ...
... 3, Theorems 3.26 and 3.28] and prove that if Ω is of class C m,α , then W [∂Ω, a, S a , ·] is bounded from C m−1,α ′ (∂Ω) to C m,α ′ (∂Ω) for α ′ ∈]0, α[. In case n = 3 and Ω is of class C 2 and if P [a, D] is the Helmholtz operator, Colton and Kress [4] have developed previous work of Günter [13] and Mikhlin [23] and proved that the operator W Ω [a, S a , ·] is bounded from C 0,α (∂Ω) to C 1,α (∂Ω) and that accordingly it is compact in C 1,α (∂Ω). ...
We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of integral operators and a generalization of a result for integral operators on differentiable manifolds.
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [14], Kupradze, Gegelia, Basheleishvili and Burchuladze [20], Mikhlin [23].) ...
... In case n = 3 and Ω is of class C 2 and if P [a, D] is the Helmholtz operator, Colton and Kress [2] have developed previous work of Günter [14] and Mikhlin [23] and proved that the operator w[∂Ω, a, S a , ·] |∂Ω is bounded from C 0,α (∂Ω) to C 1,α (∂Ω) and that accordigly it is compact in C 1,α (∂Ω). ...
We prove the validity of regularizing properties of a double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces by exploiting an explicit formula for the tangential derivatives of the double layer potential itself. We also introduce ad hoc norms for kernels of integral operators in order to prove continuity results of integral operators upon variation of the kernel, which we apply to layer potentials.
... The double-and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. For example, the representation of the solution of the Dirichlet problem for the Laplace equation is sought as a double-layer potential with unknown density and an application of certain property leads to a Fredholm equation of the second kind for determining the density function (see [22] and [33]). ...
... 22),(4.23),(5.4) and(5.5) ...
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with several singular coefficients are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using a decomposition formula and other identities for the Lauricella's hypergeometric function in many variables, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet problem for the generalized singular elliptic equation in the some part of the multidimensional ball.
... Furthermore, with the help of the property of simple layer potentials ( Fabrikant, 1997 ;Gunter, 1967 ) ...
... which have the following property ( Gunter, 1967 ) ...
Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.
... The definition of χ u shows that the right hand side ∂ 0 χ u (ξ 0 , x 1 , x 2 ) of (6.3) is defined by the second order derivative of the functions u 1 and u 2 . In view of the theorem in the Appendix III of the book [8] of Günter, the right hand side ∂ 0 χ u (ξ 0 , x 1 , x 2 ) turns out to be Hölder-continuous (possibly with a smaller Hölder exponent). ...
... Using a two-dimensional version of the results of Günter[8] (Appendix III), this can be proved as follows: the second order derivatives are Hölder-continuous and so they can be extended Hölder-continuously to the boundary. Consequently, the difference of the first order derivatives of Ξ at two points can be estimated by the distance|x − x | = |x − x | α · |x − x | 1−α ≤ c|x − x | α .And a same argument is applicable to Ξ itself. ...
Starting from distinguishing boundaries for monogenic functions (see Tutschke in Adv. Appl. Clifford Algebr 25:441–451, 2015), one can solve boundary value problems for monogenic functions (see Dao in Boundary value problems for monogenic functions in higher dimensions. Ph.D Thesis, Hanoi University of Science and Technology, Vietnam, 2019). A boundary value problem for monogenic function in is the following: prescribe the vector components and on the whole boundary, the bi-vector component on a (one-dimensional) curve on the boundary and the real part at one point. The goal of the present paper is to reduce the same boundary value problems for more general (linear or fully non-linear) first order systems to fixed-point problems for an operator containing the monogenic solution of the boundary value problem under consideration (provided the given system is written in its Clifford-analytic normal form). In case of the existence of a uniquely determined solution of the boundary value problem is proved as fixed-point of the corresponding fixed-point problem in case the right hand side is Lipschitz-continuous with sufficiently small Lipschitz constants.
... where l and l are doublet and source intensities, respectively. Based on potential theory [13], the perturbation potential inside the body is equal to zero. In order to simulate the oscillating body (a combination of heave and pitch motions), the relative velocity of each panel in the targeted computations should be considered. ...
... In this paper, an experimental formulation (Eqs. 13,14) for drag force on smoothed flat plate is used, because it has good similarity in shape to an airfoil [3]: ...
Interaction of an oscillating foil in uniform flow would produce a forwarding thrust. The efficiency of this phenomenon is very dependent on the heave and pitch frequencies, and especially different phase angles. It is also well known that in the case of two foils in tandem arrangement, the frequency of heave and pitch motion of forward foil affects the lift/drag force of aft ward foil and thus the shape of emanating vortex sheet. In this paper, the thrust coefficient and efficiency of two oscillating foils in tandem arrangement are investigated. The interference of airfoils frequencies is investigated. A three-dimensional unsteady code is developed, based on boundary element method, and applied to simulate all the targeted cases. The trailing edge vortex sheet is also modeled using vortex filament and time stepping method. This approach is adopted due to its powerful ability for estimating the hydrodynamic forces on the lifting bodies. In order to model the heave/pitch motion, it is imperative to solve the 2-DOF equations for each foil in conjunction with BEM solver. For validation purpose, the developed program is applied to a single stationary and flapping foil and the thrust coefficient is compared against existing experimental data. Subsequently, the developed code is used to analyze tandem flapping foils. The induced wake shape behind foils is compared with the single flapping foil. Based on the obtained results, the effects of forward flapping foil are clearly observed on the aft ward foil wake shape. It is also demonstrated that for better efficiency, the optimum angle of attack for the aft ward foil is zero. © 2018, The Brazilian Society of Mechanical Sciences and Engineering.
... An explicit expression of v I n is found from the gradient of the velocity potential (3.6) having therefore an integral term. This integral term is known as a harmonic double-layer potential with density Ω (Gunter 1967), which is defined on a subdomain of R 3 \ S p . This integral term becomes singular if it is evaluated on the surface S p , however, it can be continuously extended on the surface for each side and the value depends on the side by which we approach the surface. ...
We investigate the three-dimensional (3-D) flow around and through a porous screen for various porosities at high Reynolds number Re = O(10^4). Historically, the study of this problem has been focused on two-dimensional cases and for screens spanning completely or partially a channel. Since many recent problems have involved a porous object in a 3-D free flow, we present a 3-D model initially based on Koo & James (J. Fluid Mech., vol. 60, 1973, pp. 513–538) and Steiros & Hultmark (J. Fluid Mech., vol. 853, 2018 pp. 1–11) for screens of arbitrary shapes. In addition, we include an empirical viscous correction factor accounting for viscous effects in the vicinity of the screen. We characterize experimentally the aerodynamic drag coefficient for a porous square screen composed of fibres, immersed in a laminar air flow with various solidities and different angles of attack. We test various fibre diameters to explore the effect of the space between the pores on the drag force. Using PIV and hot wire probe measurements, we visualize the flow around and through the screen, and in particular measure the proportion of fluid that is deviated around the screen. The predictions from the model for drag coefficient, flow velocities and streamlines are in good agreement with our experimental results. In particular, we show that local viscous effects are important: at the same solidity and with the same air flow, the drag coefficient and the flow deviations strongly depend on the Reynolds number based on the fibre diameter. The model, taking into account 3-D effects and the shape of the porous screen, and including an empirical viscous correction factor that is valid for fibrous screens may have many applications including the prediction of water collection efficiency for fog harvesters.
... It suffices to quote the pioneers' greetings to the Eighteenth Congress of the All-Union Communist Party (Bolsheviks): 6) MEETING 16 Svetik Sheinman. We will become such a polar explorer as Papanin, 7) such a pilot as Chkalov, 8) such a mathematician as Sobolev, such a miner as Stakhanov, 9) and such a poet as Mayakovsky. 10) (Applause.) ...
This is a brief overview of the worldline and memes of Sergei Sobolev (1908–1989), a cofounder of distribution theory.
... The role of the double layer potential in the solution of boundary value problems for the operator P [a, D] is well known (cf. e.g., Günter [10], Kupradze, Gegelia, Basheleishvili and Burchuladze [14], Mikhlin [19], Mikhlin and Prössdorf [20]). In order to prove that under suitable assumptions W Ω [a, S a , ·] maps a Hölder space in ∂Ω to a Schauder space of differentiable functions on ∂Ω, one needs to estimate the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential. ...
In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in H\"{o}lder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.
... In the elasticity theory, relation (18) is a counterpart of the classical Lyapunov-Tauber theorem for the harmonic double layer potential (see [14,Ch. 2], [18, Ch. 5, Section 8]). ...
We consider a special approach to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of three-dimensional bounded domain Ω⊂R3, when the boundary surface S=∂Ω is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which do not contain neither extensions of the Dirichlet or Neumann data, nor the Steklov–Poincaré type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [W21(Ω)]3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the Lp-based Besov spaces with 43<p<4, which under appropriate boundary data implies Cα-Hölder continuity of the solution to the mixed BVP in the closed domain Ω‾ with α=12−ε, where ε>0 is an arbitrarily small number.
... The double-and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. For example, the representation of the solution of the Dirichlet problem for the Laplace equation is sought as a double-layer potential with unknown density and an application of certain property leads to a Fredholm equation of the second kind for determining the density function (see [1] and [2]). ...
Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet problem for the multidimensional singular elliptic equation in the half of the ball.
... Furthermore, by the Lyapunov-Tauber theorem (cf. [15,13] and more references therein), ...
A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [26] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.
... Thus, p e (x) contains 2 double layer potentials. Because the jump of a double layer potential is equal to its density, 25 ...
Scattering of monochromatic waves on an isolated inhomogeneity (inclusion) in an infinite poroelastic medium is considered. Wave propagation in the medium and the inclusion are described by Biot's equations of poroelasticity. The problem is reduced to 3D-integro-differential equations for displacement and pressure fields in the region occupied by the inclusion. Properties of the integral operators in these equations are studied. Discontinuities of the fields on the
inclusion boundary are indicated. The case of a thin inclusion with low permeability is considered. The corresponding scattering problem is reduced to a 2D integral equation on the middle surface of the inclusion. The unknown function in this equation is the pressure jump in the transverse direction to the inclusión middle surface. An inclusion with a thin layer of low permeability on its interface is considered. The appropriate boundary conditions on the inclusion
interface are pointed out. Methods of numerical solution of the volume integral equations of the scattering problems of poroelasticity are discussed.
... Furthermore, by the Lyapunov-Tauber theorem (cf. [15,13] and more One of the main differences with respect the bounded domain case is that the integrands of the operators V , W , P and R and their corresponding direct values and conormal derivatives do not always belong to L 1 . In these cases, the integrals should be understood as the corresponding duality forms (or their their limits of these forms for the infinitely smooth functions, existing due to the density in corresponding Sobolev spaces). ...
A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. Boundary-domain integral equations are formulated in terms of parametrix-based potential type integral operators defined on the boundary and the domain. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.
... us consider the fluid flow domain or region as an open domain that are assumed smooth in the sense of Lyapunov(Gunter, 1967), everywhere except from the trailing edge. The first componentB D refers to the surface of the foil and the second respect to an inertial reference frame. ...
Out of the numerous applications of biomimetic, aquatic inspired devices based on oscillating hydrofoils are able to achieve high levels of efficiency either for propulsion or for tidal energy extraction in nearshore and coastal regions. The ability to account and properly design for flexibility effects has the potential to further enhance the overall performance of such systems.
In the present work, a hydro-elastic model is proposed for investigating the effects of chord-wise flexibility on the performance of flapping foils with variable flexural rigidity, and whose structural response is actuated by unsteady pressure field caused by the prescribed harmonic motion of the hydro-mechanical system. A fluid-structure interaction numerical method has been developed to simulate the time-dependent structural response of the oscillating hydrofoil. We present a low order boundary element panel method (BEM) for the unsteady hydrodynamics, coupled with a finite element method (FEM) for the cylindrical bending of thin elastic plates, based on the classical Kirchhoff-Love theory.
Numerical results are presented concerning the performance of the system over a range of design and operation parameters, including Strouhal number, heaving and pitching amplitudes and effective angle of attack. To further illustrate the capabilities of the developed BEM-FEM coupled model, we validate the numerical scheme with experimental data, for the case of a chord-wise flexible thin plate under enforced heaving motion excited at the leading edge. The present model could serve as a useful tool in the design, assessment and control of biomimetic systems for renewable energy extraction.
... Furthermore, by the Lyapunov-Tauber theorem (cf. [14,16] and more references therein), L + ∆ ρ = L − ∆ ρ = L ∆ ρ. These relations are closely related with the parametrix. ...
A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix different from the one widely used in the literature by the authors Chkadua, Mikhailov and Natroshvili. Mapping properties of the surface and volume parametrix-based potential-type operators are analysed. Invertibility of the single layer potential is also studied in detail in appropriate Sobolev spaces. We show that the system of boundary-domain integral equations derived is equivalent to the Dirichlet problem prescribed and we prove the existence and uniqueness of solution in suitable Sobolev spaces of the system obtained by using arguments of compact-ness and Fredholm Alternative theory. A discussion of the possible applications of this new parametrix is included.
... The representation of the of the (first) boundary value problem solution is sought as a double-layer potential with unknown density and the function is CONTACT A. S. Berdyshev berdyshev@mail.ru determined by applying certain property leading to a Fredholm equation of the second kind (see [1,2]). Method of complex analysis (based upon analytic functions), has been applied by Gilbert [3] to construct an integral representation of solutions for the following generalized bi-axially symmetric Helmholtz equation: ...
In earlier papers, the double-layer potential has been successfully applied in solving boundary value problems for elliptic equations. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known [Complex Var Elliptic Equ. 2007;52(8):673–683], while the potential theory was constructed only for the first one [Sohag J Math. 2015;2(1):1-10]. Here, in this paper, our goal is to construct theory of double-layer potentials corresponding to the next fundamental solution. We used some properties of one of Appell's hypergeometric functions with respect to two variables to prove the limiting theorems, while integral equations concerning the denseness of double-layer potentials are derived.
... On one hand, the reducing of boundary value problems by means of double layer potential to integral equations is convenient for theoretical studies on solvability and uniqueness of solutions to boundary value problems. On the other hand, this gives an opportunity for an effective numerical solving of boundary value problems for domains of complicated shapes [1,2]. ...
... The double-layer potential plays an important role in solving boundary value problems of elliptic equations. The representation of the solution of the (first) boundary value problem is sought as a doublelayer potential with unknown density and an application of certain property leads to a Fredholm equation of the second kind for determining the function (see [18] and [29]). ...
The double-layer potential plays an important role in solving boundary value problems for elliptic equations. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the next fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.
... Then the integral (2) exists in the sense of the Cauchy principal value, the function V (x) is continuous on the surface S, and the following estimate holds: 1) ...
We prove the existence of the derivative of the acoustic single layer potential and study some properties of the operator generated by this derivative in generalized Hölder spaces.
... The double-layer and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. For example, the representation of the solution of the (first) boundary value problem is sought as a double-layer potential with unknown density and an application of certain property leads to a Fredholm equation of the second kind for determining the function (see [10] and [14]). Let be m R the half-space 1 were studied by many authors (see [1][2][3], [5][6][7], [11,12], [16,17], [19]). ...
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the potentials of the double- and simple-layers for this equation, with the help of which limit theorems are proved and integral equations containing in the kernel the density of the above potentials are derived.
... Thus, p e (x) contains 2 double layer potentials. Because the jump of a double layer potential is equal to its density, 25 ...
Scattering of monochromatic waves on an isolated inhomogeneity (inclusion) in an ininite
poroelastic medium is considered. Wave propagation in the medium and the inclusion are described by Biot´s equations of poroelasticity. The problem is reduced to 3D-integro-diifferential equations for displacement and pressure fields in the region occupied by the inclusion. Properties of the integral operators in these equations are studied. Discontinuities of the fields on the inclusion boundary are indicated. The case of a thin inclusion with low permeability is considered.
The corresponding scattering problem is reduced to a 2D-integral equation on the middle surface of the inclusion. The unknown function in this equation is the pressure jump in the transverse direction to the inclusion middle surface. An inclusion with a thin layer of low permeability on its interface is considered. The appropriate boundary conditions on the inclusion interface are pointed out. Methods of numerical solution of the volume integral equations of the scattering problems of poroelasticity are discussed.
UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.
Starting from microscopic N particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a -homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) 2-point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.
The aim of this paper is to prove a theorem of C.~Miranda for the single and double layer potential corresponding to the fundamental solution of a second order differential operator with constant coefficients in Schauder spaces in the limiting case in which the open set is of class and the densities are of class for the single layer potential and of class for the double layer potential for some nonzero natural number m. The treatment of the limiting case requires generalized Schauder spaces.
We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential that is associated to the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Hölder and Schauder spaces on the boundary of a bounded open subset of . The purpose is two-fold. On one hand we try present in a single paper the known continuity results on the topic with the best known exponents in a Hölder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by García-Cuerva and Gatto in the frame of Hölder spaces and later by the author.
A mixed boundary value problem for the bi-Laplacian equation in a thin layer around a surface with the boundary is investigated. We track what happens in -limit when the thickness of the layer converges to zero. It is shown how the mixed type boundary value problem (BVP) for the bi-Laplace equation in the initial thin layer transforms in the -limit into an appropriate Dirichlet BVP for the bi-Laplace-Beltrami equation on the surface. For this we apply the variational formulation and the calculus of Günter’s tangential differential operators on a hypersurface and layers. This approach allow global representation of basic differential operators and of corresponding BVPs in terms of the standard cartesian coordinates of the ambient Euclidean space .
We prove the validity of a regularizing property on the boundary of the double layer potential associated with the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M. Günter, S. Mikhlin, V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, U. Heinemann and extends the work of A. Kirsch who has considered the case of the Helmholtz operator.
In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ( Ω ) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number.
This chapter analyzes volume potentials and the solvability of the Poisson equation by means of such volume potentials. To go into detail, we first consider the case of weakly singular kernels, and then the case in which the first order partial derivatives of the kernel are also weakly singular. That been done, we turn to singular kernels and, after that, to weakly singular kernels with singular first order partial derivatives. Finally, we prove the basic properties of the corresponding volume potentials in spaces of Hölder continuous functions, in Schauder spaces, and in the Roumieu classes. Although similar properties in Schauder spaces can be found in the classical monographs of Günter (Potential theory and its applications to basic problems of mathematical physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing Co., New York, 1967) and of Kupradze, Gegelia, Basheleishvili, and Burchuladze (Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland series in applied mathematics and mechanics, vol 25, Russian edition. North-Holland Publishing Co., Amsterdam, New York, 1979. Edited by V. D. Kupradze), here we prove the corresponding statements with optimal Hölder exponents. To do so, we follow a proof of Miranda (Atti Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez I (8), 7, 303–336, 1965) and we exploit a known lemma (cf. Majda and Bertozzi, Vorticity and incompressible flow. Cambridge texts in applied mathematics, vol 27, Prop. 8.12, pp 348–350. Cambridge University Press, Cambridge, 2002) for which we provide a proof of Mateu, Orobitg, and Verdera (J Math Pures Appl (9), 91(4), 402–431, 2009). To conclude the chapter, we use the properties proven for the volume potentials to study the solvability of the basic boundary value problems for the Poisson equation.
The double- and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. The desired solution represents a potential of a certain layer with unknown density. One finds it with the help of the theory of Fredholm integral equations of the second kind. The potential, in turn, is expressed via the fundamental solution to the given elliptic equation. At present, fundamental solutions to Helmholtz multidimensional singular equations are known, nevertheless, the potential theory is constructed only for two-dimensional degenerate equations. In this paper, we study the mentioned potentials for a three-dimensional elliptic equation with one singular coefficient and apply the obtained results to solving the Dirichlet problem.
The Cauchy problem considered for a system of nonlinear integral equations of Volterra type that describes, with the use of Lagrangian coordinates, the motion of a finite mass of an ideal self-gravitating gas limited by a free boundary. A theorem of existence and uniqueness of the solution of the problem of motion of a rarefied gas in the space of infinitely differentiable functions is formulated and proved. The solution is constructed as a series with recursively calculated coefficients. The solutions obtained are used to study the dynamics of the free boundary.
The new two-phase model for compressible fluid flows in nonlinear poroelastoplastic media is presented. The derivation of the model is based on the symmetric hyperbolic thermodynamically compatible systems theory, which is developed with the use of the first principles and fundamental laws of irreversible thermodynamics. The governing PDEs form the first-order hyperbolic system and can be used for studying a wide variety of processes in saturated porous medium, including small-amplitude wave propagation. The theory predicts the three types of waves: fast and slow pressure waves and a shear wave, as it is in Biot’s model. The material constants of the model are fully determined by the properties of the solid and fluid constituents, and unlike Biot’s model do not contain empirical parameters. The governing PDEs for small-amplitude wave propagation in a saturated porous medium are presented and studied by means of the efficient numerical method, which is based on the finite difference fourth-order staggered-grid discretization.
We construct a fundamentally new, unsaturated numerical algorithm for solving the Dirichlet–Neumann problem for Laplace’s equation in smooth axisymmetric domains of a rather general shape. The distinctive feature of this algorithm is the absence of the leading error term, which, as a result, enables us to automatically adjust to arbitrary extra (extraordinary) supplies of smoothness of the sought solutions. In the case of -smoothness, the solutions are constructed with exponential estimate for error.
This article can be considered as the first version of a book which the author plans to write about half-range problems in operator theory. It consists of two parts. The first part is based on lectures which the author delivered at University of Calgary and Lomonosov Moscow State University. The main attention in this part is paid to the selection of waves which are involved in the formulation of the Mandelstamm radiation principle (the eigen-pairs, corresponding to the real eigenvalues) and to the factorization problems of self-adjoint and dissipative, quadratic and polynomial operator pencils. There is a dramatic difference between finite dimensional and infinite dimensional cases. It is shown that in the finite dimensional case the factorization problems can be solved completely. In the second part we consider abstract models for concrete problems of mechanics. We demonstrate the methods how concrete problems can be represented in an abstract form. The main results concern the factorization of elliptic operator pencils satisfying the resolvent growth condition in a double sector containing the real axis and the investigation of the semi-group properties of a divisor. Using Pontrjagin space methods we obtain a criterium for the stability in the celebrated Sobolev problem about a rotating top with a cavity filled with a viscous liquid.
We consider steady three-dimensional gravity–capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.
In this chapter we collect well-known concepts and results of classical potential theory that are necessary for the understanding of BEM.
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