Generalization of the Neumann problem to harmonic functions outside cuts on the plane

We consider a boundary value problem for harmonic functions outside cuts on the plane. The jump of the normal derivative and a linear combination of the normal derivative on one side with the jump of the unknown function are given on each cut. The problem is considered with three conditions at infinity, which lead to distinct results on the existence and number of solutions. We obtain an integral representation of the solution in the form of potentials whose density satisfies a uniquely solvable Fredholm integral equation of the second kind.     

The jump problem for the laplace equation

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified of the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single-layer and angular potentials. The singularities at the ends of the cuts are investigated.     

The jump problem for the Helmholtz equation and singularities at the edges

The boundary value problem for the Helmholtz equation outside several cuts in a plane is studied. The jump of the solution of the Helmholtz equation and the jump of its normal derivative are specified of the cuts. The unique solution of this problem is constructed in the explicit form by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.     

The modified jump problem for the Laplace equation and singularities at the tips

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified of the cuts. The unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The singularities at the ends of the cuts are investigated.     

On the electric current from electrodes in a magnetized semiconductor film

The problem on the electric current from electrodes in a magnetizedsemiconductor film is reduced to the skew derivative problemfor the Laplace equation outside cuts in a plane. The problemfor the Laplace equation is studied under different conditionsat infinity, which have a certain physical meaning. With thehelp of potential theory, the skew derivative problem is reducedto a Fredholm integral equation of the second kind, which isuniquely solvable. The Neumann problem for the Laplace equationin the exterior of cuts in a plane is a particular case of ourproblem.     

The modified jump problem for the Helmholtz equation

The boundary value problem for the Helmholtz equation outside several cuts in a plane is studied. The 2 boundary conditions are given on the cuts. One of them specifies the jump of the unkown function. Another one contain the jump of the normal derivative of an unknown function and a limit value of this function on the cuts. The unique solution of this problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.

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Sunto Nel presente lavoro si studia il problema al contorno per l'equazione di Helmholtz all'esterno di più tagli nel piano. Le due condizioni al contorno sono assegnate sui tagli. Una di queste prescrive il salto della funzione incognita, l'altra contiene il salto della derivata normale di una funzione incognita ed un valore limite di questa funzione sui tagli. La soluzione univoca di questo problema è ricondotta all'equazione di Fredholm di seconda specie ed indice zero, univocamente risolubiles, per mezzo dei potenziali di singolo strato ed angolare. Si studiano, inoltre, le singolarità agli estremi dei tagli.

     

Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.     

A problem for the Helmholtz equation outside cuts on the plane with the dirichlet condition and the oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the Helmholtz equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition with pure imaginary coefficient of the tangential derivative is posed on the other side. We prove the uniqueness of the solution. The solvability of the problem is proved for the case in which the above-mentioned pure imaginary coefficient is less than unity in absolute value. In this case, we obtain an integral representation of the solution of the problem in the form of potentials. The densities of the potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. The boundary value problem considered here generalizes the mixed Dirichlet-Neumann problem.     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Integral-equation method in the mixed oblique derivative problem for harmonic functions outside cuts on the plane

The mixed problem for the Laplace equation outside cuts on the plane is considered. As boundary conditions, the value of the desired function on one side of each of the cuts and the value of its oblique derivative on the other side are prescribed. This problem generalizes the mixed Dirichlet-Neumann problem. By using the potential method, the problem reduces to a uniquely solvable Fredholm integral equation of the second kind. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 115–135, 2006.     

Boundary value problem for the Laplace equation outside cuts on the plane with different conditions of the third kind on opposite sides of the cuts

We consider a boundary value problem for the Laplace equation outside cuts on a plane. Boundary conditions of the third kind, which are in general different on different sides of each cut, are posed on the cuts. We show that the classical solution of the problem exists and is unique. We obtain an integral representation for the solution of the problem in the form of potentials whose densities are found from a uniquely solvable system of Fredholm integral equations of the second kind.     

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

Longitudinal Shear of Layered Anisotropic Media with Band-Type Inhomogeneities

Using the method of jump functions, we solve the antiplane problem of elasticity theory for a stack of anisotropic strips containing plane band-type inhomogeneities. We model these inclusions by jumps of the stress vector and the derivative of the displacement vector at the middle surfaces. Applying the Fourier integral transformation, we obtain the dependence of the components of the stress tensor and displacement vector on the external load and unknown jump functions. Taking into account the conditions of the interaction between a thin inclusion and an anisotropic medium, we reduce the problem to a system of singular integral equations for the jump functions. A specific example is considered as well.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

Solvability of the linearized problem on the motion of a drop in a liquid flow

In the paper one establishes the solvability in the anisotropic Sobolev-Slobodetskii spaces of the linear problem, generated by the problem of the nonstationary motion of a drop in a fluid medium. In the formulation of the problem one takes into account the surface tension, which occurs in the noncoercive integral term in the conditions for the jump of the normal stresses. In the general case the velocity vector need not be solenoidal but its divergence must be represented in a special form. The proof of the solvability is carried out first in the Sobolev-Slobodetskii spaces and is based on a priori estimates for the solutions in these spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 53–65, 1989.     

On the properties of solutions of the harmonic Dirichlet problem in a two-dimensional domain with cuts

We consider the Dirichlet problem for the Laplace equation in a plane domain with smooth cuts of arbitrary form for the case in which the solution is not continuous at the endpoints of the cuts. We present a well-posed statement of the problem, prove the existence and uniqueness theorems for the classical solution, obtain an integral representation of the solution, and use it to analyze the properties of the solution. We show that, as a rule, the Dirichlet problem in this setting has no weak solutions, even though there exists a classical solution.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated.     

The heat-conduction problem for a shell with a system of diathermanous cuts

Using the two-dimensional Fourier transform and the elementary theory of distributions, we solve the heat-conduction problem for shells with a system of diathermanous cuts. We take account of heat exchange according to Newton's law on the lateral surfaces of the shells. For a spherical shell with two cuts of identical length we carry out numerical studies of the influence of the thermophysical properties of one cut on the jump in temperature of the adjacent cut. Three figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 86–89.