Asymptotics of the solution of a dirichlet problem in an angular domain with a periodically changing boundary

Translated from Matematicheskie Zametki, Vol. 49, No. 5, pp. 86–96, May, 1991.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

On the inverse problem for a logarithmic potential

Siberian Mathematical Journal -     

Generalized dirichlet problem in nonlinear potential theory

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA (x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting. Part of this research was performed in 1988–1989 while a visitor at Indiana University, Bloomington, Indiana.     

Representation of the solution to a boundary-value problem as a continuum integral

A Feynman integral representation is obtained for the solution of the Dirichlet and Neumann problems for a second-order parabolic equation in a Riemannian space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 6–11, 1974.In conclusion, the author wishes to thank V. S. Buslaev for his attention to this work.     

On the characteristic billiard and statements of the uniqueness problem of solution to the dirichlet problem for the vibrating string equation in a domain

The paper proposes a statement of the uniqueness problem of solution to the Dirichlet problem for the vibrating string equation in an unbounded domain containing vertical and/or horizontal strips that is convex with respect to families of characteristics of the vibrating string equation. The author proves a sufficient condition for the uniqueness of solution to the Dirichlet problem in the new statement based on the behavior of the F. John mapping realizing the characteristic billiard. Also, the author discusses the complex characteristic billiard and the statement of the uniqueness problem of solution to the Dirichlet problem for the vibrating string equation corresponding to it.     

The dirichlet problem in a domain whose boundary is partly degenerated

Summary In this paper we bring a formulation of the Dirichlet problem for strongly elliptic equations in domains vhose boundaries may include manifolds of different dimensions. It is shown that, under certain regularity conditions, this problem is equivalent to the generalized Dirichlèt problem, with respect to existence and uniquennes of solutions. This paper represents part of a thesis submitted to the Senate of the Technion, Israel Institute of Technology, in partial fulfillment of the requirements for the degree of Doctor of Science. The author wishes to tank ProfessorS. Agmon for his gudance and help in the preparation of this work.     

The dirichlet problem for a loaded mixed-type equation in a rectangular domain

We establish necessary and sufficient conditions for the unique solvability of the first boundary problem for a loaded equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We obtain a solution to the stated problem as the sum of the eigenfunction series for the corresponding one-dimensional problem with respect to eigenvalues. We prove the stability of the solution with respect to boundary functions.     

Asymptotic solution of a problem with one and two cusp points

The perturbation method and the Langer transform method are applied to obtain a second approximate solution of the problem with one and two cusp points. As an example, Langer type asymptotic formulas are derived for Bessel functions.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 42–50, 1989.     

Representation of the solution to a model problem in semiconductor physics

We study a mixed type problem for the Poisson equation arising in the modeling of charge transport in semiconductor devices [V. Romano, 2D simulation of a silicon MESFET with a non-parabolic hydrodynamical model based on the maximum entropy principle, J. Comput. Phys. 176 (2002) 70-92; A.M. Blokhin, R.S. Bushmanov, A.S. Rudometova, V. Romano, Linear asymptotic stability of the equilibrium state for the 2D MEP hydrodynamical model of charge transport in semiconductors, Nonlinear Anal. 65 (2006) 1018-1038]. Unlike well-studied elliptic boundary-value problems in domains with smooth boundaries (see, for example, [O.A. Ladyzhenskaya, N.N. Uralceva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1973; D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983]), our problem has two significant features: firstly, the boundary is not a smooth curve and, secondly, the type of boundary conditions is mixed (the Dirichlet condition is satisfied on the one part of the boundary whereas the Neumann condition on the other part). The well-posedness of the problem in Hölder and Sobolev spaces is proved. The representation of the solution to the problem is obtained in an explicit form.     

Unimprovable derivative estimates of a solution of the dirichlet problem for a biharmonic equation in bounded domains

Mathematical Notes -     

用单层位势求解尖角区域上的Dirichlet外问题

采用Kress变换以及处理第一类奇异核的积分方法,运用Nystrom方法利用单层位势求解尖角区域上的Dirichlet外问题.给出具体的算法和数值例子,通过数值例子可以看出用单层位势求解尖角区域上的Dirichlet外问题与用单双层结合求解所得的结果基本上一致,说明这种方法是有效的和可行的.     

Finite solutions to the inverse problem of the logarithmic potential

In this paper we deduce a new solution set of inverse problems of the logarithmic potential in the form of a logarithmic function of the ratio of equal degree polynomials and give examples of the finite solvability of inverse problems.     

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R ^m , m?>?2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.     

The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.