The Dirichlet Problem for the Equation Δu - k2u = 0 in the Exterior of Nonclosed Lipschitz Surfaces
نویسنده
چکیده
We study the Dirichlet problem for the equation Δu − k2u = 0 in the exterior of nonclosed Lipschitz surfaces in R. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.
منابع مشابه
The Helmholtz Equation on Lipschitz Domains
We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ( +k)u = 0 on a Lipschitz domain for all wave number k 2C with Imk 0. Following the approach for the case of smooth boundary [3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothness...
متن کاملThe Helmholtz Equation on LipschitzDomainsChangmei
We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ((+k 2)u = 0 on a Lipschitz domain for all wave number k 2C with Im k 0. Following the approach for the case of smooth boundary 3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothne...
متن کاملThe Stationary Navier-stokes System in Nonsmooth Manifolds: the Poisson Problem in Lipschitz and C Domains
In this paper we study the linearized version of the stationary Navier-Stokes equations on a fixed subdomain Ω of a smooth, compact Riemannian manifold M . Let Tr denote the trace on ∂Ω. With Def standing for the deformation tensor and with d denoting the exterior derivative operator on M , set L = 2 Def∗Def, δ = d∗. We consider the Dirichlet problem for the (modified) Stokes system L...
متن کاملA RESEARCH NOTE ON THE SECOND ORDER DIFFERENTIAL EQUATION
Let U(t, ) be solution of the Dirichlet problem y''+( t-q(t))y= 0 - 1 t l y(-l)= 0 = y(x), with variabIe t on (-1, x), for fixed x, which satisfies the initial condition U(-1, )=0 , (-1, )=1. In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigen values has been investigated . Furthermore, the leading term of the asymptotic formula for ...
متن کاملWavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering
We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-to-Dirichlet map when one of c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2013 شماره
صفحات -
تاریخ انتشار 2013