共查询到20条相似文献,搜索用时 31 毫秒
1.
声波的散射问题中,如果散射体由不可穿透障碍物和可穿透裂缝两部分组成,障碍物表面分别满足第一类和第三类边界条件,裂缝两边满足不同的第二类边界条件,通过位势理论,可以将此混合问题转化为边界积分方程,通过Fredholm算子理论可以得到这个边界积分方程解的存在性和唯一性,从而获得原问题解的存在和唯一性. 相似文献
2.
In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B in R2 with boundary ∂B that consists of two disjoint closed curves Γ and Γ0. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ are obtained by using Riesz–Fredholm theory. 相似文献
3.
W. Arendt 《Journal of Differential Equations》2011,251(8):2100-2124
We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D0 on L2(Γ) by form methods. The operator −D0 is self-adjoint and generates a contractive C0-semigroup S=(St)t>0 on L2(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H1(Ω) which Ω may or may not have. 相似文献
4.
Jussi Behrndt 《Journal of Differential Equations》2010,249(11):2663-319
In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L2(Ω)⊕(L2m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations. 相似文献
5.
Guozheng Yan 《Mathematical Methods in the Applied Sciences》2011,34(15):1813-1821
We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R2 as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
6.
We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part ΓI⊂∂Ω of the boundary of a bounded, planar domain Ω from one Cauchy data pair (u,∂u/∂ν) of a harmonic potential u in Ω collected on an accessible boundary subset ΓA⊂∂Ω. We prove Fréchet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem. 相似文献
7.
Let Ω⊂RN(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|∇u|q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as dist(x,∂Ω)→0,f is Γ-varying at ∞. Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution. 相似文献
8.
A.T. Peplow S.N. Chandler-Wilde 《Journal of Mathematical Analysis and Applications》2008,345(1):305-321
The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem. 相似文献
9.
Ross G. Pinsky 《Journal of Functional Analysis》2003,200(1):177-197
Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆Rd with oblique reflection at ∂D if D≠Rd. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities. 相似文献
10.
S. Engleder 《Journal of Mathematical Analysis and Applications》2007,331(1):396-407
In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations. 相似文献
11.
Takeshi Taniguchi 《Journal of Differential Equations》2011,251(12):3329-3362
In this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2-dimensional non-Lipschitz Navier-Stokes equation perturbed by the cylindrical Wiener process W(t) in a bounded or unbounded domain D with the smooth boundary ∂D or D=R2:
12.
Rym Chemmam Habib Mâagli Syrine Masmoudi 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(5):1555-1576
The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)α/2, 0<α<2, in a bounded C1,1 domain D in Rn. In particular, we introduce a new Kato class Kα(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian. 相似文献
13.
14.
We study the boundary value problems for Monge-Ampère equations: detD2u=e−u in Ω⊂Rn, n?1, u|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD2u=e−tu in Ω, u|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. 相似文献
15.
A waveguide occupies a domain G in ? n+1, n ? 1, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter µ. As an approximation to a row of the scattering matrix S(µ) we suggest a minimizer of a quadratic functional J R (·, µ). To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance R, the waveguide outlets to infinity. It is proved that, if a finite interval [µ1, µ2] of the continuous spectrum contains no thresholds, then, as R → ∞, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to µ ∈ [µ1, µ2]. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity. 相似文献
16.
Let Γ denote a smooth simple curve in ? N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions. 相似文献
17.
In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc in R2 as the cross section. We assume that the arc is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance λ. Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of a solution to the system by using Fredholm theory. 相似文献
18.
Tatiana V. Savina 《Journal of Mathematical Analysis and Applications》2010,370(2):716-1163
The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R2 subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ0:={y=0}, reflections are point-to-point when the given on Γ0 conditions are u=n∂u=0, u=Δu=0 or n∂u=n∂Δu=0, and point to a continuous set when u=n∂Δu=0 or n∂u=Δu=0 on Γ0. 相似文献
19.
We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method. 相似文献
20.
Ahmed Hamydy 《Journal of Mathematical Analysis and Applications》2010,371(2):534-169
Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions. 相似文献