摘要:

AbstractWe study the large‐time asymptotics for solutionsu(x,t) of the wave equation with Dirichlet boundary data, generated by a time‐harmonic force distribution of frequency ω, in a class of domains with non‐compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω0≔ ℝ2× (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. In particular, we prove thatuis bounded ast→ ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω0at the frequencies ω = πk(k= 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU+ κ2U= −fin Ω,U= 0 on ∂Ω is reduc

ISSN:0170-4214    

DOI:10.1002/mma.1670120102

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

摘要:

AbstractThe paper considers Dirichlet (or Neumann type) boundary value problems of generalized potential theory\documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm d}\alpha = f,\;\delta \varepsilon \left(\alpha \right) = g\,{\rm in}\,M, $$\end{document}\documentclass{article}\pagestyle{empty}\begin{document}$$ \alpha = 0\,{\rm on}\,\partial M $$\end{document}on Lipschitz manifolds with boundary. Here ϵ denotes a permissible non‐linearity. The existence theory is developed in the framework of monotone operators. The approach covers a variety of applications including fluid dynamics and electro‐ and magneto‐statics. Only fairly weak regularity assumptions are required (e.g. Lipschitz boundary,L∞‐coefficients). As a by‐product we obtain a non‐linear Hodge theorem generalizing a result by L. M. Sibner and R. J. Sibner (‘A non‐linear Hodge‐DeRham theorem’,Acta Ma

ISSN:0170-4214    

DOI:10.1002/mma.1670120103

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

摘要:

AbstractWe consider operator‐valued Riccati initial‐value problems of the formR′(t) +TR(t) +R(t)T=TA(t) +TB(t)R(t) +R(t)TC(t) +R(t)TD(t)R(t),R(0) =R0. HereAtoDandR0have values as non‐negative bounded linear operators inL1(μ), where μ is a finite measure, andTis a closed non‐negative operator inL1(μ) satisfying additional technical conditions. For such problems the notion ofstrongly mildsolutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomoge

ISSN:0170-4214    

DOI:10.1002/mma.1670120104

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

摘要:

AbstractThe scattering of a ‘plane wave’ off a submerged body situated in an ocean of finite depth is investigated. The index of refraction is considered to be depth‐independent. It is shown that the far field is not unique; hence, the problem of determining the shape of an object from its far field is not well‐posed. If solutions are sought among a restricted class of problems the ‘dense set’ property implies that the problem can be made

ISSN:0170-4214    

DOI:10.1002/mma.1670120105

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

摘要:

AbstractThis paper is concerned with the effect of perturbing Burgers' equation by a small term ϵ2Utt. It is shown by means of an energy estimate that the solution of Burgers' equation provides a uniformO(ϵ) approximation of the solution of the full hyperbolic problem. Existence and uniqueness of classical solutions for both problems is proved. A related linear problem is first addressed using the Faedo–Galerkin method to obtain key estimates. Important for the hyperbolic problem is the introduction of an ϵ‐dependent energy in order to track the order‐ϵ behaviour of various higher‐order derivatives. Subsequent use of Schauder technique and Banach contraction mapping principle yields solutions of the semiline

ISSN:0170-4214    

DOI:10.1002/mma.1670120106

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

摘要:

AbstractIt is shown that there exist domains Ω ⊂ ℝN, which outside of some ball coincide with the strip ℝN− 1× (0, π) and for which the Dirichlet Laplacian – Δ has eigenvalues within the subinterval (1, 4) of the essential sp

ISSN:0170-4214    

DOI:10.1002/mma.1670120107

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY

ISSN:0170-4214    

DOI:10.1002/mma.1670120101

出版商:John Wiley&Sons, Ltd    

年代:1990

数据来源: WILEY