3,758 research outputs found

    Concentration of Solutions for a Singularly Perturbed Neumann Problem in non smooth domains

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    We consider the equation −ϵ2Δu+u=up-\epsilon^{2}\Delta u + u = u^ {p} in a bounded domain Ω⊂R3\Omega\subset\R^{3} with edges. We impose Neumann boundary conditions, assuming 1<p<51<p<5, and prove concentration of solutions at suitable points of ∂Ω\partial\Omega on the edges.Comment: 24 pages. Second Version, minor changes. To appear in Annales de l'Institut Henri Poincar\'e - Analyse non lin\'eair

    hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

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    Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R3\mathbb{R}^3 exhibit singular behavior from the edges and corners. We present quasi-optimal hphp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hphp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.Comment: 41 pages, 11 figure

    Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems

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    When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -- the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue -- whether one or both of the steady weak and strong shocks are physically admissible solutions -- has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.Comment: 19 pages; 8 figures; accepted by Science China Mathematics on February 22, 2017 (invited survey paper). doi: 10.1007/s11425-016-9045-

    Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

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    We provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of the criteria involve a comparison with the spectral radii of some associated linear operators. We apply our results to prove the existence of multiple nonzero radial solutions for some systems of elliptic boundary value problems subject to nonlocal boundary conditions. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1404.139
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