Journal ArticleDOI
Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains
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In this paper, it was shown that the concentration of solutions occurs at some geodesics of ∂Ω when ǫ → 0, where ∆ ≥ 0.Abstract:
We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain \(\Omega \subseteq \mathbb{R}^3 \) and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.read more
Citations
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Journal ArticleDOI
Concentration on curves for nonlinear Schrödinger Equations
TL;DR: In this article, the existence of a solution concentrating along the whole of Γ, exponentially small in e at any positive distance from it, provided that e is small and away from certain critical numbers.
Journal ArticleDOI
On Bound States Concentrating on Spheres for the Maxwell--Schrödinger Equation
Teresa D'Aprile,Juncheng Wei +1 more
TL;DR: This system describes standing waves for the nonlinear Schrodinger equation interacting with the electrostatic field: the unknowns v and $\phi$ represent the \emph{wave function} associated to the particle and the electric potential, respectively.
Journal ArticleDOI
On the number of interior peak solutions for a singularly perturbed Neumann problem
TL;DR: In this article, the singularly perturbed Neumann problem is considered and a solution with energies in the order of ϵN−m is given for each m ∈ (0, N).
Journal ArticleDOI
Constant mean curvature hypersurfaces condensing on a submanifold
TL;DR: In this paper, Mahmoudi et al. proved the existence of local foliation by constant mean curvature hypersurfaces condensing to a point (which is required to be a nondegenerate critical point of the scalar curvature function).
Journal ArticleDOI
Concentration on minimal submanifolds for a singularly perturbed Neumann problem
Fethi Mahmoudi,Andrea Malchiodi +1 more
TL;DR: In this paper, the authors considered the problem of finding a solution along k-dimensional minimal submanifolds of ∂Ω, for N ⩾ 3 and for k ∈ { 1, …, N − 2 }.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
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The Chemical Basis of Morphogenesis
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Book
Differential geometry of curves and surfaces
TL;DR: This paper presents a meta-geometry of Surfaces: Isometrics Conformal Maps, which describes how the model derived from the Gauss Map changed over time to reflect the role of curvature in the model construction.
Journal ArticleDOI
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
Journal ArticleDOI
A theory of biological pattern formation.
Alfred Gierer,Hans Meinhardt +1 more
TL;DR: It is shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue, and the theory is applied to quantitative data on hydra and is shown to account for activation and inhibition of secondary head formation.
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