By Albrecht Pietsch

ISBN-10: 3540056440

ISBN-13: 9783540056447

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**Example text**

Let X denote a vector field perpendicular at Σ. Then the Leibnitz rule and the classical formula for the variation of the area yields (104) d E(zΣ ) = εk dX Σ d θk V dX d dσ dX dσ + V θk = εk ∇X V θk − V θk H · X dσ, Σ where H denotes the mean-curvature vector of Σ. We recall that, if x1 , . . , xk are local coordinates orthonormal at a point p ∈ Σ, and if F : x1 , . . , xk → Rn denotes the immersion of Σ (F (0) = p), then H(p) is given by k (105) H(p) = i=1 ∂2F |x=0 ∂x2i ⊥ . Here ⊥ denotes the component orthogonal to Σ.

On interacting bumps of semi-classical states of nonlinear Schrdinger equations. Adv. Diff. Eq. 5, no. 7-9, 899–928 (2000). : A bifurcation theorem for potential operators. J. Funct. Anal. 77, 1-8 (1988). : Adiabatic limits for some Newtonian systems in R n . Asympt. Anal. 25, 149-181 (2001). : Boundary concentration phenomena for a singularly perturbed elliptic problem. To appear on Comm. Pure Appl. Math. : Large amplitude stationary solutions to a chemotaxis systems. J. : Diffusion, cross-diffusion, and their spike-layer steady states.

From formula (104) the conditions of stationarity of Σ becomes (106) θk ∇⊥ V = V H, where ∇⊥ denotes the component of V normal to Σ. Note that when Σ is a k-dimensional sphere then, by (105), conditions (106) and Mk (r) = 0 coincide. We conjecture that, under suitable non-degeneracy assumptions, (106) is a sufficient condition for the existence of solutions of (1) concentrating on Σ. 2, requires a more delicate analysis. In particular, wee suspect that concentration occurs in general along sequences εj → 0 as in [23].

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