For the positive solutions of the Gross-Pitaevskii systemwe prove that L ∞ -boundedness implies C 0,α -boundedness, uniformly as β → +∞, for every α ∈ (0, 1). Moreover we prove that the limiting profile, as β → +∞, is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given.MSC : 35B40, 35B45, 35J55.
We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary conditions in a planar domain. We analyse the behavior of its eigenvalues as the singular pole moves in the domain. For any value of the circulation we prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the Laplacian as the pole approaches the boundary. We show that the magnetic eigenvalues depend in a smooth way on the position of the pole, as long as they remain simple. In case of half-integer circulation, we show that the rate of convergence depends on the number of nodal lines of the corresponding magnetic eigenfunction. In addition, we provide several numerical simulations both on the circular sector and on the square, which find a perfect theoretical justification within our main results, together with the ones in [5].
Given ρ > 0, we study the elliptic problem find (U, λ) ∈ H 1 0 (B1) × R such thatthe unitary ball and p is Sobolev-subcritical. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about ρ, N and p) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every ρ (in the existence range) when p is L 2 -critical and subcritical, i.e. 1 < p ≤ 1 + 4/N , while they are stable for almost every ρ in the L 2 -supercritical regime 1 + 4/N < p < 2 * − 1. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L p+1 -norm among functions having prescribed L 2 and H 1 0 -norm. arXiv:1307.3981v1 [math.AP]
In this paper we consider a stationary Schrödinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.MSC : 35J10, 35J20, 35P05, 49Q10.
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