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Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on ∂ C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Sharp bounds for the intersection of nodal lines with certain curves

Junehyuk Jung (2014)

Journal of the European Mathematical Society

Let Y be a hyperbolic surface and let φ be a Laplacian eigenfunction having eigenvalue - 1 / 4 - τ 2 with τ > 0 . Let N ( φ ) be the set of nodal lines of φ . For a fixed analytic curve γ of finite length, we study the number of intersections between N ( φ ) and γ in terms of τ . When Y is compact and γ a geodesic circle, or when Y has finite volume and γ is a closed horocycle, we prove that γ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between N ( φ ) and γ is O ( τ ) . This bound is sharp.

Sharp spectral asymptotics and Weyl formula for elliptic operators with Non-smooth Coefficients-Part 2

Lech Zielinski (2002)

Colloquium Mathematicae

We describe the asymptotic distribution of eigenvalues of self-adjoint elliptic differential operators, assuming that the first-order derivatives of the coefficients are Lipschitz continuous. We consider the asymptotic formula of Hörmander's type for the spectral function of pseudodifferential operators obtained via a regularization procedure of non-smooth coefficients.

Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer (2013)

Annales de l’institut Fourier

In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure...

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