Abstract. For a domain Ω contained in a hemisphere of the n-dimensional sphere S n we prove the optimal result λ 2 /λ 1 (Ω) ≤ λ 2 /λ 1 (Ω ) for the ratio of its first two Dirichlet eigenvalues where Ω , the symmetric rearrangement of Ω in S n , is a geodesic ball in S n having the same n-volume as Ω. We also show that λ 2 /λ 1 for geodesic balls of geodesic radius θ 1 less than or equal to π/2 is an increasing function of θ 1 which runs between the value (j n/2,1 /j n/2−1,1 ) 2 for θ 1 = 0 (this is the Euclidean value) and 2(n + 1)/n for θ 1 = π/2. Here j ν,k denotes the kth positive zero of the Bessel function Jν (t). This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of S n and having a fixed value of λ 1 the one with the maximal value of λ 2 is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ 2 /λ 1 . Various other results for λ 1 and λ 2 of geodesic balls in S n are proved in the course of our work.

We give an iterative method to estimate the disturbance of semi-wavefronts of the equation:As a consequence, we show the exponential stability, with an unbounded weight, of semiwavefronts with speed c > 2 √ 2 and h > 0. Under the same restriction of c and h, the uniqueness of semi-wavefronts is obtained.If φ c (+∞) = 1 semi-wavefronts are called wavefronts.If h = 0, the questions on existence, uniqueness, geometry and stability of wavefronts have been satisfactory responded (see, e.g.[11] and [15] and references therein). In this case the general conclusions are: (i) semi-wavefronts are indeed monotone wavefronts existing for all c ≥ 2, (ii) two wavefronts with same speed are unique up to translations and (iii) wavefronts are stable under suitable perturbations.However, for h > 0 it has been only recently established the existence of semiwavefronts on the domain {(h, c) ∈ R 2 : h ≥ 0 and c ≥ 2} (see [7] and [2]). The

A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in R d has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of R d , in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.

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We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of R 2 . Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegő type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences 1 supporting the existence of a Faber-Krahn type inequality.