Let Ω ⊂ R n be a convex domain and let f : Ω → R be a positive, subharmonic function (i.e. ∆f ≥ 0). Thenwhere cn ≤ 2n 3/2 . This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies cn ≥ n − 1. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other Ω 2 ⊂ Ω 1 ⊂ R n :

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

Abstract. We construct analytic solutions to the Euler equations with an interface between two fluids, extending work of Duchon and Robert. We also show that the estimates of Duchon and Robert yield global analytic solutions to the Muskat problem with small initial data.

We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach the maximum, uniformly with respect to the eccentricity of the domain itself. This is achieved by obtaining quantitatively sharp second derivative estimates, which are consistent with the shape of the superlevel sets. In particular, we prove that the eigenfunction is concave (rather than merely log concave) in an entire superlevel set near its maximum. By estimating the mixed second and third derivatives partial derivatives of the eigenfunction, we also determine the rate at which a degree 2 Taylor polynomial approximates the eigenfunction itself.

We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex potential V on a domain Ω. We find two length scales L1 and L2, and an orientation of the domain Ω, which determine the shape of the level sets of the eigenfunction. As an intermediate step, we also establish bounds on the first eigenvalue in terms of the first eigenvalue of an associated ordinary differential operator. arXiv:1411.7353v1 [math.AP] 26 Nov 2014We can now introduce the crucial parameters L 1 and L 2 that will appear as important length scales in our study of the first eigenfunction u(x, y). For each c ≥ 0, let us define the sublevel sets of V (x, y) bySince V (x, y) is convex, these sublevel sets Ω c are convex subsets of Ω.Definition 1.4 (The Parameter L 1 ) Let L 1 be the largest value such that the sublevel set Ω L −2 1 has inner radius at least equal to L 1 .Remark 1.5 This definition is analogous to the definition of the parameter L from [J1] described above, and roughly speaking is equal to the largest length scale L 1 on which the potential increases by at most L −2 1 from its minimum.