Morrey’s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One Convex Energy Functions

Rank-one Convex Energy under Certain Geometric Flows

hold for all Lipschitz functions w : Ω̄ → R with w|∂Ω = 0. Here Ω is a bounded Lipschitz domain in R and ∇w = (w xα) is the Jacobi matrix of w defined pointwise on Ω. There has been a considerable amount of work on Morrey’s quasiconvexity in the calculus of variations and nonlinear elasticity [3, 5, 8, 9, 10]. Integral estimates like (1.1) are also important for other problems in geometric mappi...

The “hot Spots” Conjecture for Nearly Circular Planar Convex Domains

We prove the “hot spots” conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying diam(Ω)2/|Ω| < 1.378. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points on ∂Ω. When Ω is a disk, diam(Ω)2/|Ω| t 1.273. Hence, the above condition indicates that Ω is a nea...

The Rank One Abelian Stark Conjecture

On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph G = (V, E) into a metric space (X, d) is a function x : V (G) → X such that in the embedding for every pair of non-adjacent vertices x(s), x(t) there exists another vertex x(u) adjacent to x(s) which is closer to x(t) than x(s). This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that ever...

The Log-Rank Conjecture for Read- k XOR Functions

The log-rank conjecture states that the deterministic communication complexity of a Boolean function g (denoted by D(g)) is polynomially related to the logarithm of the rank of the communication matrixMg whereMg is the communication matrix defined byMg(x, y) = g(x, y). An XOR function F : {0, 1} × {0, 1} → {0, 1} with respect to f : {0, 1} → {0, 1} is a function defined by F (x, y) = f(x⊕ y). I...