A MODEL OF COLUMNAR JOINTING

We propose to use nonlinear elasticity to model the propagation of cracks in cooling lava. In particular, our work aims to understand the enigmatic fracture process that leads to the formation of column joints. Column joints are plane fracture surfaces fragmenting a basalt flow into prismatic columns. These columns are characterized by their polygonal cross-section and usually exhibit a strikingly high degree of regularity. We present a variational model with the assumption that the fracture process seeks to minimize the total energy of the system. The expression for the elastic energy is simplified and the configuration of minimal energy is analytically determined by a rigorous derivation. Further, we study the behavior of the energy under Steiner and Schwarz symmetrization of the column cross-section. In particular, we prove that the minimum of the energy among all possible convex and bounded column cross-sections B ⊂ ℝ2 is attained when B is the two-disk. Our results thus give strong evidence supporting the conjecture that the minimal energy is attained for a regular hexagon when the column cross-section is further required to tile the plane.

Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

Extensions of the Hardy-Littlewood inequalities for Schwarz symmetrization

For a class of functionsH:(0,∞)×ℝ+2→ℝ, including discontinuous functions of Carathéodory type, we establish that∫ℝNH(|x|,u(x),v(x))dx≤∫ℝNH(|x|,u*(x),v*(x))dx, whereu*(x)andv*(x)denote the Schwarz symmetrizations of nonnegative functionsuandv.