The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.

In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

The strain gradient elasticity theory including only three independent material length scale parameters has been proposed by Zhou et al. to explain the size effect phenomena in micro scales. In this paper, the general formulations of strain gradient elasticity theory in orthogonal curvilinear coordinates are derived, and then are specified for the cylindrical and spherical coordinates for the convenience of applications in cases where orthogonal curvilinear coordinates are suitable. Two basic problems, one is the twist of a cylindrical bar and the other is the radial deformation of a solid sphere, are analyzed under the cylindrical and spherical coordinates, respectively. The results reveal that only the material length scale parameter l2 enters the torsion problem, while completely disappears in the problem of radial deformation of a sphere. The size effect of radial deformation of a solid sphere is controlled by the material length scale parameters l1 and l2. In addition, for the incompressible solid sphere especially, only the material length scale parameter l1 enters this radial deformation problem by neglecting the strain gradient terms associated with hydrostatic strains. Predictably, the present paper offers an alternative avenue for measuring the three independent material length scale parameters from bar twisting and sphere expansion tests.

This paper studies outflow of a light fluid from a point source, starting from an initially spherical bubble. This region of light fluid is embedded in a heavy fluid, from which it is separated by a thin interface. A gravitational force directed radially inward toward the mass source is permitted. Because the light inner fluid is pushing the heavy outer fluid, the interface between them may be unstable to small perturbations, in the Rayleigh–Taylor sense. An inviscid model of this two-layer flow is presented, and a linearized solution is developed for early times. It is argued that the inviscid solution develops a point of infinite curvature at the interface within finite time, after which the solution fails to exist. A Boussinesq viscous model is then presented as a means of quantifying the precise effects of viscosity. The interface is represented as a narrow region of large density gradient. The viscous results agree well with the inviscid theory at early times, but the curvature singularity of the inviscid theory is instead replaced by jet formation in the viscous case. This may be of relevance to underwater explosions and stellar evolution.