We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn . The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

The existence of a non-trivial bounded solution to the Dirichlet problem is established for a class of nonlinear elliptic equations involving a fully anisotropic partial differential operator. The relevant operator depends on the gradient of the unknown through the differential of a general convex function. This function need not be radial, nor have a polynomial-type growth. Besides providing genuinely new conclusions, our result recovers and embraces, in a unified framework, several contributions in the existing literature, and augments them in various special instances.

We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operator

is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality

holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

We give a new characterization, in the equicharacteristic case, of Teter rings by using Macaulay inverse systems. We extend the previous characterizations due to Teter, to Huneke and Vraciu and to Ananthnarayan et al., to any characteristic of the ground field and remove the hypothesis on the socle ideal. We construct and describe the variety parametrizing Teter covers and we show how to check if an Artin ring is Teter. If this is the case, we show how to compute a Teter cover.

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study is

where H 1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.

We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface that characterize these singularities.

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb ), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x 2 + y 5) has two topological orbits.