Stationary surfaces in Lorentz–Minkowski space

Abstract

A linear Weingarten surface in Euclidean space R 3 is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ R. Such a surface is said to be hyperbolic when a 2 + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in R 3 that consists of surfaces with self-intersections whose generating curves are periodic.

Non-resonant boundary value problems with singular $${\phi}$$ -Laplacian operators

“…If κ = 0 (the case κ = 0 follows immediately by a direct integration), then it is proved, by a shooting argument, in Theorems 4.2 and 6.9 from [10] that the above Neumann problem has at least one solution. Now, using the fixed point operator given in Lemma 3 from [2] and a similar strategy like in the proof of Theorem 1, one can show that the Neumann problem 1).…”

“…Now, using the fixed point operator given in Lemma 3 from [2] and a similar strategy like in the proof of Theorem 1, one can show that the Neumann problem 1). For the geometric motivation of the above problems see the paper [10].…”

Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space

“…Let u ∈ K 0 be a critical point of I . This means that u solves the variational inequality (27) with h = g(·, u) (see (25)). But, then Lemma 3 ensures that u is a solution of (23).…”

“…The equation Mv = constant is then analyzed in [31], while Mv = f (v) with a general nonlinearity f is considered in [8]. On the other hand, motivated by the study of stationary surfaces in Minkowski space, in [25] the author consider the Neumann problem…”

Space-like surfaces with free boundary in the Lorentz–Minkowski space

“…Of course, these surfaces satisfy the boundary condition on the contact angle (see pictures in [15]). More recently, the first author has studied stationary surfaces in L 3 with some assumption on the symmetry of the surface, relating geometric quantities of the surface such as its height, area and volume: [16,17].…”