Communications in Contemporary Mathematics

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with non-positive curvature. We show this result by constructing a non-properly embedded minimal plane in H³. Hence, this gives a counterexample to Calabi–Yau conjecture for embedded minimal surfaces in negative curvature case.

In this paper, we consider the elliptic systems

This paper deals with a notion of Sobolev space W^{1, p} introduced by Bourgain, Brezis and Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by Ponce to obtain a Poincaré-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressed in terms of the intrinsic distance is equivalent to the L^p-norm of the intrinsic gradient, and provide a Poincaré-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in . The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

We give recursive relations to compute the rational Khovanov bigraded homology groups of three infinite families of non-quasi-alternating 3-column pretzel links. Based on this computations, we obtain an infinite family of homologically thin but not quasi-alternating knots.

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long-time behavior of the flow and the geometric meaning is investigated.

In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂u_t) - Δu - ϵΔ∂u_t = g with u(0, ⋅) = u₀ where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and . We show the existence of a critical value ϵ₀ >0 such that: if ϵ > ϵ₀, then the problem admits a unique solution; if ϵ = ϵ₀, the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ₀, then the solution is not unique in general. Passing to the limit with ϵ to 0⁺, we prove the existence (and uniqueness) of the solution of the Barenblatt differential inclusion Δu + g ∈ f(∂u_t) for a class of maximal monotone operators f. Next, we give an extension of the main result for a stochastic perturbation of the problem and we give some numerical illustrations of the Barenblatt and the Barenblatt–Sobolev equation.

We prove the existence of at least two T-periodic solutions, not differing from each other by an integer multiple of 2π, of the sine-curvature equation

We consider minimizers of the Ginzburg–Landau energy with pinning term and zero degree Dirichlet boundary condition. Without any assumptions on the pinning term, we prove that these minimizers do not develop vortices in the limit ε → 0. We next consider the specific case of a periodic discontinuous pinning term taking two values. In this setting, we determine the asymptotic behavior of the minimizers as ε → 0.