A Modified Approach to Transconjuntival Levator Advancement Offering Intraoperative Options

Background The transconjunctival technique is a preferable and beneficial approach in mild to moderate blepharoptosis repair as without skin incision. However, accurate surgical manipulation of this method is greatly restricted by the poor intraoperative evaluation. Objectives To introduce a modified transconjunctival approach with flexible intraoperative adjustments in order to achieve more accurate ptosis correction. Methods By transconjunctival approach, the levator aponeurosis and the Müller’s muscle were folded using a square-like mattress suture for flexible adjustment and accurate correction. Results In 18 mild ptosis eyelids, 94.5% (17 eyelids) achieved adequate or normal correction. In 9 eyelids with moderate ptosis, 88.9% (8 eyelids) achieved adequate or normal correction. Amongst 24 ptosis patients, 23 (95.8%) achieved good or fair symmetry result. Conclusion We presented a modified transconjunctival technique for repair of mild to moderate ptosis, which is characterized by flexible intraoperative adjustments achieving both satisfying functional and aesthetic outcomes.

Semilinear elliptic equations involving mixed local and nonlocal operators

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ) s , with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

Symmetry-breaking in a generalized Wirtinger inequality

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the p-norm of the derivative among all functions whose q-norm is equal to 1 and whose (r − 1)-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes.