Anisotropic symmetrization and Sobolev inequalities on Finsler manifolds with nonnegative Ricci curvature

By using a sharp isoperimetric inequality and an anisotropic symmetrization argument, we establish Morrey-Sobolev and Hardy-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature; in some cases we also discuss the sharpness of these functional inequalities. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our results are also new in the Riemannian setting.Comment: 17 page

A bipolar Hardy inequality on Finsler manifolds

We establish a bipolar Hardy inequality on complete, not necessarily reversible Finsler manifolds. We show that our result strongly depends on the geometry of the Finsler structure, namely on the reversibility constant rF and the uniformity constant $l_F$. Our result represents a Finslerian counterpart of the Euclidean multipolar Hardy inequality due to Cazacu and Zuazua [3] and the Riemannian case considered by Faraci, Farkas and Krist\'aly [5]

Topological rigidity of compact manifolds supporting Sobolev-type inequalities

Let (M,g) be an n-dimensional (n≥3) compact Riemannian manifold with Ric(M,g)≥(n−1)g. If (M,g) supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere (Sn,g0), we prove that (M,g) is topologically close to (Sn,g0). Moreover, the Sobolev constants on (M,g) are precisely the optimal constants on the sphere (Sn,g0) if and only if (M,g) is isometric to (Sn,g0); in particular, the latter result answers a question of V.H. Nguyen

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold (M,g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M,g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings a la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments

Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Krist\'aly [\textit{Math.\ Ann.}, in press, doi.org/10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our results are also new in the Riemannian setting

Clinical features and management of oral lichen planus (OLP) with emphasis on the management of hepatitis C virus (HCV)-related OLP

Oral lichen planus (OLP) is a chronic inflammatory disease characterized by the occurrence of multiple, symmetrical lesions in the oral cavity. Hepatitis C virus (HCV) infection has been suggested as an etiological factor in OLP. The purpose of this review was to summarize the current literature regarding the treatment of OLP in patients with HCV infection. An electronic search of the PubMed database was conducted until January 2018, using the following keywords: OLP, HCV, corticosteroids, retinoids, immunomodulatory agents, surgical interventions, photochemotherapy, laser therapy, interferon, ribavirin, and direct-acting antivirals. We selected the articles focusing on the clinical features and treatment management of OLP in patients with/without HCV infection. Topical corticosteroids are considered the first-line treatment in OLP. Calcineurin inhibitors or retinoids can be beneficial for recalcitrant OLP lesions. Systemic therapy should be used in the case of extensive and refractory lesions that involve extraoral sites. Surgical intervention is recommended for isolated lesions. In patients with HCV, monotherapy with interferon (IFN)-α may either improve, aggravate or trigger OLP lesions, while combined IFN-α and ribavirin therapy does not significantly influence the progression of lesions. Direct-acting antiviral (DAA) therapy appears to be a promising approach in patients with HCV-related OLP, as it can improve symptoms of both liver disease and OLP, with fewer side effects. Nevertheless, for clinical utility of DAAs in OLP patients, further studies with larger sample sizes, adequate treatment duration, and long term follow-up are required

Enhancement of bone consolidation using high-frequency pulsed electromagnetic fields (HF-PEMFs): An experimental study on rats

In vitro studies showed that high-frequency pulsed electromagnetic fields (HF-PEMFs) increase the activity/expression of early and late osteogenic markers and enhance bone mineralization. The main aim of this study was to investigate the in vivo effects of HF-PEMFs on fracture healing using a rat model. A femur fracture was established by surgery in 20 male Wistar rats. Titanium nails were implanted to reduce and stabilize the fracture. After surgery, 20 rats were equally divided into untreated control and treated group (from the first postoperative day HF-PEMFs at 400 pulses/sec [pps] were applied for 10 minutes/day, for two weeks). Quantitative and qualitative assessment of bone formation was made at two and eight weeks following surgery and included morphological and histological analysis, serological analysis by ELISA, micro-computed tomography (micro-CT), and three-point bending test. At two weeks in HF-PEMF group, soft callus was at a more advanced fibrocartilaginous stage and the bone volume/total tissue volume (BV/TV) ratio in the callus area was significantly higher compared to control group (p = 0.047). Serum concentration of alkaline phosphatase (ALP) and osteocalcin (OC) was significantly higher in HF-PEMF group (ALP p = 0.026, OC p = 0.006) as well as the mechanical strength of femurs (p = 0.03). At eight weeks, femurs from HF-PEMF group had a completely formed woven bone with dense trabeculae, active bone marrow, and had a significantly higher BV/TV ratio compared to control (p = 0.01). HF-PEMFs applied from the first postoperative day, 10 minutes/day for two weeks, enhance bone consolidation in rats, especially in the early phase of fracture healing