Communications in Contemporary Mathematics

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition

We consider the following prescribed $Q$-curvature problem:

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot |$ and the Cheeger constant $h$ of planar sets. To do so, we study the so-called Blaschke–Santaló diagram of the triplet $(P,h,|\cdot |)$ for different classes of domains: simply connected sets, convex sets and convex polygons with at most $N$ sides. We completely determine the diagram in the latter cases except for the class of convex $N$-gons when $N\ge 5$ is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.

The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer $r$ we prove that if $G$ is a profinite group in which the centralizer of every nontrivial element has rank at most $r$, then $G$ is either a pro-$p$ group or a group of finite rank. Further, if $G$ is not virtually a pro-$p$ group, then $G$ is virtually of rank at most $r+1$.

In this paper, we consider the following Schrödinger–Poisson system

We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) $\beta$-Hölder continuous for some $\beta \in (1-s,1]$, where $s\in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $\beta$-Hölder continuous.

The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.

The chemotaxis system

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi: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.

Let $X$ be a compact connected orientable Cauchy–Riemann (CR) manifold with the action of a compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin–Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of $0$ by the momentum map. We also show that “quantization commutes with reduction” for Sasakian manifolds.