Communications in Contemporary Mathematics

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is $n\ge 7$ and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ${ℝ}^n$ with respect to upper Ahlfors regular measures $\nu$, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $\nu$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also present an application to the Galois theory of differential equations depending on a discrete parameter.

Let $\pi :X\to {\mathbb{P}}^n$ be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibers in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.

We study the positivity of the first Chern class of a rank $r$ Ulrich vector bundle $𝜀$ on a smooth $n$-dimensional variety $X\subseteq {\mathbb{P}}^N$. We prove that $c_1(𝜀)$ is very positive on every subvariety not contained in the union of lines in $X$. In particular, if $X$ is not covered by lines we have that $𝜀$ is big and $c_1{(𝜀)}^n\ge r^n$. Moreover we classify rank $r$ Ulrich vector bundles $𝜀$ with $c_1{(𝜀)}^2=0$ on surfaces and with $c_1{(𝜀)}^2=0$ or $c_1{(𝜀)}^3=0$ on threefolds (with some exceptions).

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local $P{C}^1$ structure.

We investigate the well-posedness in the generalized Hartree equation $i{u}_t+\mathrm{\Delta }u+(|x{|}^{-(N-\gamma )}\ast |u{|}^p)|u{|}^{p-2}u=0$, $x\in {ℝ}^N$, $0<\gamma <N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

Smale–Barden manifolds $M$ are classified by their second homology $H_2(M,\mathbb{Z})$ and the Barden invariant $i(M)$. It is an important and difficult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all $M$ with $H_2(M,\mathbb{Z})={\mathbb{Z}}^k\oplus ({\oplus }_{i=1}^r{\mathbb{Z}}_{m_i}^{2g_i})$ and $i(M)=0,\infty ,$ provided that $k\ge 1$, $m_i\ge 2,g_i\ge 1$, $m_i$ are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

In this paper, we introduce a new functional for the conformal spectrum of the conformal Laplacian on a closed manifold $M$ of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds. The functional introduced here has been investigated in the recent work [S. Perez-Ayala, Extremal eigenvalues of the conformal Laplacian under Sire–Xu normalization, preprint (2019); arXiv:2011.06018].

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space $H^N$, we have

• The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in $H^N$ raised to some power $\alpha >0$ is strictly concave;
• Let $\mathrm{\Gamma }$ be the heat kernel on $H^N$. Then $\mathrm{\Gamma }(\cdot ,y,t)$ is strictly log-concave in $H^N$ for $y\in H^N$ and $t>0$.

To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang–Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${𝒢}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so-called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.