Communications in Contemporary Mathematics

In this paper, we study a highly nonlocal parametric problem involving a fractional-type operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Our approach is of variational nature; by working in suitable fractional Sobolev spaces which encode Dirichlet homogeneous boundary conditions, and exploiting an abstract critical point theorem for smooth functionals, we derive the existence of at least three weak solutions to our problem for suitable values of the parameters. Finally, we provide a concrete estimate of the range of these parameters in the autonomous case, by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. The methods adopted here can be exploited to study different classes of elliptic problems in presence of a degenerate nonlocal term.

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter $\lambda >0$ varies. We also show the existence of a smallest positive solution ${ū}_{\lambda }$ and investigate the properties of the map $\lambda \to {ū}_{\lambda }$. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Let $\mathrm{\Omega }\subset {ℝ}^N$ be an open subset, for a positive integer $N$, and let $L\subset ℂ\otimes T\mathrm{\Omega }$ be a $C^{\infty }$-smooth locally integrable subbundle. We give a proof of the following result: If $(\mathrm{\Omega },L)$ is nowhere strictly hypoanalytically pseudoconvex (as defined in the paper) then for any sufficiently small domain $\omega ⋐\mathrm{\Omega },$ and any $f\in C^0(\omega )$ which is continuous up to the boundary such that $f$ is a solution with respect to $L$ on $\omega$, it holds true that ${\mathrm{\text{max}}}_{z\in \partial \omega }|f(z)|={\mathrm{\text{max}}}_{z\in \overline{\omega }}|f(z)|.$ We also point out a relation to Levi curvature.

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

In this paper, first we study a complete smooth metric measure space $(M^n,g,e^{-f}dv)$ with the ($\infty$)-Bakry–Émery Ricci curvature ${\mathrm{\text{Ric}}}_f\ge \frac{a}{2}g$ for some positive constant $a$. It is known that the spectrum of the drifted Laplacian ${\mathrm{\Delta }}_f$ for $M$ is discrete and the first nonzero eigenvalue of ${\mathrm{\Delta }}_f$ has lower bound $\frac{a}{2}$. We prove that if the lower bound $\frac{a}{2}$ is achieved with multiplicity $k\ge 1$, then $k\le n$, $M$ is isometric to ${\mathrm{\Sigma }}^{n-k}\times {ℝ}^k$ for some complete $(n-k)$-dimensional manifold $\mathrm{\Sigma }$ and by passing an isometry, $(M^n,g,e^{-f}dv)$ must split off a gradient shrinking Ricci soliton $({ℝ}^k,g_{\mathrm{\text{can}}},\frac{a}{4}|t{|}^2)$, $t\in {ℝ}^k$. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian ${ℒ}=\mathrm{\Delta }-\frac{1}{2}〈x,\nabla \cdot 〉$ for properly immersed self-shrinkers in the Euclidean space ${ℝ}^{n+p}$, $p\ge 1$ and show the discreteness of the spectrum of ${ℒ}$ and a logarithmic Sobolev inequality.

We discuss critical and subcritical general maximizing problems associated with a Trudinger–Moser type inequality for a class of weighted Sobolev spaces. The key of our approach relies on a concentration-compactness type result and sharp estimates for the concentration level of a Moser-type functional. In particular, we give an explicit concentrating sequence that is maximizing for the maximal concentration level. As an application, we prove the existence of a nontrivial solution for a family of quasilinear elliptic equations with critical growth.

In this paper, we extend the well-known theorem of Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate Néron–Severi group of a general hypersurface in any smooth projective variety.

Consider a Riemann surface of genus $g$ bordered by $n$ curves homeomorphic to the unit circle, and assume that $2g-2+n>0$. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in $L^2$ and $L^{\infty }$, and furthermore that the space of $(-1,1)$ differentials in $L^2\cap L^{\infty }$ decomposes as a direct sum of infinitesimally trivial differentials and $L^2$ harmonic $(-1,1)$ differentials. Thus the tangent space of this Teichmüller space is given by $L^2$ harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of $L^2$ harmonic Beltrami differentials.

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain with bounded boundary which is symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain’s geometry. Our approach is close to the Born–Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

In this paper, we study $N$-expansive homeomorphisms on surfaces. We prove that when $f$ is a $2$-expansive homeomorphism defined on a compact boundaryless surface $M$ with non-wandering set $\mathrm{\Omega }(f)$ being the whole of $M$ then $f$ is expansive. This condition on the non-wandering set cannot be relaxed: we present an example of a $2$-expansive homeomorphisms on a surface with genus $2$ with wandering points that is not expansive.