The Journal of Geometric Analysis最新文献

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo–Nirenberg and log-Caffarelli–Kohn–Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Gross-type log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on ({mathbb {R}}^n) for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

In this paper, we study the following nonlinear Maxwell–Dirac system

for (xin {mathbb {R}}^3) and (pin (2,3)), where (a > 0) is a constant, (alpha =(alpha _1,alpha _2,alpha _3)), (alpha _1,alpha _2,alpha _3) and (beta ) are (4times 4) Pauli–Dirac matrices, ({textbf{A}}=(A_1,A_2,A_3)) is the magnetic field, (phi ) is the electron field and q is the changing point-wise charge distribution. Under a local condition that V has a local trapping potential well, when (varepsilon >0) is sufficiently small, we construct an infinite sequence of localized bound state solutions concentrating around the local minimum points of V. These solutions are of higher topological type in the sense that they are obtained from a symmetric linking structure. In the second part of this paper, we consider the case in which V(x) may approach a as (|x|rightarrow infty ). This is a degenerate case as most works in the literature assume a strict gap condition (sup _{xin {mathbb {R}}^3} |V(x)|< a), which is a key condition used in setting up the linking structure as well as in dealing with the compactness issues of the variational formulation.

This paper is concerned with an inverse problem of a coupled thermoelastic plate model. Two major features are that the thermal equation has a memory effect, while the plate equation has a curved middle surface. A differential geometric approach is developed, by which we study the pointwise Carleman estimates for elliptic and hyperbolic equations. We are able to prove a key Carleman estimate for the strongly coupled system. From them, the Hölder stability in recovering the source terms and the coupling coefficient is obtained. The measurements of the plate deflection and temperature are assumed to be taken on a subdomain of the boundary.

This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are (5/2,(=2.5)) in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle–Huisken to allow for mean curvature flow with surgery in the Alexandrov immersed, 2-dimensional setting.

We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation (mathcal {S}^sharp ) known on a lateral boundary. We show that, under a non-conjugacy assumption, every defining function r(x, y) of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric; and each one of (mathcal {S}^sharp ) and r (up to an elliptic factor) determines the other uniquely. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary/lens rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.

In this paper, we introduce the pluricomplex Green function of the Monge–Ampère equation for ((n-1))-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge–Ampère and Hessian equations on a punctured domain. We prove the pluricomplex Green function is (C^{1,alpha }) by constructing approximating solutions and establishing uniform a priori estimates for the gradient and the complex Hessian. The singular solutions turn out to be smooth for the k-Hessian equations for ((n-1))-k-admissible functions.

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain (Omega ) of ({{,mathrm{mathbb {R}},}}^N). The mean curvature, that depends on the location of the solution u itself, is asked to be of the form f(x)h(u), where f is a nonnegative function in (L^{N,infty }(Omega )) and (h:{{,mathrm{mathbb {R}},}}^+mapsto {{,mathrm{mathbb {R}},}}^+) is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. (hequiv 1). This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples.

We prove an abstract structure theorem for weighted manifolds supporting a weighted f-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions.

We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the eigenvalue of Courant sharp eigenfunctions that is independent of the widths of the necks.