The Journal of Geometric Analysis最新文献

In this article, we introduce the fractional medians, provide a representation for the set of all fractional medians in terms of non-increasing rearrangements, and investigate the mapping properties of the fractional maximal operators defined by these medians. Our maximal operator is a generalization of the one introduced by Strömberg (Indiana Univ Math J 28(3):511–544, 1979). It turns out that our maximal operator is smoother than the usual fractional maximal operator. Furthermore, we provide an alternative proof of the embedding from BV to (L^{n/(n-1),1}) due to Alvino (Boll Un Mat Ital A 14(1):148–156, 1977) by using the usual medians.

This article delves into the (L_p) Minkowski problem within the framework of generalized Gaussian probability space. This type of probability space was initially introduced in information theory through the seminal works of Lutwak et al. (Ann Probab 32(1B):757–774, 2004, IEEE Trans Inf Theory 51(2):473–478, 2005), as well as by Lutwak et al. (IEEE Trans Inf Theory 58(3):1319–1327, 2012). The primary focus of this article lies in examining the existence of this problem. While the variational method is employed to explore the necessary and sufficient conditions for the existence of the normalized Minkowski problem when (p in mathbb {R} setminus {0}), our main emphasis is on the existence of the generalized Gaussian Minkowski problem without the normalization requirement, particularly in the smooth category for (p ge 1).

We prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

Let (Omega ) be a smooth, bounded, strictly convex domain in ( mathbb {R}^N , (Nge 2)). Assume (K, f) and g are smooth positive functions and K(x) may be singular near (partial Omega ). When K satisfies suitable conditions, we provide sufficient and necessary conditions on f and g for the existence of strictly convex solutions to the singular boundary blow-up Monge-Ampère problem

where (M[u]=det , (u_{x_{i}x_{j}})) is the Monge-Ampère operator and (0le q<N+1). Two nonexistence results of strictly convex solution are also considered when K has strong singularity. In addition, we analyze the boundary asymptotic behavior of such solution by finding new structure conditions on (K, f) and g. We present some examples to illustrate the applicability of our main results.

A probability measure (mu ) on ({{mathbb {R}}}^d) with compact support is called a spectral measure if it possesses an exponential orthonormal basis for (L^2(mu )). In this paper, we establish general criteria for determining whether a probability measure is spectral or not. As applications of these criteria, we provide a straightforward proof for the Lebesgue measure restricted to ([0, 1]^d) or ([0, 1]cup [a, a+1]cup [b, b+1]) to be a spectral measure. Furthermore, we investigate the spectrality of Cantor–Moran measure

generated by an admissible sequence ({(A_n,{{mathcal {D}}}_n)}_{n=1}^{infty }). It is noteworthy that our general criteria can be applied to establish numerous known and novel results.

Let ((mathcal {M},g)) be a smooth compact Riemannian manifold of dimension (Nge 8). We are concerned with the following elliptic system

where (Delta _g=div_g nabla ) is the Laplace–Beltrami operator on (mathcal {M}), h(x) is a (C^1)-function on (mathcal {M}), (varepsilon >0) is a small parameter, (alpha ,beta >0) are real numbers, ((p,q)in (1,+infty )times (1,+infty )) satisfies (frac{1}{p+1}+frac{1}{q+1}=frac{N-2}{N}). Using the Lyapunov–Schmidt reduction method, we obtain the existence of multiple blowing-up solutions for the above problem.

In this paper, we are concerned with the existence and properties of ground states for the Schrödinger–Poisson system with combined power nonlinearities

having prescribed mass

in the Sobolev critical case. Here ( a>0), and (gamma >0), (mu >0) are parameters, (lambda in {mathbb {R}}) is an undetermined parameter. By using Jeanjean’ theory, Pohozaev manifold method and Brezis and Nirenberg’s technique to overcome the lack of compactness, we prove several existence results under the (L^2)-subcritical, (L^2)-critical and (L^2)-supercritical perturbation (mu |u|^{q-2}u), under different assumptions imposed on the parameters (gamma ,mu ) and the mass a, respectively. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions of a Sobolev critical Schrödinger–Poisson problem perturbed with a subcritical term in the whole space ({mathbb {R}}^3).

Let ({rho _nu }_{nu in (0,nu _0)}) with (nu _0in (0,infty )) be a (nu _0)-radial decreasing approximation of the identity on (mathbb {R}^n), (X(mathbb {R}^n)) a ball Banach function space satisfying some extra mild assumptions, and (Omega subset mathbb {R}^n) a (W^{1,X})-extension domain. In this article, the authors prove that, for any f belonging to the inhomogeneous ball Banach Sobolev space ({W}^{1,X}(Omega )),

where (Gamma ) is the Gamma function and (pin [1,infty )) is related to (X(mathbb {R}^n)). Using this asymptotics, the authors further establish a characterization of (W^{1,X}(Omega )) in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation and some recently found profound properties of (W^{1,X}(mathbb {R}^n)) to overcome those difficulties caused by that the norm of (X(mathbb {R}^n)) has no explicit expression and (X(mathbb {R}^n)) might not be translation invariant. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which are all new. Particularly, when (X(Omega ):=L^p(Omega )) with (pin (1,infty )), this characterization coincides with the celebrated results of J. Bourgain, H. Brezis, and P. Mironescu in 2001 and H. Brezis in 2002; moreover, this characterization is also new even when (X(Omega ):=L^q(Omega )) with both (qin (1,infty )) and (pin [1,q)cup (q,frac{n}{n-1}]). In addition, the authors give several specific examples of (W^{1,X})-extension domains as well as (dot{W}^{1,X})-extension domains.

We consider compact hypersurfaces with boundary in ({mathbb {R}}^N) that are the critical points of the fractional area introduced by Paroni et al. (Commun Pure Appl Anal 17:709–727, 2018). In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold (Gamma ) of dimension (N-2) and lies in a hyperplane (H subset {mathbb {R}}^N). Then we show that the critical points coincide with a smooth manifold ({mathcal {N}}subset H) of dimension (N-1) with (partial {mathcal {N}}= Gamma ). Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds (Gamma _1) and (Gamma _2) of dimension (N-2) and suppose that (Gamma _1) lies in a hyperplane H perpendicular to the (x_N)-axis and that (Gamma _2 = Gamma _1 + d , e_N) ((d >0) and (e_N = (0,cdots ,0,1) in {mathbb {R}}^N)). Then, assuming that (Gamma _1) has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds ({mathcal {N}}_1 subset H) and ({mathcal {N}}_2 subset H + d , e_N) of dimension (N-1) with (partial {mathcal {N}}_i = Gamma _i) for (i in {1,2}). Moreover, the interior of the critical points does not intersect the boundary of the convex hull in ({mathbb {R}}^N) of (Gamma _1) and (Gamma _2), while this can occur in the codimension-one situation considered by Dipierro et al. (Proc Am Math Soc 150:2223–2237, 2022). We also obtain a quantitative bound which may tell us how different the critical points are from ({mathcal {N}}_1 cup {mathcal {N}}_2). Finally, in the same setting as in the second case, we show that, if d is sufficiently large, then the critical points are disconnected and, if d is sufficiently small, then (Gamma _1) and (Gamma _2) are in the same connected component of the critical points when (N ge 3). Moreover, by computing the fractional mean curvature of a cone whose boundary is (Gamma _1 cup Gamma _2), we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.

Let (X = {X_0,ldots ,X_m}) be a family of smooth vector fields on an open set (Omega subseteq mathbb {R}^N). Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets (Omega ), we find necessary and sufficient conditions on X for the existence of a Lie group ((Omega ,*)) such that the operator (L=sum _{i = 1}^mX_i^2+X_0) is left-invariant with respect to the operation (*). Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.