The Journal of Geometric Analysis最新文献

We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As an application, we prove a quantitative symmetry result for the reverse Serrin problem, which we introduce for the first time in this paper. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition.

We consider numerical shape optimization of a fluid–structure interaction model. The constrained system involves multiscale coupling of a two-dimensional unsteady Navier–Stokes equation and a one-dimensional ordinary differential equation for fluid flows and structure, respectively. We derive shape gradients for both objective functionals of least-squares type and energy dissipation. The state and adjoint state equations are numerically solved on the time-dependent domains using the Arbitrary-Lagrangian–Eulerian method. Numerical results are presented to illustrate effectiveness of algorithms.

In this paper, we study a class of strongly degenerate problems with critical exponential growth in (mathbb {R}^N), (Nge 2). We do not assume ellipticity condition on the operator and thus the maximum principle given by Lieberman (Commun Partial Differ Equ 16:311–361, 1991) can not be accessed. Therefore, a careful and delicate analysis is necessary and some ideas can not be applied in our scenario. The arguments developed in this paper are variational and our main result completes the study made in the current literature about the subject. Moreover, when (N=2) or (N=3) the solutions model the slow steady-state flow of a fluid of Prandtl-Eyring type.

In this paper, we are concerned with the Cauchy problem of 3D viscous and heat-conducting micropolar fluids with far field vacuum. Compared with the case of non-vacuum at infinity (Huang and Li in Arch Ration Mech Anal 227:995–1059, 2018; Huang et al. in J Math Fluid Mech 23(1):50, 2021), due to ((rho (t, x), theta (t, x))rightarrow (0, 0)) as (|x|rightarrow infty ), we don’t have useful energy equality (or inequality), which is very important to establish a priori estimates in Huang and Li (Arch Ration Mech Anal 227:995–1059, 2018) and Huang et al. (J Math Fluid Mech 23(1):50, 2021). Thus, a new assumption of a priori estimates and more complicated calculations will be needed. On the other hand, we need to deal with some strong nonlinear terms which come from the interactions of velocity and micro-rotation velocity. Finally, we show that the global existence and uniqueness of strong solutions provided that the initial energy is suitably small. In particular, large-time behavior and a exponential decay rate of the strong solution are obtained, which generalizes the incompressible case (Ye in Dicret Contin Dyn Syst Ser B 24:6725–6743, 2019) to the full compressible case.

Applying a family of mass-capacity related inequalities proved in Miao (Peking Math J 2023, https://doi.org/10.1007/s42543-023-00071-7), we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity.

In this article, we consider a complete, non-compact almost Hermitian manifold whose curvature is asymptotic to that of the complex hyperbolic space. Under natural geometric conditions, we show that such a manifold arises as the interior of a compact almost complex manifold whose boundary is a strictly pseudoconvex CR manifold. Moreover, the geometric structure of the boundary can be recovered by analysing the expansion of the metric near infinity.

This paper provides a new definition of the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg–Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin–Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

Bernstein problem for affine maximal type equation

has been a core problem in affine geometry. A conjecture (Version I in Section 1) initially proposed by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph with (N=2, theta =3/4) and then was strengthened by Trudinger-Wang (Invent. Math., 140, 2000, 399-422) to its full generality (Version II), which asserts that any Euclidean complete, affine maximal, locally uniformly convex (C^4)-hypersurface in ({mathbb {R}}^{N+1}) must be an elliptic paraboloid. At the same time, the Chern’s conjecture was solved completely by Trudinger-Wang in dimension two. Soon after, the Affine Bernstein Conjecture (Version III) for affine complete affine maximal hypersurfaces was also shown by Trudinger-Wang in (Invent. Math., 150, 2002, 45-60). Thereafter, the Bernstein problem has morphed into a broader conjectures for any dimension (Nge 2) and any positive constant (theta >0). The Bernstein theorem of Trudinger-Wang was then generalized by Li-Jia (Results Math., 56 2009, 109-139) to (N=2, theta in (3/4,1]) (see also Zhou (Calc. Var. PDEs., 43 2012, 25-44) for a different proof). In the past twenty years, much effort was done toward higher dimensional issues but not really successful yet, even for the case of dimension (N=3). Recently, counter examples were found in (J. Differential Equations, 269 (2020), 7429-7469), toward the Full Bernstein Problem IV for (Nge 3,theta in (1/2,(N-1)/N)) and using a much more complicated argument. In this paper, we will construct explicitly various new Euclidean complete affine maximal type hypersurfaces which are not elliptic paraboloid for the improved range

In this paper, we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge–Ampère equation. As a consequence, we find that in the space of Kähler potentials any two points close to each other in (C^2) norm can be connected by a geodesic along which the associated metrics do not degenerate.