We study complete stable minimal hypersurfaces in Riemannian manifolds under various curvature assumptions. In an (n+1) -dimensional complete oriented manifold with nonnegative scalar curvature, we prove that no complete oriented noncompact stable minimal hypersurface can be conformally equivalent to a bounded domain in an n-dimensional manifold with nonpositive scalar curvature. As a consequence, there exists no complete stable minimal hypersurface in ℝ^n+1 that is conformally equivalent to a bounded domain in ℝ^n . We also show that compact stable minimal hypersurfaces in manifolds with nonnegative scalar curvature have nonnegative smooth Yamabe invariant and we characterize the equality case. Next, we consider complete two-sided stable minimal hypersurfaces in ambient manifolds with pinched sectional curvature. Under an additional curvature condition that makes the second fundamental form a Codazzi tensor, we obtain a rigidity result under an L^2 -condition on the second fundamental form and derive an upper bound for the first eigenvalue of the Laplacian. Finally, we establish that any complete noncompact two-sided minimal hypersurface immersed in a warped product manifold is stable, provided that the angle function is positive and the second derivative of the warping function is nonnegative.
In this paper we consider fully nonlinear second-order elliptic partial differential equations, with measurable, possibly singular and unbounded lower order coefficients, in unbounded domains Ω⊂ℝ^n . We prove the Alexandroff–Bakelman–Pucci maximum principle for L^p- viscosity solutions under suitable assumptions on the geometry of Ω , satisfied by a class of very large unbounded domains including the complement of hypersurfaces, such as the cut plane in dimension n=2 and the closure of (n-1) -dimensional open convex cones in higher dimensions, as well as their subdomains.
We investigate the massless cubic Dirac equations in two and three space dimensions, referred to as the Soler model. The solutions to this model are known as wave functions, which have unit L^2 norm. We identify a class of data that can be large in L^2 , which we call dispersed large data, and show global existence and long-time dynamics, including pointwise decay and scattering, for the cubic Dirac model with this set of large data. To achieve these goals, we carefully balance the decay rates and the largeness of the solutions within the framework of the vector-field method.
In 2017, Weisz introduced the concept of ω -Walsh–Lebesgue points and proved that the Fejér means σ _n (f) of the Walsh–Fourier series of a d-dimensional integrable function f converge to the function at each ω -Walsh–Lebesgue point, provided that n →∞ and n lies within a cone. In this paper, we aim to generalize these results to the Vilenkin–Fejér means and other summability methods of Vilenkin–Fourier series. We also present convergence theorems for θ -summations as well as for matrix transform and Nörlund summations.
We introduce a class of integral operators that are closely related to the Hankel transform, spherical averages, square functions, and maximal functions. Specifically, we consider the operator T_ν ,ρ ,Ω(f)(x)=( ∫ _0^∞| ∫ _ℝ^nf(x-y) Ω (y) J_ν(s|y|)/ |y|^ n-ρ dy| ^2ds/s^1-2ρ) ^1/2, where ν ,ρ∈ℝ with ν >-12 , where J_ν denotes the Bessel function of the first kind of order ν , and Ω : ℝ^n→ℝ is a homogeneous function of degree zero whose restriction to the unit sphere 𝕊^n-1 has mean value zero. By exploiting a precise relationship between T_ν ,ρ ,Ω , the Hankel transform, and spherical means operators, we develop a framework for the analysis of its mapping properties. Assuming that the kernel Ω belongs to the Orlicz space L(log L)^1/2(𝕊^n-1) , we establish L^p bounds for T_ν ,ρ ,Ω . Moreover, we prove that appropriately truncated versions of the operator satisfy additional L^p inequalities, leading to new L^p inequalities that are unavailable in the absence of truncation. Our results provide new L^p estimates for oscillatory singular integral operators with rough kernels involving radial functions of Schlömilch-series type, which naturally arise in problems of radial Fourier analysis, spectral theory, and integral transforms involving Bessel kernels. These results highlight previously unexplored boundedness properties of the general operator T_ν ,ρ ,Ω .
In this paper, we consider the asymptotic behavior of solutions for a non-autonomous p-Laplacian equation with polynomial growth nonlinearity of arbitrary order driven by nonlinear colored noise in ℝ^N. Due to the lack of Lipschitz continuity of diffusion and nonlinear terms, we cannot ensure the uniqueness of solutions for such a system, such that the long-time behavior of solutions can be only considered by the theory of multi-valued non-autonomous random dynamical system. Thus, we first establish the asymptotic compactness of the corresponding multi-valued non-autonomous cocycle by carrying out the tail-estimates to overcome the non-compactness of Sobolev embeddings on unbounded domains. Then the measurability of random attractors is derived by proving the weak upper semi-continuity of the corresponding cocycle. Finally, we prove the existence and uniqueness of pullback random attractors.
Let Ω⊂ℝ^d , d≤ 3 , be a bounded domain with C^4 boundary, let n≥ 1 be an integer, and let κ >1 . We study the L^2 -sphere constrained modified Swift–Hohenberg equation ∂ _tu+Δ ^2u+2Δ u+|u|^2n-2u+(κ -α (u))u=0 in (0,∞ )×Ω , subject to ‖ u(t)‖ _L^2(Ω )=1, u=Δ u=0 on ∂Ω , u(0)=u_0. Here α (u) is the scalar multiplier determined by the constraint. It removes the normal component of the ambient gradient and makes the vector field tangent to the unit sphere of L^2(Ω ) . For every u_0∈ℳ∩ V , where ℳ={u∈ L^2(Ω ):‖ u‖ _L^2=1} and V=H^2(Ω )∩ H_0^1(Ω ) , we prove existence and uniqueness of a global strong solution u∈ C([0,∞ );V) ∩ L^2_loc(0,∞ ;E) ∩ H^1_loc(0,∞ ;L^2(Ω )), where E={u∈ H^4(Ω ):u=Δ u=0 on ∂Ω} . The proof uses normalized spectral Galerkin approximations, exact preservation of the constraint, and a Galerkin energy identity. The energy satisfies the exact dissipation law ℰ(u(t))+∫ _s^t‖∂ _ru(r)‖ _L^2^2 dr=ℰ(u(s)), 0≤ s≤ t. Consequently, the long-time dynamics are formulated on the invariant energy sublevels 𝒳_M={u∈ℳ∩ V:ℰ(u)≤ M}. For each nonempty 𝒳_M , the restricted semiflow has a compact global attractor relative to 𝒳_M in the L^2 -metric. No compact attractor on the full space ℳ∩ V is claimed. We also characterize equilibria as constrained critical points of ℰ , prove existence of constrained ground states, and identify the tangent-space linearization with the constrained second variation of the energy. This linearization result is structural: no stability, convergence-to-equilibrium, or pattern-selection theorem is claimed.
We establish weighted mixed-norm estimates for maximal operators, higher-order Riesz transforms, square functions, and spectral multipliers associated with the Laguerre operator ℒ_ν on ℝ_+^n , for all ν∈ (-1,∞ )^n .
We study the Cauchy problem for the short-pulse equation on the circle, addressing global well-posedness, finite-time singularity formation, and global weak solutions. For small initial data we prove global existence in H^2(𝕊) via a sharp criterion expressed through two conserved quantities arising from the geometric correspondence with the sine-Gordon equation. For large data we exhibit a new class of solutions developing finite-time singularities, derived through Lagrangian coordinates. Finally, for any zero-mean initial datum in L^4(𝕊) we construct a unique global entropy weak solution, and we show that any weak solution in L^∞ ((0,T);𝒞_b(𝕊)) —not necessarily satisfying an entropy condition-is unique.
We study the problem of classifying locally strongly convex centroaffine Tchebychev hypersurfaces with constant sectional curvature in ℝ^n+1 . As the main results, we solve the problem for n=2,3 . Moreover, in terms of a centroaffine invariant function, we establish a new centroaffine geometric characterization for the hypersurface x_n+1=12x_1(x_2^2+⋯ +x_n^2)+x_1ln x_1 .
In this paper, we prove that the solution to a locally constrained flow in hyperbolic space introduced by Hu et al. [14] exists for a long time and converges exponentially fast to a geodesic sphere centered at the origin if the initial hypersurface is strictly horo-convex. Unlike previous results, we do not require the initial hypersurface to be star-shaped. As a consequence, we refine an Alexandrov–Fenchel type weighted geometric inequality proved by them so that both sides are isometry-invariant.
Let G=ℝ^n⋊ℝ, (x,u)· (x',u')=(x+e^ux',u+u'), with right Haar measure dx, du, and let L=-∂ _u^2-e^2uΔ _x. Martini proved the L^p -boundedness of the first-order Riesz transforms associated with L and left open whether the adjoint vertical Riesz transform R_0^* is of weak type (1, 1). We prove directly, for every n≥ 1 , that it is not. By Martini’s reduction, this is equivalent to showing that an explicit convolution operator is not bounded from L^1(G) to L^1,∞(G) .
In this paper, we study the Berezin transform B(f)(z)=∫ _𝔻(1-|z|^2)^2/|1-ω̅z|^4f(ω )dA(ω ). When p>1, r≥ 1 , we prove that B: L^P→ L^p(μ ) is r -summing if and only if the Carleson embedding J_μ from Bergman spaces 𝒜^p to L^p(μ ) is r -summing.
We establish a rigidity result to Berenstein problem on domain with holes in ℝ^N+1 . Let Ω be a bounded domain with a hole containing the origin in ℝ^N+1 . We shall show that, if the following overdetermined elliptic problem -Δ u=λ u in Ω , u=0 on ∂Ω , ∂ u/∂ν =c^± on ∂Ω ^± has a sequence of eigenfunctions, then the outer boundary ∂ ^+Ω and inner boundary ∂ ^-Ω are spheres, where c^± are non-zero constants. Similarly, if Ω has multiple holes, all connected boundaries of Ω are spheres.
In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant α=-1 (or +1), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.
Mainly motivated by the mirror symmetry considerations, we investigate the Dirichlet problem for the Lagrangian phase operator with supercritical phase on a general almost complex manifold (M,J,ω ) . Under the key assumption of smooth subsolution existence, we establish a priori C^2,α estimates for solutions to this nonlinear elliptic equation. Building on these regularity results, we resolve the Dirichlet problem by proving the existence and uniqueness of smooth solutions.
This paper investigates the existence, uniqueness, and exponential stability of asymptotically almost periodic mild solutions to the magneto-micropolar fluid equations on three-dimensional real hyperbolic manifolds ℍ^3(ℝ) , after reformulating the original system as a pressure-free abstract evolution equation through the Ebin-Marsden viscous operator and the Kodaira-Hodge projection. Within this geometric framework, we establish the existence and uniqueness of asymptotically almost periodic mild solutions for the linear problem, together with the existence, uniqueness, and exponential stability of small solutions to the nonlinear system, by combining semigroup estimates on real hyperbolic manifolds with a Massera-type principle, a fixed-point argument, and cone inequalities. Finally, numerical simulations of a reduced scalar model on a bounded truncated Poincaré half-plane qualitatively illustrate perturbation decay, bounded recurrence induced by asymptotically almost periodic forcing, and hyperbolic dissipation stabilization.
