Positive Smooth Function Research Articles

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

For a Riemannian manifold M, possibly with boundary, we consider the Riemannian product M×Rk with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with non-empty boundary and contained within certain regions of M×Rk with suitable weights. Our results include half-space and Bernstein-type theorems in weighted cylinders. We also generalize to this setting some classical properties about the confinement of a compact minimal hypersurface to certain regions of Euclidean space according to the position of its boundary. Finally, we show interesting situations where the statements are applied, some of them in relation to the singularities of the mean curvature flow.

A fully nonlinear locally constrained anisotropic curvature flow

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+1 (n≥2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.

Constant angle surfaces in the Lorentzian warped product manifold $I \times_{f} \mathbb E^2_1$

Let $I \times_{f} \mathbb E^2_1$ be a 3-dimensional Lorentzian warped product manifold with the metric $\tilde g = dt^2 + f^2(t) (dx^2 - dy^2)$, where $I$ is an open interval, $f$ is a strictly positive smooth function on $I,$ and $\mathbb E^2_1$ is the Minkowski 2-plane. In this work, we give a classification of all space-like and time-like constant angle surfaces in $I \times_{f} \mathbb E^2_1$ with nonnull principal direction when the surface is time-like. In this classification, we obtain space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we have some results on constant angle surfaces of the anti-de Sitter space $ \mathbb H^3_1(-1)$.

Deforming convex bodies in Minkowski geometry

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers

We consider the problemε2div(∇a(y)u)−V(y)u+up=0,u>0in Ω,∇a(y)u⋅ν=0on ∂Ω, where Ω is a bounded domain in R2 with smooth boundary, the exponent p is greater than 1, ε>0 is a small parameter, V is a uniformly positive smooth potential on Ω¯, and ν denotes the outward normal of ∂Ω. For two positive smooth functions a1(y),a2(y) on Ω¯, the operator ∇a(y) is given by∇a(y)u=(a1(y)∂u∂y1,a2(y)∂u∂y2).(1). Let Γ⊂Ω¯ be a smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ is a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], where σ=p+1p−1−12. By assuming some additional constraints on the functions a(y), V(y) and the curves Γ, ∂Ω, we prove that there exists a sequence of ε such that the problem has solutions uε with clustering concentration layers directed along Γ, exponentially small in ε at any positive distance from it.(2). If Γ˜ is a simple closed smooth curve in Ω (not touching the boundary ∂Ω), which is also a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], then a similar result of concentrated solutions is still true.

New eigenvalue estimates involving Bessel functions

Given a compact Riemannian manifold (Mn, g) with boundary ∂M, we give an estimate for the quotient R ∂M f dµg R M f dµg , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Indepen[1]dently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

Morse inequalities at infinity for a resonant mean field equation

In this paper, we study the following mean field type equation: [Formula: see text] where [Formula: see text] is a closed oriented surface of unit volume [Formula: see text] = 1, [Formula: see text] positive smooth function and [Formula: see text], [Formula: see text]. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters [Formula: see text]. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. [Formula: see text].

Radial solutions of the elliptic Liouville equation

In this paper, we derive new closed-form radial solutions for the following elliptic Liouville equation △ϕ+φ(x)eϕ=0inΩ⊂Rn, and its generalized equation △ϕ+φ(x)eϕ=f(x)inΩ⊂Rn, where △ is the Laplace operator, φ and f are smooth positive functions.

Interpolation of Spaces of Functions of Positive Smoothness on a Domain

Interpolation of Spaces of Functions of Positive Smoothness on a Domain

Simplest Test for the Three-Dimensional Dynamical Inverse Problem (The BC-Method)

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

Stable and isoperimetric regions in some weighted manifolds with boundary

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming local convexity of the boundary and certain behavior of the Bakry–Émery–Ricci tensor we deduce rigidity properties for stable sets by using deformations constructed from parallel vector fields tangent to the boundary. As a consequence, we completely classify the stable sets in some Riemannian cylinders Ω×R with product weights. Finally, we also establish uniqueness results showing that any minimizer of the weighted perimeter for fixed weighted volume is bounded by a horizontal slice Ω×{t}.

Global boundedness in a chemotaxis quasilinear parabolic predator–prey system with pursuit-evasion

We analyze some configurations of the general chemotaxis predator–prey model with pursuit-evasion dynamics ∂tu−∇⋅(Fu(u)∇u)+∇⋅(Fp(u)∇p)=uF1(w)−F2(u)∂tw−∇⋅(Fw(w)∇w)−∇⋅(Fq(w)∇q)=wF3(w)−δuwin Ω×(0,T) with Neumann boundary condition and non-negative initial data, where p and q are the predator’s and the prey’s pheromone, respectively, modeled by parabolic or elliptic equations, and Ω⊂Rd, with d≥1, is a smooth bounded domain. We assume Fu and Fw to be smooth positive functions satisfying kusp1≤Fu(s) and kwsp2≤Fw(s) when s≥s0>1, Fp,Fq smooth non-negative functions such that kp1sp0≤Fp(s)≤kp1sp0 when s≥s0 and Fq(s)≡kqsq0 for all s≥0, with q0=1 or q0≥2. We also assume F1,F2 and F3 to be smooth with F1(0)=F2(0)=F3(0)=0, F2≥0, F1(s)≤k1sθ, F2(s)≥k2s1+b, b≥0, F3(s)≤k3−k4sa, for s≥s0, a>0, k3≥0, k4>0. We prove that for θ,a,b,q0 and p0 satisfying some relation there exists a unique classical solution to the system which is global in time and bounded. The result in independent on p1,p2∈R.

Interpolation of Spaces of Functions of Positive Smoothness on a Domain

An interpolation theorem for spaces of functions of positive smoothness on a domain with flexible cone condition is established.

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

The planar Lp dual Minkowski problem

In this paper we study the Lpq-dual Minkowski problem for the case p < 0 < q. We prove for any positive smooth function f on $$\mathbb{S}^{1}$$, there exists an F: ℝ+ → ℝ−, such that if F(q) < p < 0 or 0 < q < −F(−p) then there is a smooth and strictly convex body solving the planar Lpq-dual Minkowski problem.

A Moser–Bernstein problem for Riemannian warped products

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has classically been studied in the the Euclidean n-dimensional space and it is known as the Moser–Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precisely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $$M=P\times _h{\mathbb {R}}$$, where P is a complete Riemannian parabolic manifold and h a positive smooth function on P.

A double mean field equation related to a curvature prescription problem

We study a double mean field–type PDE related to a prescribed curvature problem on compacts surfaces with boundary:(0.1){−Δu=2ρ(Keu∫ΣKeu−1|Σ|)in Σ,∂νu=2ρ′(heu2∫∂Σheu2−1|∂Σ|)on ∂Σ.Here ρ and ρ′ are real parameters, K,h are smooth positive functions on Σ and ∂Σ respectively and ν is the outward unit normal vector to ∂Σ.We provide a general blow–up analysis, then a Moser–Trudinger inequality, which gives energy–minimizing solutions for some range of parameters. Finally, we provide existence of min–max solutions for a wider range of parameters, which is dense in the plane if Σ is not simply connected.

Simplest Test for the Two-Dimensional Dynamical Inverse Problem (BC-Method)

The dynamical system$$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla \ln \rho \cdot \nabla u=0& in\kern1em {\mathbb{R}}_{+}^2\times \left(0,T\right),\\ {}u\left|{}_{t=0}\right.={u}_t\left|{}_{t=0}\right.=0& in\kern1em {\mathbb{R}}_{+}^2,\\ {}{u}_y\left|{}_{y=0}\right.=f& for\kern1em 0\le t\le T,\end{array}} $$ is under consideration, where $$ {\mathbb{R}}_{+}^2:= \left\{\left(x,y\right)\in {\mathbb{R}}^2\left|y\right.>0\right\} $$; ρ = ρ(x, y) is a smooth positive function; f = f(x, t) is a boundary control; u = uf(x, y, t) is a solution. With the system one associates a response operator $$ R:f\mapsto {u}^f\left|{}_{y=0}\right. $$. The inverse problem is to recover the function ρ via the response operator. A short presentation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way of making use of them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains

We establish an embedding theorem for spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in spaces of the same type and obtain some closely related results.

Curvature of $$C_5\oplus C_{12}$$-Manifolds

The Chinea–Gonzalez class $$C_5\oplus C_{12}$$ consists of the almost contact metric manifolds that are locally described as double-twisted product manifolds $$I\times _{(\lambda _1,\lambda _2)}{\widehat{M}}$$ , $$I\subset {\mathbb {R}}$$ being an open interval, $${\widehat{M}}$$ a Kahler manifold and $$\lambda _1,\lambda _2$$ smooth positive functions. In this article, we investigate the behavior of the curvature of $$C_5\oplus C_{12}$$ -manifolds. Particular attention to the N(k)-nullity condition is given and some local classification theorems in dimension $$2n+1\ge 5$$ are stated. This allows us to classify $$C_5\oplus C_{12}$$ -manifolds that are generalized Sasakian space forms. In addition, we provide explicit examples of these spaces.