Positive Smooth Function Research Articles

On positive solutions of the nonlocal Yamabe type equation in compact Riemannian manifold

In this paper we deal with a nonlocal curvature equation involving a critical nonlinearity and a lower order perturbation f on a compact Riemannian N-manifold (M,g) (with or without boundary):(−Δg)psu+a(x)up−1=K(x)ups⁎−1+f(x,u) in Mu>0 in Mu=0 on ∂M where (−Δg)ps is the fractional p-Laplacian on (M,g) for p>1 and the fractional exponent 0<s<1 with ps<N and ps⁎:=NpN−ps. Here a(x) is a non-trivial function in L∞(M) and K is a positive smooth function on M. Using critical point theory on the manifold, we prove the existence of a positive solution under certain assumptions.

An inverse problem for the minimal surface equation in the presence of a riemannian metric

In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold (Rn,g) where the metric is of the form g(x)=c(x)(g^⊕e) . Here g^ is a simple Riemannian metric on Rn−1 , e is the Euclidean metric on R and c a smooth positive function. We show that if the associated Dirichlet-to-Neumann maps corresponding to metrics g and c~g agree, then the Taylor series of the conformal factor c~ at xn=0 is equal to a positive constant. We also show a partial data result when n = 3.

Hierarchical Robust Generalized Nash Equilibrium Seeking of High-Order Uncertain Nonlinear Systems.

This article investigates the distributed generalized Nash equilibrium (GNE) seeking problem of noncooperative games (NGs) for high-order strict-feedback nonlinear multiagent systems (MASs). In particular, the feasible action set of each agent is not only subject to local set and inequality constraints but also coupled through an equality constraint with other agents. This constraint structure is more general and covers most of the constraints in the GNE seeking literature. To accomplish the concerned GNE seeking objective, we propose a novel hierarchical GNE seeking approach in this article to decouple the distributed GNE seeking algorithm design into two layers. First, we construct a distributed primary-dual GNE estimator to generate virtual reference signals that converge to the GNE. Then, with the output of the estimator as the reference signal, we develop an adaptive tracking controller to solve the resultant tracking problems under output constraints. To overcome the negative effects of the disturbances, novel compensating terms associated with smooth functions and positive integrable time-varying functions are incorporated in the controller design, which thereby realizes the exact GNE seeking in the presence of nonvanishing mismatched disturbances. At last, an example is given to support the theoretical analysis of the proposed algorithms.

Existence of solutions for an approximation of the Paneitz problem on spheres

This paper is devoted to studying the nonlinear problem with subcritical exponent $(S_{\varepsilon}) : \Delta^{2}u-c_n\Delta u+d_nu = Ku^{\frac{n+4}{n-4}-\varepsilon}$, $u$ on $ S^n$, where $n\geq5$, $ \varepsilon$ is a small positive parameter and $K$ is a a smooth positive function on $S^n$. We construct some solutions which blow up at $q$ different critical points of $K$.

Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u&gt;0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Maximum principles and consequences for γ$\gamma$‐translators in Rn+1${\mathbb {R}}^{n+1}$

AbstractIn this paper, we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flow, where the deformation speed is given by a homogeneous smooth symmetric positive function defined in a symmetric open cone. The main result of this paper is about the uniqueness of ‐translators in the class of complete graphs defined on a ball.

Chern Flat and Chern Ricci-Flat Twisted Product Hermitian Manifolds

Let (M1,g) and (M2,h) be two Hermitian manifolds. The twisted product Hermitian manifold (M1×M2f,G) is the product manifold M1×M2 endowed with the Hermitian metric G=g+f2h, where f is a positive smooth function on M1×M2. In this paper, the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold are derived. The necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature are obtained. Under the condition that the logarithm of the twisted function is pluriharmonic, it is proved that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat, if and only if M1,g and M2,h are Chern flat or Chern Ricci-flat, respectively.

Integral Representations and Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain

Integral Representations and Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain

Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres

This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents (S_{pm varepsilon}): Delta ^{2}u-c_{n}Delta u+d_{n}u = Ku^{ frac{n+4}{n-4}pm varepsilon}, u>0 on S^{n}, where ngeq 5, ε is a small positive parameter and K is a smooth positive function on S^{n}. We construct some solutions of (S_{-varepsilon}) that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation (S_{+varepsilon}).

Boundary blow-up solutions to singular k-Hessian equations with gradient terms: Sufficient and necessary conditions and asymptotic behavior

Let Ω be a smooth, bounded and strictly (k−1)-convex domain in RN(N≥2). Assume that b and f are smooth positive functions and b may be singular near ∂Ω. We provide sufficient and necessary conditions on b, f and q for the existence of k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u)=b(x)f(u)|∇u|q for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. The asymptotic behavior of such solutions is also investigated. Moreover, when b has strong singularity near ∂Ω, we study the existence and nonexistence of boundary blow-up solutions.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Interior Bubbling Solutions for an Elliptic Equation with Slightly Subcritical Nonlinearity

In this paper, we considered the Neumann elliptic equation (Pε): −Δu+K(x)u=u(n+2)/(n−2)−ε, u&gt;0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥6, ε is a small positive real and K is a smooth positive function on Ω¯. Using refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional, we constructed simple and non-simple interior bubbling solutions of (Pε) which allowed us to prove multiplicity results for (Pε) provided that ε is small. The existence of non-simple interior bubbling solutions is a new phenomenon for the positive solutions of subcritical problems.

Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain in Lebesgue Spaces

Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain in Lebesgue Spaces

A class of inverse curvature flows and 𝐿^{𝑝} dual Christoffel-Minkowski problem

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space R n + 1 \mathbb {R}^{n+1} with speed ψ u α ρ δ f − β \psi u^\alpha \rho ^\delta f^{-\beta } , where ψ \psi is a smooth positive function on unit sphere, u u is the support function of the hypersurface, ρ \rho is the radial function, f f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When ψ = 1 \psi =1 , we prove that the flow exists for all time and converges to infinity if α + δ + β ⩽ 1 \alpha +\delta +\beta \leqslant 1 , and α ⩽ 0 &gt; β \alpha \leqslant 0&gt;\beta , while in case α + δ + β &gt; 1 \alpha +\delta +\beta &gt;1 , α , δ ⩽ 0 &gt; β \alpha ,\delta \leqslant 0&gt;\beta , the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered at the origin. In particular, the results of Gerhardt [J. Differential Geom. 32 (1990), pp. 299–314; Calc. Var. Partial Differential Equations 49 (2014), pp. 471–489] and Urbas [Math. Z. 205 (1990), pp. 355–372] can be recovered by putting α = δ = 0 \alpha =\delta =0 . Our previous works [Proc. Amer. Math. Soc. 148 (2020), pp. 5331–5341; J. Funct. Anal. 282 (2022), p. 38] and Hu, Mao, Tu and Wu [J. Korean Math. Soc. 57 (2020), pp. 1299–1322] can be recovered by putting δ = 0 \delta =0 and α = 0 \alpha =0 respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to L p L^p -Minkowski problem and L p L^p -Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the L p L^p dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to L p L^p dual Minkowski problem and L p L^p dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the long time existence and convergence of a class of anisotropic flows (i.e. for general function ψ \psi ). The final result not only gives a new proof of many previously known solutions to L p L^p dual Minkowski problem, L p L^p -Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to L p L^p dual Christoffel-Minkowski problem with some conditions.

Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

This paper deals with the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source, (0.1)ut=Δu−χ∇⋅(uv∇v)+u(a(t,x)−b(t,x)u),x∈Ω0=Δv−μv+νu,x∈Ω∂u∂n=∂v∂n=0,x∈∂Ω,where Ω⊂RN is a smooth bounded domain, a(t,x) and b(t,x) are positive smooth functions, and χ, μ and ν are positive constants. In recent years, it has been drawn a lot of attention to the question of whether logistic kinetics prevents finite-time blow-up in various chemotaxis models. In the very recent paper (Kurt and Shen, 2021), we proved that for every given nonnegative initial function 0⁄≡u0∈C0(Ω̄) and s∈R, (0.1) has a unique globally defined classical solution (u(t,x;s,u0),v(t,x;s,u0)) with u(s,x;s,u0)=u0(x), which shows that, in any space dimensional setting, logistic kinetics prevents the occurrence of finite-time blow-up even for arbitrarily large χ. In Kurt and Shen (2021), we also proved that globally defined positive solutions of (0.1) are uniformly bounded under the assumption (0.2)ainf>μχ24if0<χ≤2μ(χ−1)ifχ>2.In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption (0.2). Among others, we provide some concrete estimates for ∫Ωu−p and ∫Ωuq for some p>0 and q>2N and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a “rectangular” type bounded invariant set (in Lq) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution (u∗(t,x),v∗(t,x)), which is periodic in t if a(t,x) and b(t,x) are periodic in t and is independent of t if a(t,x) and b(t,x) are independent of t.

Volume-preserving right-handed vector fields are conformally Reeb

Right-handed and Reeb vector fields are two rich classes of vector fields on closed, oriented three-manifolds. Prior work of Dehornoy and Florio–Hryniewicz has produced many examples of Reeb vector fields which are right-handed. We prove a result in the other direction. We show that the closed two-form associated with a volume-preserving right-handed vector field is contact-type. This implies that any volume-preserving right-handed vector field is equal to a Reeb vector field after multiplication by a positive smooth function. Combining our result with theorems of Ghys and Taubes shows that any volume-preserving right-handed vector field has a global surface of section.

The boundary value problem for the mean field equation on a compact Riemann surface

Let (Σ, g) be a compact Riemann surface with smooth boundary ∂E, ∆g be the Laplace-Beltrami operator, and h be a positive smooth function. Using a min-max scheme introduced by Djadli and Malchiodi (2008) and Djadli (2008), we prove that if Σ is non-contractible, then for any ρ Σ (8kπ, 8(k +1)π) with k Σ ℕ*, the mean field equation $$\left\{ {\matrix{{{\Delta _g}u = \rho {{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }}} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$ has a solution. This generalizes earlier existence results of Ding et al. (Ann Inst H Poincaré Anal Non Linéaire, 1999) and Chen and Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If h is a positive smooth function, then for any ρ ∈ (4kπ, 4(k + 1)π) with k ∈ ℕ*, the mean field equation $$\left\{ {\matrix{{{\Delta _g}u = \rho \left( {{{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }} - {1 \over {\left| \Sigma \right|}}} \right)} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {\partial u/\partial v = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$ has a solution, where v denotes the unit normal outward vector on ∂Σ. Note that in this case we do not require the surface to be non-contractible.

Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Let M 1 , g and M 2 , h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M 1 , g and M 2 , h is the product manifold M 1 × M 2 endowed with the warped product Hermitian metric G = f 2 2 g + f 1 2 h , where f 1 and f 2 are positive smooth functions on M 1 and M 2 , respectively. In this paper, the formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold are derived in terms of the corresponding objects of its components. We also prove that if the warped function f 1 and f 2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M 1 , g and M 2 , h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.

On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations

AbstractLet (X,D) be a polarized log variety with an effective holomorphic torus action, and Θ be a closed positive torus invariant (1,1) ‐current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge‐Ampère equations that correspond to generalized and twisted Kähler‐Ricci g‐solitons. We prove a version of the Yau‐Tian‐Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ‐twisted g‐Ding‐stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler‐Ricci/Mabuchi solitons or Kähler‐Einstein metrics. © 2022 Wiley Periodicals, Inc.

Existence results for the mean field equation on a closed symmetric Riemann surface

Let (S,g) be a closed Riemann surface, G be a finite isometric group acting on it and ♯G(x) be the number of all distinct points in the set G(x) for x∈S. If there exists some ℓ∈N⁎ satisfying ♯G(x)≡ℓ for all x∈S, then we show that for any ρ∈(8πkℓ,8π(k+1)ℓ) with k∈N⁎, the mean field equationΔgu=ρ(heu∫Sheudvg−1|S|) has a G-invariant solution, where h is a strictly positive and G-invariant smooth function. Our method is a modification of the min-max scheme due to Ding-Jost-Li-Wang (1999), Djadli-Malchiodi (2008) and Djadli (2008).