Positive Smooth Function Research Articles

Adaptive output feedback tracking control for a class of nonlinearly parameterised uncertain systems

ABSTRACT In this paper, an adaptive output feedback tracking control scheme is proposed for a class of uncertain nonlinear systems where unknown parameters appear in the form of nonlinear parameterisation. Specific filters are constructed with the output signal and parameter estimates to make state estimation. The nonlinearly parameterised uncertainties are approximated by the Taylor expansion technique and compensation for remainder terms is realised with a series of high-order derivable functions to play the role of sign functions. A smooth positive function and parameter adaption technique are combined to deal with the effect made by state estimation error in the process of stability analysis of the closed-loop system. It is proved that all the signals are guaranteed to be bounded and the tracking error will converge to a small bounded set which is adjustable by tuning certain design constants. Numerical simulation is conducted to verify the effectiveness of the proposed control scheme.

The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation

In this paper we consider the existence and asymptotic behavior of k-convex solution to the boundary blow-up k-Hessian problemSk(D2u)=H(x)f(u) for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0, where k∈{1,2,⋯,N}, Sk(D2u) is the k-Hessian operator, Ω is a smooth, bounded, strictly convex domain in RN(N≥2), H∈C∞(Ω) is positive in Ω, but is not necessarily bounded on ∂Ω, and f is a smooth positive function that satisfies the so-called Keller-Osserman condition. Further results are obtained for the special case that Ω is a ball. Our approach to show the existence and asymptotic behavior, exploits the method of sub- and super-solutions and Karamata regular variation theory.

Embedding of spaces of functions of positive smoothness on irregular domains

For spaces of functions of positive smoothness defined on irregular domain of n-dimensional Euclidean space, an embedding theorem into spaces of the same type and related results are presented.

On the Neumann Laplacian in nonuniformly collapsing strips

Consider the Neumann Laplacian in the region below the graph of [Formula: see text], for a positive smooth function [Formula: see text] with both [Formula: see text] and [Formula: see text] bounded. As [Formula: see text] such region collapses to [Formula: see text] and an effective operator is found, which has Robin boundary conditions at [Formula: see text]. Then, we recover (under suitable assumptions in the case of unbounded [Formula: see text]) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for [Formula: see text] diverging less than exponentially.

A new result for global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

We consider the following chemotaxis model{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+μ(u−u2),x∈Ω,t>0,vt−Δv=−uv,x∈Ω,t>0,(D(u)∇u−χu⋅∇v)⋅ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω on a bounded domain Ω⊂RN(N≥1), with smooth boundary ∂Ω,χ and μ are positive constants. Here, D is supposed to be smooth positive function satisfying D(u)≥CD(u+1)m−1 for all u≥0 with some CD,m>0. Besides appropriate smoothness assumptions, in this paper it is only required thatm>{1−μ2χ[1+max1≤s≤2⁡λ0(s)‖v0‖L∞(Ω)23]ifN≤2,1ifN≥3, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where λ0 is a positive constant which is corresponding to the maximal Sobolev regularity. The results of this paper extends the results of Jin (J. Differential Equations 263 (9) (2017) 5759–5772), who proved the possibility of boundness of weak solutions, in the case m>1 and N=3. To the best of our knowledge, this is the first result which gives the relationship between m and μχ that yields to the boundedness of the solution.

Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem $$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: $$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $$ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).

Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains

В работе устанавливается теорема вложения пространств функций положительной гладкости, определенных на нерегулярных областях $n$-мерного евклидова пространства, в пространства того же типа, а также некоторые близкие результаты. Библиография: 13 названий.

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.

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem−Δu=λa(x)eu−4πNδ0 in Ω,u=0 on ∂Ω, where Ω is a smooth and bounded domain in R2 such that 0∈Ω, a(x) is a positive smooth function, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution uλ blowing up at 0 and satisfying λ∫Ωa(x)euλ→8π(N+1) as λ→0+.

Deforming a hypersurface by principal radii of curvature and support function

We study the motion of smooth, closed, strictly convex hypersurfaces in $$\mathbb {R}^{n+1}$$ expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature $$\sigma _k$$ and support function h. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known $$L_p$$ -Christoffel–Minkowski problem $$\varphi h^{1-p}\sigma _k=c$$ . Here $$\varphi $$ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For $$1\le k\le n-1, p\ge k+1$$ , assuming the spherical hessian of $$\varphi ^{\frac{1}{p+k-1}}$$ is positive definite, we prove the $$C^{\infty }$$ convergence of the normalized flow to a homothetic self-similar solution. One of the highlights of our arguments is that we do not need the constant rank theorem/deformation lemma of Guan and Ma (Invent Math 151:553–577, 2003) and thus we give a partial answer to a question raised in Guan and Xia (Calc Var Part Differ Equ 57:69, 2018. https://doi.org/10.1007/s00526-018-1341-y . Moreover, for $$k=n, p\ge n+1$$ , we prove the $$C^{\infty }$$ convergence of the normalized flow to a homothetic self-similar solution without imposing any condition on $$\varphi .$$ In the final section of the paper, for $$1\le k<n$$ , we will give an example that spherical hessian of $$\varphi ^{\frac{1}{p+k-1}}$$ is negative definite at some point and the solution to the flow loses its smoothness.

Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

We consider the nonlinear problem of inhomogeneous Allen–Cahn equationϵ2Δu+V(y)u(1−u2)=0inΩ,∂u∂ν=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, ϵ is a small positive parameter, ν denotes the unit outward normal of ∂Ω, V is a positive smooth function on Ω¯. Let Γ be a curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ∫ΓV1/2. We can prove that there exists a solution uϵ such that: as ϵ→0, uϵ approaches +1 in one part of Ω, while tends to −1 in the other part, except a small neighborhood of Γ.

Embeddings of Weighted Spaces of Functions of Positive Smoothness on Irregular Domains in Lebesgue Space

Embeddings of Weighted Spaces of Functions of Positive Smoothness on Irregular Domains in Lebesgue Space

Volume minimization and conformally Kähler, Einstein–Maxwell geometry

Let $M$ be a compact complex manifold admitting a Kahler structure. A conformally Kahler, Einstein–Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde g$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^{2} \tilde g$ being a Kahler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kahler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kahler metric $g$ in the fixed Kahler class gives a cKEM metric $\tilde g = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kahler–Ricci solitons and Sasaki–Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kahler–Ricci soliton case and Sasaki–Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov–Maschler.

A subcritical approximation of the Paneitz problem on spheres

This paper is concerned with the following subcritical approximation of a fourth order conformal invariant (the Paneitz curvature) on spheres \((S_\varepsilon ) : \Delta ^{2}u-c_n\Delta u+d_nu = K|u|^{\frac{8}{n-4}-\varepsilon }u\), in \( S^n\), where \(n\ge 5\), \( \varepsilon \) is a small positive parameter and K is a smooth positive function on \(S^n\). We construct some sign-changing solutions which blow up at two different critical points of K. Furthermore, we construct sign-changing solutions of \((S_\varepsilon )\) having two bubbles and blowing up at the same critical point of K.

Uniqueness of closed self-similar solutions to $$\varvec{\sigma _k^{\alpha }}$$-curvature flow

By adapting the test functions introduced by Choi–Daskaspoulos (Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487, 2016) and Brendle–Choi–Daskaspoulos (Acta Math 219(1):1–16, 2017) and exploring the properties of the k-th elementary symmetric function \(\sigma _{k}\) intensively, we show that for any fixed k with \(1\le k\le n-1\), any strictly convex closed hypersurface in \({\mathbb {R}}^{n+1}\) satisfying \(\sigma _{k}^{\alpha }=\langle X,\nu \rangle \), with \(\alpha \ge \frac{1}{k}\), must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in \({\mathbb {R}}^{n+1}\) satisfying \(F+C=\langle X,\nu \rangle \), where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is a constant.

Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group

Let $(M,g)$ be a compact K\"ahler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the K\"ahler form $\omega_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell K\"ahler metric if the conformal metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell K\"ahler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature K\"ahler manifolds. More generally we consider extensions of Calabi functional and extremal K\"ahler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal K\"ahler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.

The coexistence of quasi-periodic and blow-up solutions in a superlinear Duffing equation

In this paper, we construct a continuous positive periodic function $p(t)$ such that the corresponding superlinear Duffing equation possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient $a(x)$ is an arbitrary positive smooth periodic function defined in the whole real axis.

On the principal Ricci curvatures of a Riemannian 3-manifold

Abstract We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)2m−(aD)2m] for every smooth function a. In the case of a positive a, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for Zm(a) in the case of a real function a. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the C∞-topology. We also describe all real functions a satisfying Zm(a)=0.

Bubbling solutions for an anisotropic planar elliptic problem with exponential nonlinearity

Let Ω be a bounded domain in R2 with smooth boundary, we study the following anisotropic elliptic problem −∇(a(x)∇υ)=a(x)[eυ−sϕ1−h(x)]inΩ,υ=0on∂Ω,where a(x) is a positive smooth function, h∈C0,α(Ω¯), s>0 is a large parameter and ϕ1 is a positive first eigenfunction of the problem −∇(a(x)∇ϕ)=λa(x)ϕ under Dirichlet boundary condition in Ω. We construct solutions of this problem which exhibit multiple concentration behavior around maximum points of a(x)ϕ1 in the domain as s→+∞.