Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)$P(D)$ be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on Rⁿ${\mathbb{R}}^n$ and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e^−tH$e^{-tH}$ generated by H=P(D)+V$H=P(D)+V$ with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e^−t(Δ+V)$e^{-t(\mathrm{\Delta }+V)}$ as V∈_K2(Rⁿ)$V\in K_2({\mathbb{R}}^n)$, the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e^−tP(D)$e^{-tP(D)}$ on L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. As a consequence of the Gaussian upper bound, the L^p$L^p$-spectral independence of H is concluded.     

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if −A   generates a _C0$C_0$-semigroup on a Hilbert space, then for each τ>0$\tau >0$ the operator A   has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy f(z)=O(e^−τRe(z))$f(z)=O(e^{-\tau \mathrm{Re}(z)})$ as |z|→∞$|z|\to \infty$. The bound of this calculus grows at most logarithmically as τ↘0$\tau \searrow 0$. As a consequence, f(A)$f(A)$ is a bounded operator for each holomorphic function f (on a right half-plane) with polynomial decay at ∞. Then we show that each semigroup generator has a so-called (strong) m  -bounded calculus for all m∈N$m\in \mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called γ-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.     

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

A note on value distribution of difference polynomials of meromorphic functions

In this paper, we investigate the value distribution of the difference counterpart Δf(z)-afn(z)of f^′(z) -afn(z)$\Delta f(z)-af{(z)}^n\text{o}\text{f}\hspace{0.17em}{f}^{\prime }(z)\hspace{0.17em}-af{(z)}^n$ and obtain an almost direct difference analogue of result of Hayman.     

Solutions of a system of forced burgers equation in terms of generalized laguerre polynomials

In this article, we obtain explicit solutions of a linear PDE subject to a class of radial square integrable functions with a monotonically increasing weight function |x|ⁿ⁻¹eβ|x|²/2,${\left|x\right|}^{n-1}{e}^{\beta {\left|x\right|}^2}/2,$, β≥0, x ∈?n$\beta \ge 0,\hspace{0.17em}x\hspace{0.17em}\in {?}^n$. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n > 1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.     

Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ (μ>0$\mu >0$) in C^n+m${\mathbf{C}}^{n+m}$ is defined by the inequality ‖w‖²<e^{−μ‖z‖2}${\Vert w\Vert }^2<e^{-\mu {\Vert z\Vert }^2}$, where (z,w)∈Cⁿ×C^m$(z,w)\in {\mathbf{C}}^n\times {\mathbf{C}}^m$, which is an unbounded non-hyperbolic domain in C^n+m${\mathbf{C}}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock–Bargmann–Hartogs domains in terms of the polylogarithm functions and Kim–Ninh–Yamamori determined the automorphism group of the domain _Dn,m(μ)$D_{n,m}(\mu )$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock–Bargmann–Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ with m≥2$m\ge 2$ must be an automorphism.     

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r   points in very general position on a smooth elliptic quartic curve B⊂P³$B\subset {\mathbb{P}}^3$ and let L∈Pic(X)$L\in \mathrm{Pic}(X)$ be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H⁰(X,L)$H^0(X,L)$.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations

One of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form P(x)=P(_gp(x,Λ)?_cp,p=1,…,_Nc)$P(x)=\mathbb{P}(g_p(x\text{,}\Lambda )?c_p\text{,}p=1\text{,}\dots \text{,}{N}_c)$. Here x∈R^_Nx$x\in {\mathbb{R}}^{N_x}$ is the vector of physical parameters, Λ∈R^_NΛ$\Lambda \in {\mathbb{R}}^{N_{\Lambda }}$ is a random vector describing the uncertainty of the model, g:R^_Nx×R^_NΛ→R^_Nc$g:{\mathbb{R}}^{N_x}\times {\mathbb{R}}^{N_{\Lambda }}\to {\mathbb{R}}^{N_c}$ is the constraints mapping, and c∈R^_Nc$c\in {\mathbb{R}}^{N_c}$ is the vector of constraint levels. In this paper specific Monte Carlo tools for the estimations of the gradient and Hessian of P(x)$P(x)$ are proposed when the input random vector Λ$\Lambda$ has a multivariate normal distribution and small variances. Using the small variance hypothesis, approximate expressions for the first- and second-order derivatives are obtained, whose Monte Carlo estimations have low computational costs. The number of calls of the constraints mapping g   for the proposed estimators of the gradient and Hessian of P(x)$P(x)$ is only 1+2_Nx+2_NΛ$1+2N_x+2N_{\Lambda }$.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.