Heat kernel bounds and desingularizing weights

We study the parabolic operator ∂_t−Δ_x+V(t,x), in d?1, with a potential V=V+−V−,V±?0 assumed to be from a parabolic Kato class, and obtain two-sided Gaussian bounds on the associated heat kernel. The constraints on the Kato norms of V⁺ and V⁻ are completely asymmetric, as they should be. Further improvements to our heat kernel bounds are obtained in the case of time-independent potentials.If V has singularities of the type ±c|x|⁻² with a suitably small constant c, we obtain new lower and (sharp) upper weighted heat kernel bounds. The rate of growth of the weights depends (explicitly) on the constant c. The standard bounds and methods (estimates in L^p-spaces without desingularizing weights) fail for singular potentials.     

Note on forced oscillation of nth-order sublinear differential equations

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t₀,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x^″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673].     

Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

This paper studies the scattering matrix S(E;?) of the problem

−?²ψ^″(x)+V(x)ψ(x)=Eψ(x)     

A constructive procedure to recover asymptotics of short-range or long-range potentials

We study an inverse scattering problem for a pair of Hamiltonians (H,H₀) on L²(Rⁿ), where H₀=-Δ and H=H₀+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets.     

Invariance of quasi-arithmetic means with function weights

Let I⊂R be a non-trivial interval and let . We present some results concerning the following functional equation, generalizing the Matkowski-Sutô equation,

λ(x,y)φ⁻¹(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ⁻¹(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,     

On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue λ₁(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C¹ compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ₁(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ₁(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

Isolated singularities for weighted quasilinear elliptic equations

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|^p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b₀(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities

We obtain Sobolev inequalities for the Shcrödinger operator −Δ−V, where V has critical behaviour V(x)=((N−2)/2)²|x|⁻² near the origin. We apply these inequalities to obtain point-wise estimates on the associated heat kernel, improving upon earlier results.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces

We give the boundedness on Triebel-Lizorkin spaces for oscillatory singular integral operators with polynomial phases and rough kernels of the form eiP(x)Ω(x)|x|⁻ⁿ, where Ω∈Llog⁺L(Sn−1) is homogeneous of degree zero and satisfies certain cancellation condition.     

A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|^−n−α(α>0) at the origin, an oscillating factor ei|y|^−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions x^α|ψ⁽ⁱ⁾(x+β)| and α|ψ⁽ⁱ⁾(x+β)|−x|ψ⁽ⁱ⁺¹⁾(x+β)| respectively to be monotonic and completely monotonic on (0,∞), where i∈N, α>0 and β≥0 are scalars, and ψ⁽ⁱ⁾(x) are polygamma functions.     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.