Lower Order Perturbation Research Articles

Determination of lower order perturbations of a polyharmonic operator in two dimensions

Determination of lower order perturbations of a polyharmonic operator in two dimensions

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.

Gauge invariance of radiative jet functions in the position-space formulation of SCET

In subleading powers of soft-collinear effective theory (SCET), the Lagrangian contains couplings between soft quarks and hard-collinear quarks. Matrix elements of the hard-collinear parts of these couplings are radiative jet functions. In the position-space formulation of SCET, the Lagrangians are constructed from operators that appear to be gauge invariant. Nevertheless, we find violations of gauge invariance arise in the hard-collinear sector because gauge transformations can shift the momentum of a hard-collinear quark field from the hard-collinear sector to the soft sector, where the hard-collinear fields, by definition, have no support. The violations of gauge invariance are manifested in perturbation theory in the hard-collinear sector through the absence of certain Feynman diagrams that would be present in full QCD. A consequence of the absence of these diagrams is that the radiative jet functions that follow directly from the position-space Lagrangians are not gauge invariant, and we demonstrate this through explicit calculations in lower-order perturbation theory. We obtain gauge-invariant Lagrangians by adding to existing position-space Lagrangians terms that are proportional to the soft-quark equation of motion. These gauge-invariant Lagrangians are valid for nonzero, as well as zero, quark masses. We also remark briefly on the gauge invariance of certain Lagrangians that have been constructed in the label-momentum formulation of SCET. Published by the American Physical Society 2024

Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equations

We study a general class of focusing L 2 L^2 -critical nonlinear Schrödinger equations with lower order perturbations, in the possible absence of the pseudo-conformal symmetry and the conservation law of energy. In dimensions one and two, we construct multi-bubble Bourgain-Wang type blow-up solutions, which behave like a sum of pseudo-conformal blow-up solutions that concentrate at K K distinct singularities, 1 ≤ K &gt; ∞ 1\leq K&gt;{\infty } , and a regular profile. Moreover, we obtain the uniqueness in the energy class where the convergence rate is within the order ( T − t ) 4 + (T-t)^{4+} , for t t close to the blow-up time T T . These results in particular apply to the canonical nonlinear Schrödinger equations and, through the pseudo-conformal transform, yield the existence and conditional uniqueness of non-pure multi-solitons, which behave asymptotically as a sum of multi-solitons and a dispersive part. Thus, the results provide new examples of the mass quantization conjecture and the soliton resolution conjecture. Furthermore, through a Doss-Sussman type transform, we obtain multi-bubble Bourgain-Wang solutions for stochastic nonlinear Schrödinger equations, where the driving noise is taken in the sense of controlled rough path.

Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation

Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation

On Hamiltonian systems with critical Sobolev exponents

In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain Ω⊂RN, with N≥3, inspired by the classical results of Brezis and Nirenberg [4]. We solve the problem of finding a positive solution for all dimensions N≥4. For the critical dimension N=3 we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all 1-homogeneous or subcritical superlinear perturbations and parts where there are no solutions for some of those perturbations.

Effect of lower order perturbation on maximization problem associated with Trudinger–Moser inequality

Effect of lower order perturbation on maximization problem associated with Trudinger–Moser inequality

Construction of minimal mass blow-up solutions to rough nonlinear Schrödinger equations

We study the focusing mass-critical rough nonlinear Schrödinger equations, where the stochastic integration is taken in the sense of controlled rough path. In both dimensions one and two, the minimal mass blow-up solutions are constructed, which behave asymptotically like the pseudo-conformal blow-up solutions near the blow-up time. Furthermore, the global well-posedness is obtained if the mass of initial data is below that of the ground state. These results yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrödinger equations with lower order perturbations, particularly in the absence of the standard pseudo-conformal symmetry and the conservation law of energy.

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C2-smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold.

Strichartz and Local Smoothing Estimates for Stochastic Dispersive Equations with Linear Multiplicative Noise

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schrodinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schrodinger equations with variable coefficients and lower order perturbations, integrability of global solutions to stochastic nonlinear Schrodinger equations with constant coefficients. As another consequence, we prove as well the large deviation principle for the small noise asymptotics.

Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients

We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheĭm-Klibanov method based on Carleman estimates.

On a viscoelastic plate equation with a polynomial source term and p(x,t)-Laplacian operator in the presence of delay term

In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

Infinite order $$\Psi $$DOs: composition with entire functions, new Shubin-Sobolev spaces, and index theorem

We study global regularity and spectral properties of power series of the Weyl quantisation \(a^w\), where \(a(x,\xi ) \) is a classical elliptic Shubin polynomial. For a suitable entire function P, we associate two natural infinite order operators to \(a^{w}\), \(P(a^w)\) and \((P\circ a)^{w},\) and prove that these operators and their lower order perturbations are globally Gelfand–Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to \(\infty \) for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of f-\(\Gamma ^{*,\infty }_{A_p,\rho }\)-elliptic symbols, where f is a function of ultrapolynomial growth and \(\Gamma ^{*,\infty }_{A_p,\rho }\) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov–Hörmander integral formula.

Nonparametric estimation for linear SPDEs from local measurements

The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.

On a variational inequality for a plate equation with p-Laplacian end memory terms

ABSTRACT In this paper we investigate the unilateral problem for a plate equation with memory terms and lower order perturbation of p-Laplacian type where Ω is a bounded domain of , g>0 is a memory kernel and is a nonlinear perturbation. Using the penalty and Faedo–Galerkin's methods, we establish results on existence and uniqueness of weak solutions.

Minimal collision arcs asymptotic to central configurations

<p style='text-indent:20px;'>We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree <inline-formula><tex-math id="M1">\begin{document}$ -\alpha $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \alpha\in(0,2) $\end{document}</tex-math></inline-formula> and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the <i>collision manifold</i> having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

We study optimal control problems for stochastic nonlinear Schrodinger equations in both the mass subcritical and critical case. For general initial data of the minimal $$L^2$$ regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type $$U^2$$ and $$V^2$$ , adapted to evolution operators related to linear Schrodinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland’s variational principle.

Teukolsky formalism for nonlinear Kerr perturbations

We develop a formalism to treat higher order (nonlinear) metric perturbations of the Kerr spacetime in a Teukolsky framework. We first show that solutions to the linearized Einstein equation with nonvanishing stress tensor can be decomposed into a pure gauge part plus a zero mode (infinitesimal perturbation of the mass and spin) plus a perturbation arising from a certain scalar (‘Debye–Hertz’) potential, plus a so-called ‘corrector tensor’. The scalar potential is a solution to the spin −2 Teukolsky equation with a source. This source, as well as the tetrad components of the corrector tensor, are obtained by solving certain decoupled ordinary differential equations involving the stress tensor. As we show, solving these ordinary differential equations reduces simply to integrations in the coordinate r in outgoing Kerr–Newman coordinates, so in this sense, the problem is reduced to the Teukolsky equation with source, which can be treated by a separation of variables ansatz. Since higher order perturbations are subject to a linearized Einstein equation with a stress tensor obtained from the lower order perturbations, our method also applies iteratively to the higher order metric perturbations, and could thus be used to analyze the nonlinear coupling of perturbations in the near-extremal Kerr spacetime, where weakly turbulent behavior has been conjectured to occur. Our method could also be applied to the study of perturbations generated by a pointlike body traveling on a timelike geodesic in Kerr, which is relevant to the extreme mass ratio inspiral problem.

Deep optical imaging within complex scattering media

Optical imaging has had a central role in elucidating the underlying biological and physiological mechanisms in living specimens owing to its high spatial resolution, molecular specificity and minimal invasiveness. However, its working depth for in vivo imaging is extremely shallow, and thus reactions occurring deep inside living specimens remain out of reach. This problem originates primarily from multiple light scattering caused by the inhomogeneity of tissue obscuring the desired image information. Adaptive optical microscopy, which minimizes the effect of sample-induced aberrations, has to date been the most effective approach to addressing this problem, but its performance has plateaued because it can suppress only lower-order perturbations. To achieve an imaging depth beyond this conventional limit, there is increasing interest in exploiting the physics governing multiple light scattering. New approaches have emerged based on the deterministic measurement and/or control of multiple-scattered waves, rather than their stochastic and statistical treatment. In this Review, we provide an overview of recent developments in this area, with a focus on approaches that achieve a microscopic spatial resolution while remaining useful for in vivo imaging, and discuss their present limitations and future prospects. Optical microscopy is limited to shallow in vivo imaging depths owing to the exponential extinction of single-scattered waves by multiple light scattering. In this Review, we survey methodologies for deep optical imaging that maintain microscopic resolution by making deterministic use of multiple-scattered waves.

Maximization problem on Trudinger-Moser inequality involving Lebesgue norm

We study a maximization problem on the Trudinger-Moser inequality and clarify the effect of lower order perturbation on the attainability of the best constant. In particular, we show the explicit forms of threshold nonlinearity involving lower order perturbation. Our results are based on the estimates of the Dirichlet energy of maximizers.