Lower Order Perturbation Research Articles

Harmonic fields on the extended projective disk and a problem in optics

The Hodge equations for 1-forms are studied on Beltrami’s projective disk model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1-fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the noncharacteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptic-hyperbolic boundary-value problems is formulated for those equations as well. For both classes of boundary-value problems, an arbitrarily small lower-order perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f ∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f ∗ contains a one-parametric ordered family F ∗ . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family F ∗ . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in R with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in F ∗ . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.

Universal Matrix Perturbation Method for Structural Dynamic Reanalysis of General Damped Gyroscopic Systems

We investigate an effective matrix perturbation method for structural dynamic reanalysis of general damped gyroscopic systems. By using the complex eigensubspace condensation and the or thogonal decomposition procedures, two greatly reduced generalized eigenvalue equations are obtained. The lower-order perturbations of eigensolutions (i.e. complex eigenvalues and the corresponding left and right eigenvectors) are then determined by solving the two reduced eigenvalue problems. The higher-order perturbations of eigensolutions are obtained by executing a singular value decomposition procedure for a complex matrix. The proposed method is a universal perturbation method, for it is universally applicable to the reanalysis of general damped gyroscopic systems with all three cases of complex eigenvalues: distinct, repeated, and closely spaced eigenvalues. Numerical examples corresponding to the three different cases of eigenvalues are presented. The perturbed eigensolutions are computed using the present method and compared with the exact solutions.

On the Location of Blow up Points of Least Energy Solutions to the Brezis-Nirenberg Equation

In this paper, we study the asymptotic behavior of solutions to the semilinear elliptic problem involving critical Sobolev exponent with lower order perturbation, which was considered by Han and Rey before. We focus our attention on the least energy solutions obtained by the method of Brezis and Nirenberg, and show that the blow up point of the least energy solutions is a minimum point of the (positive) Robin function on the domain. This additional characterization extends the former result of Han and Rey in this case.

Non-symmetric perturbations of symmetric Dirichlet forms

We provide a path-space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local Dirichlet form. The representation is a combination of Feynman–Kac and Girsanov formulas, and extends previously known results in the framework of symmetric diffusion processes through the use of the Hardy class of smooth measures, which contains the Kato class of smooth measures.

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in R N , with N⩾5, a>0, α⩾0 and 2 ∗= 2N N−2 . We show that the exponent q= 2(N−1) N−2 plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem − Δu+au=u 2 ∗−1 −αu q−1 in Ω, u>0 in Ω, ∂u ∂ν =0 on ∂Ω. Namely, we prove that when q= 2(N−1) N−2 there exists an α 0>0 such that the problem has a least energy solution if α< α 0 and has no least energy solution if α> α 0.

Lower order perturbations of Dirichlet processes

Lower order perturbations of Dirichlet processes

A universal matrix perturbation technique for complex modes

A universal matrix perturbation technique for complex modes is presented. This technique is applicable to all the three cases of complex eigenvalues: distinct, repeated and closely spaced eigenvalues. The lower order perturbation formulas are obtained by performing two complex eigensubspace condensations, and the higher order perturbation formulas are derived by a successive approximation process. Three illustrative examples are given to verify the proposed method and satisfactory results are observed.

Assessment of higher order correlation effects with the help of Moller-Plesset perturbation theory up to sixth order

For 30 molecules and two atoms, MPn correlation energies up to n = 6 are computed and used to analyse higher order correlation effects and the initial convergence behaviour of the MPn series. Particularly useful is the analysis of correlation contributions EXY (n) ...(n = 4,5,6; X, Y,... = S, D, T, Q denoting single, double, triple, and quadruple excitations) in the form of correlation energy spectra. Two classes of system are distinguished, namely class A systems possessing well separated electron pairs and class B systems which are characterized by electron clustering in certain regions of atomic and molecular space. For class A systems, electron pair correlation effects as described by D, Q, DD, DQ, QQ, DDD, etc., contributions are most important, which are stepwise included at MPn with n = 2,...,6. Class A systems are reasonably described by MPn theory, which is reflected by the fact that convergence of the MPn series is monotonic (but relatively slow) for class A systems. The description of class B systems is difficult since three- and four-electron correlation effects and couplings between two-, three-, and four-electron correlation effects missing for lower order perturbation theory are significant. MPn methods, which do not cover these effects, simulate higher order with lower order correlation effects thus exaggerating the latter, which has to be corrected with increasing n. Consequently, the MPn series oscillates for class B systems at low orders. A possible divergence of the MPn series is mostly a consequence of an unbalanced basis set. For example, diffuse functions added to an unsaturated sp basis lead to an exaggeration of higher order correlation effects, which can cause enhanced oscillations and divergence of the MPn series.

Universal Perturbation Technique for Reanalysis of Non-Self-Adjoint Systems

A novel matrix perturbation technique for modal dynamic reanalysis of non-self-adjoint systems is presented. This technique is established first by performing complex eigensubspace condensation and orthogonal decomposition procedure. The lower-order perturbations of eigensolutions (i.e., complex eigenvalues, complex left and right eigenvectors) are obtained by solving two reduced eigenvalue problems. The higher-order perturbations of eigensolutions are then derived by implementing the successive approximation process. The proposed technique is a kind of universal dynamic reanalysis method because it is universally applicable to systems with all of the three cases: distinct, repeated, and closely spaced eigenvalues. Illustrative examples are given to verify the present method, and satisfactory results are observed.

On a model system for the oblique interaction of internal gravity waves

We give local and global well-posedness results for a system of two Kadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhu to model the oblique interaction of weakly nonlinear, two dimensional, long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves. We use the Fourier transform restriction norms introduced by J. Bourgain and the Strichartz estimates for the linear KP group. Finally we extend the result of [3] for lower order perturbation of the system in the absence of transverse effects.

Characterization of the homogeneous polynomials P for which (P + Q)(D) admits a continuous linear right inverse for all lower order perturbations Q

Characterization of the homogeneous polynomials P for which (P + Q)(D) admits a continuous linear right inverse for all lower order perturbations Q

Stationary profiles of degenerate problems when a parameter is large

The structure of positive solutions to nonlinear diffusion problems of the form $-\text{div}\ (|{\nabla} u|^{p-2}{\nabla} u) = \lambda f(u)$, in $\Omega$, $u = 0$ on $\partial \Omega$, $p > 1$, $\Omega \subset \mathbb R^N$ a bounded, smooth domain, is precisely studied as $\lambda \to +\infty $, for a class of logistic-type nonlinearities $f(u)$. By logistic it is understood that $f(u)/u^{p-1}$ is decreasing in $u > 0$, $f(u) \sim m u^{p-1}$, $m > 0$, as $u\to 0+$, while $f$ has a positive zero $u = u_0$ of order $k$. It is shown that the positive solution ${u_\lambda}$ homogenizes towards $u_0$ as $\lambda \to +\infty $, and develops a boundary layer near $\partial\Omega$ whose width is exactly measured. On the other hand, the arising of ``dead cores" $\{{u_\lambda} = u_0\}$ for $\lambda$ large is shown in the parameters regime $k < p-1$, the distance $\text{dist}(\{{u_\lambda} = u_0\},\partial\Omega)$ to $\partial \Omega$ being also exactly estimated as $\lambda \to +\infty $. Thus, earlier results in [12], [22] are substantially sharpened. In addition, suitable lower-order perturbations at infinity of the problem are studied.

PROBABILISTIC REPRESENTATIONS AND HYPERBOUND ESTIMATES FOR SEMIGROUPS

In this paper, we study lower order perturbations of a symmetric second-order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the, in general, not sub-Markovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coefficients.

A topological approach to evaluation of non-degenerate Schrödinger perturbation energy based on a circular scale of time

A recurrence formalism is given according to which any high-order Rayleigh–Schrödinger perturbation energy of non-degenerate state can be represented in terms of the lower-order perturbation energies. The formalism is developed with the aid of a circular topology of the scale of time.

Some results on $p$-Laplace equations with a critical growth term

Equations involving the $p$-Laplacian with a term in the critical growth range are considered. Existence results are obtained under minimal assumptions on the lower order perturbation. The problem is studied by means of variational methods; in particular, a problem with linking geometry is treated thanks to the orthogonalization technique introduced in [13].

Critical growth problems for polyharmonic operators

We prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower-order perturbations of the critical term. The results highlight the phenomenon of bifurcation of the critical dimensions discovered by Pucci and Serrin; moreover, we show that another bifurcation seems to appear for ‘nonresonant’ dimensions.

Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations

The solvability of semilinear elliptic equations with nonlinearities in the critical growth range depends on the terms with lower-order growth. We generalize some known results to a wide class of lower-order terms and prove a multiplicity result in the left neighborhood of every eigenvalue of $-\Delta$ when the subcritical term is linear. The proofs are based on variational methods; to assure that the considered minimax levels lie in a suitable range, special classes of approximating functions having disjoint support with the Sobolev ``concentrating" functions are constructed.

Existence theorems on the Dirichlet problem for the equation Δu + f(x, u)=0

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofup at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofaij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have\(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=up possesses a positive solution inH01(Ω).