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We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semiclassical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one-dimensional setting, we obtain the complete asymptotic expansion, in powers of $h$, of each eigenvalue. In two dimensions we obtain the left margin of the spectrum, under some additional assumptions.

This paper is concerned with the rigorous analysis of the zero electron mass limit of the full Navier--Stokes--Poisson. This system has been introduced in the literature by Anile and Pennisi (see [Phys. Rev. B, 46 (1992), pp. 13186--13193]) in order to describe a hydrodynamic model for charge-carrier transport in semiconductor devices. The purpose of this paper is to prove rigorously zero electron mass limit in the framework of general ill-prepared initial data. In this situation the velocity field and the electronic fields develop fast oscillations in time. The main idea we will use in this paper is a combination of formal asymptotic expansion and rigorous uniform estimates on the error terms. Finally we prove the strong convergence of the full Navier--Stokes--Poisson system toward the incompressible Navier--Stokes equations.

The Helmholtz equation with rapidly oscillating absorption coefficient and constant scattering (diffusion) coefficient is considered. It models the light absorption in a tissue containing a periodic set of thin blood vessels; it is assumed that the absorption takes place within these vessels only. So, the scattering coefficient is supposed to be constant while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is equal to the large parameter $\omega$. Two other parameters appear in the problem: $\varepsilon$ is the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and $\delta$, which is the ratio of the thickness of thin vessels and the period. Both parameters $\varepsilon$ and $\delta$ are small. The one-dimensional setting is considered. The classical high order homogenization method is applicable only in the case $\varepsilon^2\omega\delta\to 0, \omega\delta\to \infty$, w...

The equation studied is $u^{\prime\prime}+\frac{n-1}{r}u^{\prime}+\varepsilon u\,u^{\prime}+ku^{\prime2}=0$, with boundary conditions $u\left(1\right)=0$, $u\left(\infty\right) =1$. This model equation has been studied by many authors since it was introduced in the 1950s by P. A. Lagerstrom. We use an elementary approach to show that there is an infinite series solution which is uniformly convergent on $1\leq r 0$, $k\geq0$, and $n\geq1$.

We consider the orbital stability of single-spike bound states of semiclassical nonlinear Schrodinger equations with critical nonlinearity and a trap potential. Due to the effect of the trap potential, we derive the asymptotic expansion formulas and obtain the necessary conditions for orbital stability and instability of single-spike bound states. Our argument is applied to two-component systems of nonlinear Schrodinger equations with a common trap potential, cubic nonlinearity in two spatial dimensions. The orbital stability of bound states with spikes of these systems is investigated. Our results show the existence of stable spikes in two-dimensional Bose–Einstein condensates.

Effective boundary conditions (wall laws) are commonly employed to approximate PDEs in domains with rough boundaries, but it is neither easy to design such laws nor to estimate the related approximation error. A two-scale asymptotic expansion based on a domain decomposition result is used here to mitigate such difficulties, and as an application we consider the Poisson equation. The proposed scheme considers rough curved boundaries and allows a complete asymptotic expansion for the solution, highlighting the influence of the boundary curvature. The derivation and estimation of high order effective conditions is a corollary of such development. Sharp estimates for first and second order wall law approximations are considered for different Sobolev norms and show superior convergence rates in the interior of the domain. A numerical test illustrates several of the results obtained here.

A problem of homogenization of a divergence‐type second order uniformly elliptic operator is considered with arbitrary bounded rapidly oscillating periodic coefficients, either with periodic “outer” boundary conditions or in the whole space. It is proved that if the right‐hand side is Gevrey regular (in particular, analytic), then by optimally truncating the full two‐scale asymptotic expansion for the solution one obtains an approximation with an exponentially small error. The optimality of the exponential error bound is established for a one‐dimensional example by proving the analogous lower bound.

In this paper the vanishing Debye length limit (space charge neutral limit) of bipolar time-dependent drift-diffusion models for semiconductors with p-n junctions (i.e., with a fixed bipolar background charge) is studied in one space dimension. For general sign-changing doping profiles, the quasi-neutral limit (zero-Debye-length limit) is justified rigorously in the spatial mean square norm uniformly in time. One main ingredient of our analysis is the construction of a more accurate approximate solution, which takes into account the effects of initial and boundary layers, by using multiple scaling matched asymptotic analysis. Another key point of the proof is the establishment of the structural stability of this approximate solution by an elaborate energy method which yields the uniform estimates with respect to the scaled Debye length.

Established here is the uniquenes of solutions for the traveling wave problem cU � (x )= U (x+1)+U (x−1)−2U (x)+f (U (x)), x ∈ R, under the monostable nonlinearity: f ∈ C 1 ((0, 1)) ,f (0) = f (1) =0

We consider, on an interval of arbitrary length, global minimizers of a class of energy functionals containing a small parameter $\varepsilon$ and a long-range interaction. Such functionals arise from models for phase separation in diblock copolymers and from stationary solutions of FitzHugh--Nagumo systems. We show that every global minimizer is periodic with a period of order $\varepsilon^{1/3}$. Also, we identify the number of global minimizers and provide asymptotic expansions for the periods and global minimizers.

This work is concerned with diffusion processes having fast and slow components. It was known that under suitable assumptions the slow component can be approximated by the Markov process with averaged characteristics. In this work, asymptotic expansions for the solutions of the Kolmogorov backward equations are constructed and justified. Certain probabilistic conclusions and examples are also provided.

In the author's paper [SIAM J. Math. Anal., 44 (2012), pp. 3786--3805], the author employs the decay estimates of the fundamental solution of the linear dissipative equation to derive the asymptotic expansion for the dissipative equation with the potential term. However, these decay estimates contain an error. The corrected estimates published here prove the main theorems in the original paper.

We rigorously derive the leading order term in the asymptotic expansion of the scattering amplitude of a collection of a finite number of dielectric inhomogeneities of small diameter. We then apply this asymptotic formula for the purpose of identifying the location and certain properties of the shapes of the small inhomogeneities from scattering amplitude measurements at a fixed frequency. Our main idea is to reduce this reconstruction problem to the calculation of an inverse Fourier transform.

In this paper we study the stability of integrable Hamiltonian systems under small perturbations, proving a weak form of the KAM/Nekhoroshev theory for viscosity solutions of Hamilton--Jacobi equations. The main advantage of our approach is that only a finite number of terms in an asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved. An application of these results is to show that Diophantine invariant tori and Aubry--Mather sets are stable under small perturbations.

One-dimensional free energy local minimizers are viewed as three-dimensional lamellar-type critical points in a box. To determine whether they model the lamellar phase of diblock copolymers in the strong segregation region, we analyze their spectra. We obtain the asymptotic expansions of their eigenvalues and eigenfunctions. Consequently we find that they are stable, i.e., are local minimizers in space, only if they have sufficiently many interfaces. Interestingly the one-dimensional global minimizer is near the borderline of three-dimensional stability.

We derive high-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for inhomogeneities with Lipschitz boundaries and those with extreme conductivities.

The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, the well-developed geometric theory does not apply. We present a method based on blow-up techniques, which leads to a rigorous geometric analysis of these problems. A detailed analysis of the extension of slow manifolds past fold points and canard points in planar systems is given. The efficient use of various charts is emphasized.

Let {pn(x)}_{n\ge 0}$ be the set of orthonormal polynomials with respect to the exponential weight w(x)=e-v(x) , where v(x)=x2m + ... is a monic polynomial of degree 2m with $m \ge 2$ and is even. An asymptotic approximation is obtained for pn(x), as $n \to \infty$, which holds uniformly for $0 \le x \le O(n^{1/2m})$. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

Here we investigate the limits and the boundary layers of the three-dimensional displacement in thin elastic plates as the thickness tends to zero in each of the eight main types of lateral boundary conditions on their edges: hard and soft clamped, hard and soft simple support, friction conditions, sliding edge, and free plates. Relying on construction algorithms [M. Dauge and I. Gruais, Asymptotic Anal., 13 (1996), pp. 167--197], we establish an asymptotics of the displacement combining inner and outer expansions. We describe the two first terms in the outer expansion: these are Kirchhoff--Love displacements satisfying prescribed boundary conditions that we exhibit. We also study the first boundary layer term: when the transverse component is clamped, it has generically nonzero transverse and normal components, whereas when the transverse component is free, the first boundary layer term is of bending type and has only its nonzero in-plane tangential component.

When the thickness of the interface (denoted by $\eps$) tends to zero, any stable stationary internal layered solutions to a class of reaction--diffusion systems cannot have a smooth limiting interfacial configuration. This means that if the limiting configuration of the interface has a smooth limit, it must become unstable for small $\eps$, which makes a sharp contrast with the one-dimensional case. This suggests that stable layered patterns must become very fine and complicated in this singular limit. In fact we can formally derive that the rate of shrinking of stable patterns is of order $\eps^{1/3}$. Using this scaling, the resulting rescaled reduced equation determines the morphology of magnified patterns. A variational characterization of the critical eigenvalue combined with the matched asymptotic expansion method is a key ingredient for the proof, although the original linearized system is not of self-adjoint type.

It is known that existence of a formal power series solution $\hat y(x)$ to a system of nonlinear ordinary differential equations (ODEs) with analytic or infinitely smooth coefficients at an irregular singular point implies the existence of an actual solution y(x), which possesses the asymptotic expansion $\hat y(x)$. In the present paper we extend this result for systems with finitely smooth coefficients. In this case one cannot speak about a formal power series solution $\hat y(x)$; it has therefore to be replaced by the requirement of existence of an "approximate" solution y0 (x). The existence of a corresponding actual solution is a subject of certain conditions that link the smoothness of the system, the "accuracy" of the approximation y0 (x), and the "degeneracy" of the system, linearized with respect to y0 (x). As applications, problems of reduction of linear time dependent systems of ODEs into diagonal and triangular forms, as well as some other problems, are considered. In particular, the well-kn...

Let an integer $m\ge0$ be fixed. Let ${\cal X}_m$ be the space of functions $f\in C^{\infty}({\Bbb R}^{n})$ that admit an asymptotic expansion $f(r\beta)\sim \sum_{k=m}^{\infty} {\psi_k(\beta)}/{r^{n+k}}, r\to\infty$, $\psi_k\in C^{\infty}(S^{n-1})$, and the expansion can be differentiated with respect to $x=r\beta$ any number of times. In this paper, we derive a precise characterization of the range of the Radon transform R acting on ${\cal X}_m$; that is, we explicitly describe the space ${\cal Z}_m=R{\cal X}_m$. The conditions which describe the space ${\cal Z}_m$ are easily verifiable.

We study the asymptotic behavior of radially symmetric solutions of the nonlocal equation $$ \varepsilon\phi_t- \varepsilon\Delta\phi +\frac{1}{\varepsilon}W'(\phi)-\lambda_\varepsilon (t) =0 $$ in a bounded spherically symmetric domain $\Omega\subset\RN$, where $\lambda_\varepsilon (t)=\frac{1}{\varepsilon} \int_{\Omega}{\!\!\!\!\!\!\!\!-} \ W'(\phi)\, dx$, with a Neumann boundary condition. The analysis is based on "energy methods" combined with some a priori estimates, the latter being used to approximate the solution by the first two terms of an asymptotic expansion. We only need to assume that the initial data as well as their energy are bounded. We show that, in the limit as $\varepsilon\to 0$, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a by-product of our analysis, we obtain an $L^2$ estimate on the "Lagrange multiplier" $\lambda_\varepsilon (t)$, which holds in the nonradial case as well. In addition, we show rigorously (in general geometry) that the nonlocal G...

We consider here the problem of describing the melting of an ice ball surrounded by water. The corresponding mathematical model consists of the Stefan problem with radial symmetry. We obtain asymptotic expansions for the radius of the melting ball which turn out to be of a different nature according to the cases N greater than or equal to 3 and N = 2, N being the space dimension. The methods employed combine matched asymptotic expansion techniques, a priori estimates, and topological results.

Asymptotic expansions are obtained for the coefficients of capacity and induction of two spheres which hold as the distance $\varepsilon $ between the spheres tends to zero, starting from expressions in terms of definite integrals obtained earlier [A. H. Van Tuyl, Electrostatic problems for two conducting spheres, SIAM J. Math. Anal., 20 (1989), pp. 1293–1320]. Bounds for the remainders are given which hold uniformly for all ratios of the radii of the spheres. These asymptotic expansions are used to obtain an asymptotic expansion for the capacity of the spheres with respect to the infinite sphere with a uniform bound for the remainder and to find the asymptotic behavior of the capacity of two spheres. The asymptotic expansion for the capacity of two spheres with respect to the infinite sphere is also found directly from a definite-integral representation involving elliptic functions, leading to a smaller uniform bound for the remainder. Finally, the asymptotic behavior of the coefficients of potential is ...

A uniform asymptotic expansion in the inertial constant e is given for one-dimensional linear thermoelasticity by a two-timing method. The result shows that the usual coupled/quasi-static approximation $(\epsilon = 0)$ is not uniformly asymptotically correct even to lowest order. The interaction between diffusion and wave propagation is clearly displayed. The proof of uniform asymptotic validity to order $\epsilon ^2 $ is accomplished by energy estimates.

An asymptotic expression for the large n behavior of Pollaczek polynomials of degree n and argument $\cos ({\theta /{\surd n}})$ is derived. This is applied to find the second term in the asymptotic expansion of the zeros of the Pollaczek polynomials and it also solves a problem of Richard Askey. Some new analytic properties of a confluent Horn function are also proved.

A new method for representing the remainder and coefficients in Airy-type expansions of integrals is given. The quantities are written in terms of Cauchy-type integrals and are natural generalizations of integral representations of Taylor coefficients and remainders of analytic functions. The new approach gives a general method for extending the domain of the saddle-point parameter to unbounded domains. As a side result the conditions under which the Airy-type asymptotic expansion has a double asymptotic property become clear. An example relating to Laguerre polynomials is worked out in detail. How to apply the method to other types of uniform expansions, for example, to an expansion with Bessel functions as approximants, is explained. In this case the domain of validity can be extended to unbounded domains and the double asymptotic property can be established as well.

There has been renewed interest in both formal and rigorous theories of exponentially-small contributions to asymptotic expansions. In particular, a generalized asymptotic expansion was obtained for the confluent hypergeometric function $U(a,a - b + 1,z)$ in which the parameters a and b are complex constants, and z is a large complex variable. This expansion is expressed in terms of generalized exponential integrals and has a larger region of validity and greater accuracy than conventional expansions of Poincare type. The expansion was established by transformations and a re-expansion of an integral representation of $U(a,a - b + 1,z)$. In this paper it is shown how the same result can be achieved by a direct differential-equation approach, thereby laying the foundation for a rigorous theory of generalized asymptotic solutions of linear differential equations.

Explicit and realistic error bounds are constructed for a one-term and a two-term asymptotic approximation of the Jacobi function $\varphi _\mu ^{(\alpha ,\beta )} (t)$ as $\mu \to \infty $, uniformly for $t \in (0,\infty )$. A similar result is obtained for the zeros $t_{\mu ,k} $ of this function as $\mu \to \infty $, which holds uniformly with respect to unbounded k. Exponentially decaying error bounds are also given for asymptotic approximations of $\varphi _\mu ^{(\alpha ,\beta )} (t)$ as $t \to \infty $ and of $t_{\mu ,k} $ as $k \to \infty $, which are uniform for $\mu \geqq \delta > 0$.

Error bounds are obtained for an asymptotic expansion of the ratio of two gamma functions ${{\Gamma (z + a)} / {\Gamma (z + b)}}$ when a and b are complex constants and $|z|$ is large. These bounds reduce to earlier bounds for the real case when a, b, and z are real. Properties of completely monotonic functions are used to provide error bounds in the complex case, as in the earlier real case.

The asymptotic behaviour of the “swallowtail integral” is examined, a generalized Airy function, given by \[ S(x,y,z) = \int_{ - \infty }^{ + \infty } {\exp \left\{ {i\left( {{{t^5 } / {5 + {{xt^3 ...

By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “exponential improvement.” It is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions, and also in terms of the error function.Inter alia, the results supply a rigorous foundation for some of the recent work of M. V. Berry on a smooth interpretation of the Stokes phenomenon.

A procedure is presented for obtaining the complete asymptotic expansion of a class of fractional integrals (of Riemann–Liouville type), in which the integrand contains the product of two derivatives of the Fermi–Dirac integral. The procedure uses two-sided Laplace transforms and Abelian asymptotics of the inverse Laplace transform. The fractional integrals considered arise in various problems from statistical mechanics and solid state physics.

This paper discusses the relevance of the asymptotic behavior of the coefficients of asymptotic power series for the study of Stokes phenomena. By way of illustration a connection problem is considered in the theory of linear difference equations.

Asymptotic expansions are derived for the double integral \[ \iint_D {g(x,y)\exp (iNf(x,y))dx\,dy}, \] as $N \to + \infty $, where $f(x,y)$ has a stationary point at a corner of the boundary of D. Two different methods are given, one for the case of local extrema and one for saddlepoints. In the case of saddlepoints, our method allows the boundary of D to be tangent to a level curve of f.