SIAM Digital Library
 
 
 

SIAM J. on Mathematical Analysis

BROWSE VOLUMES

Year Range: 

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

1971

Volume 2, Issue 4, pp. 483-625


On Weak Solutions of a Mildly Nonlinear Dirichlet Problem

M. B. Rosenzweig

SIAM J. Math. Anal. 2, pp. 483-495 (13 pages)

Full Text: | Download PDF

Show Abstract
This paper investigates minimal conditions on functions of the type $F(x,v(x))$ and on the boundary of a domain $R$ to insure the existence and uniqueness of a weak solution to the mildly nonlinear Dirichlet problem \[ \begin{gathered} \Delta u(x) = F(x,u(x)),\quad x \in R, \hfill \\ u(x) = 0,\quad x \in \partial R, \hfill \\ \end{gathered} \] The principal technique is the construction of an operator by means of a linearization of the nonlinear equation. The operator is shown to satisfy the Schauder–Tikhonov theorem and the resulting fixed point is shown to be a solution of the mildly nonlinear problem.

A Recurrence Concerning Rayleigh Functions

N. Liron

SIAM J. Math. Anal. 2, pp. 496-499 (4 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
L. Carlitz has suggested the problem of evaluating $a(n,k) = \sum_{r = 1}^{n - 1} {r^k \sigma _2 (\nu )\sigma _{n - r} (\nu )} $, where $\sigma _r (v)$ are the Rayleigh functions. The special cases $k = 1,2,3$ were given by N. Kishore.
Both N. Kishore and L. Carlitz gave recurrence relations for $a(n,2k + 1,\nu )$ which involve $a(n,l,\nu )$, $l = 0,1,2, \cdots ,2k$. For $a(n,2k,v)$ their formulas lead to nothing new, and therefore are not sufficient to evaluate $a(n,k,v)$. In this paper we give a recurrence relation for $b_n (z,\nu ) = \sum_{r = 1}^{n - 1} {\sigma _r (\nu )\sigma _{n - r} (\nu )e^{rz} } $, which leads immediately to a recurrence relation for $a(n,k,\nu )$. The recurrence relation is valid for all $k$ and $n$.

A Variation of Hadamard’s Finite Part Integrals

Paul B. Bailey

SIAM J. Math. Anal. 2, pp. 500-504 (5 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
A variation of Hadamard’s finite part integrals is described which can be used to solve a class of problems in a direct manner, eliminating the “ascent-descent” technique which Hadamard used. It is shown that the Green’s formula derived from the standard singular solution for the scalar wave equation yields the Kirchhoff formulas, if, when the space dimension is odd, one retains the logarithmically infinite part of the principal value integral.

Existence Theorem and Convergence of Minimizing Sequences in Extremum Problems

Zita Poracká-Diviš

SIAM J. Math. Anal. 2, pp. 505-510 (6 pages)

Full Text: | Download PDF

Show Abstract
In the paper, a necessary and sufficient condition is given for the strong convergence of minimizing sequences for a differentiable convex functional in a reflexive Banach space.

Bounded Solutions for a Second Order Nonlinear Equation

H. Arthur Dekleine

SIAM J. Math. Anal. 2, pp. 511-520 (10 pages)

Full Text: | Download PDF

Show Abstract
The boundedness of all solutions for the second order nonlinear equation \[ \left[ {p(t)u'} \right]^\prime + \sum\limits_{k = 1}^N {a_k (t)f_k (u)} = e(t)\] is studied. It is assumed that the product $p(t)a_k (t)$ is, for each $k$, a locally integrable perturbation of a continuous function, is locally of bounded variation and has a small negative variation. By appealing to a Stieltjes version of Gronwall’s inequality, bounds are obtained for the energy integral associated with a particular solution, and consequently for the solution.

Pointwise Bounds on Derivatives of Solutions to Ordinary Differential Equations

Geoffrey Butler and Thomas Rogers

SIAM J. Math. Anal. 2, pp. 521-528 (8 pages)

Full Text: | Download PDF

Show Abstract
Various pointwise estimates on the derivatives of solutions to $n$th order linear differential equations in terms of an associated Taylor polynomial are derived. These estimates are used to obtain a necessary and sufficient condition for a function which satisfies a sequence of linear differential equations on an interval to be regular on that inverval.

Trigonometric Approximation in the Sobolev Spaces $W^{r,2} [ - \pi ,\pi ]$ with Constant Weights

Edgar A. Cohen, Jr.

SIAM J. Math. Anal. 2, pp. 529-535 (7 pages)

Full Text: | Download PDF

Show Abstract
It is shown that, although sines and cosines are not, in general, complete in $W^{r,2} [ - \pi ,\pi ]$, they are complete in the subspace of those functions whose first $r - 1$ derivatives are periodic of period $2\pi $. Also, the sequence of sines and cosines is extended to a complete sequence.

Disconjugacy Tests for Singular Linear Differential Equations

D. Willett

SIAM J. Math. Anal. 2, pp. 536-545 (10 pages) | Cited 4 times

Full Text: | Download PDF

Show Abstract
Special disconjugacy tests of the de la Vallée Poussin type for a closed interval $[\alpha ,\beta ]$, which need not be bounded, are derived for linear differential equations with continuous coefficients on the open interval $(\alpha ,\beta )$. The method applies, in general, to linear perturbations of disconjugate linear equations. The results include a precise first-term asymptotic description at both $\alpha $ and $\beta $ of a fundamental system of solutions.

The Asymptotic Distribution of Eigenvalues for the Boundary Value Problem $y''(x) - \lambda ^2 p(x)y(x) = 0,y \in L_2 ( - \infty , + \infty )$

Laurence Weinberg

SIAM J. Math. Anal. 2, pp. 546-566 (21 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
In this paper we study the problem $y''(x) - \lambda ^2 p(x)y(x) = 0$, $y \in L_2 ( - \infty , + \infty )$. Sufficient conditions are given on $p(x)$ to insure the existence of eigenvalues and to enable one to compute the asymptotic distribution of all large positive eigenvalues. The equation considered is of a form of interest in wave mechanics and particle scattering.

The Behavior as $\varepsilon \to + 0$ of Solutions to $\varepsilon \nabla ^2 w = {{\partial w} / {\partial y}}$ in $| y | \leqq 1$ for Discontinuous Boundary Data

L. Pamela Cook and G. S. S. Ludford

SIAM J. Math. Anal. 2, pp. 567-594 (28 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
The title problem is considered for boundary data $w(x, - 1) = f(x)$, $w(x,1) = g(x)$. Here $f$, $g$ are infinitely differentiable except at $x = 0$, a respectively, where they have right- and left-hand derivatives of all orders. With $g = 0$ five regions are distinguished : the core $0 < x_0 \leqq | x |$ and the free layer $\varepsilon ^{ - {1 / 2}} | x | \leqq X_\infty $, excluding$0 < x_0 \leqq | x |$ and the free layer $\varepsilon ^{ - {1 / 2}} | x | \leqq X_0$, $| {y + 1} | \leqq y_{ - 1} $, in $ - 1 \leqq y \leqq y_1 < 1$; their boundary layers $\varepsilon ^{ - 1} (1 - y) \leqq Y_\infty $; and the excluded region $\varepsilon ^{ - 1} | x | \leqq X_{ * \infty } $, $\varepsilon ^{ - 1} (1 + y) \leqq y_{ * \infty } $. The solution for $f = 0$ is asymptotically zero everywhere except in the boundary layer, where $0 < x_a \leqq | {x - a} |$ is distinguished from the transition zone $\varepsilon ^{ - 1} | {x - a} | \leqq X_{ * \infty } $. By means of Fourier transforms it is shown that the method of matched asymptotic expansions gives approximations to all orders in each of the regions, and that the latter can be extended to overlap. For the excluded region, which gives birth to the “parabolic” free layer, this contradicts what has previously been supposed. Of particular interest is the transition zone, which resolves a breakdown in the “hyperbolic” boundary layer. The expansion in the core is determined independently of the others, but not that in the free layer. As a consequence, the odd powers of $\varepsilon ^{{1 / 2}} $ which appear in the free layer are absent in the core. Other assumptions concerning $f$ and $g$ are also considered.

The Summation of Series

M. Lawrence Glasser

SIAM J. Math. Anal. 2, pp. 595-600 (6 pages)

Full Text: | Download PDF

Show Abstract
A number of formulas are presented for representing a variety of infinite series in terms of rapidly convergent definite integrals. Several new explicit summations are obtained by these methods.

Uncertainty Inequalities for Hankel Transforms

Patrick C. Bowie

SIAM J. Math. Anal. 2, pp. 601-606 (6 pages) | Cited 3 times

Full Text: | Download PDF

Show Abstract
In this paper an uncertainty inequality for Hankel transforms is obtained.
Let $\nu > 0$ be fixed. We set \[ d\mu _\nu (x) = c_\nu ^{ - 1} x^{2v} dx,\quad c_\nu = 2^{{{\nu - 1} / 2}} \Gamma (\nu + \frac{1}{2}),\] and \[ {\bf J}_\nu (x) = c_\nu x^{ - \nu + {1 / 2}} J_{\nu - {1 / 2}} (x),\] where $J_{\nu - {1 / 2}} (x)$ is a Bessel function of the first kind of order $\nu - \frac{1}{2}$. We define \[ f^ \wedge (t;\nu ) = \int_0^\infty {f(x)J_\nu (xt)d\mu _\nu (x)} .\]
A probability frequency function with respect to $d\mu _\nu $, is defined as a nonnegative function in $L_\nu ^1 (0,\infty )$ with norm one, and the generalized variance of a probability frequency function $F(x)$ is defined by \[ V_\nu [F] = \int_0^\infty {x^2 F(x)d\mu _\nu (x)} .\] Let $f(x)$ belong to $L_\nu ^2 (0,\infty )$ with norm one. By Parseval’s equality $| {f(x)} |^2 $ and $| {f^\wedge (x;v)} |^2 $ can be considered as probability frequency functions. The uncertainty inequality \[ V_\nu \left[ {| {f(x)} |^2 } ]V_\nu [ {| {f^ \wedge (x;\nu )} |^2 } \right] \geqq (\nu + \frac{1}{2})^2 \] is proved, and the constant $(\nu + \frac{1}{2})^2 $ is shown to be the best possible.

Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

R. A. Hunt and M. H. Taibleson

SIAM J. Math. Anal. 2, pp. 607-625 (19 pages) | Cited 6 times

Full Text: | Download PDF

Show Abstract
It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field $K$. This includes the case where $K$ is a $p$-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjölin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of the valuation) of the local field case over the trigonometric case have been utilized. Consequently many purely technical elements of the trigonometric case have disappeared and one is left only with elements of the proof which bear on the central idea. For this reason the proof given can be used to obtain a clearer understanding of the proof for trigonometric Fourier series.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Fast Algorithms for Image Reconstruction with Application to Partially Parallel MR Imaging
  2. Image Denoising Using Mean Curvature of Image Surface
  3. A Variational Approach for Sharpening High Dimensional Images
  4. Contour Stencils: Total Variation along Curves for Adaptive Image Interpolation
  5. Directional Multiscale Amplitude and Phase Decomposition by the Monogenic Curvelet Transform
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close