Inequalities and Applications

The following sections are included:

• Contributors

• PREFACE

• CONTENTS

The following sections are included:

• The line method

• Qualitative results

• Blowup or quenching

• Parabolic-functional equations

• Strongly coupled parabolic equations

• Lotka-Volterra systems

• Invariant sets

• Periodic solutions

• Differential inequalities in Banach spaces

• Multiple solutions for the Dirichlet problem

• Miscellany

• Editorships and books

• List of Publications

We study free-boundary problems involving generalized capacitary potentials in annular domains. Given λ > 0 and the convex exterior boundary component, we seek a convex interior boundary component Γ_λ such that the capacitary potential has a normal derivative of λ on Γ_λ. In this context, we study the existence of monotonically and continuously-varying families of solution surfaces, parametrized by λ. These solution families are the basis for a (non-global) uniqueness theory.

The general solution of a system of functional equations which seems fundamental in the theory of price and quantity indices is determined. Then conditions for the monotonicity of this solution (that is, inequalities) are given.

Consider the Dirichlet problem for Poisson's equation −Δu = f for a family C of right hand sides f, whose distribution-function is prescribed, i.e. f ≥ 0 and for every t ∊ R the size of {x ∊ Ω | f(x) ≥ t} is given. How does one have to distribute f in Ω such as to render certain norms of u as small as possible?

In this paper we establish continuous dependence on the initial time geometry for solutions of the backward heat equation under rather weak constraint hypotheses on solutions. Similar results can be derived for more general heat operators.

Please refer to full text.

We prove that solutions of quasilinear elliptic problems with blowup at the boundary are convex in convex domains. We first derive asymptotic estimates by means of a method proposed in [10]. These bounds together with Korevaar's maximum principle [9] are then used to establish the convexity. The last part contains gradient estimates.

The inequalities we will investigate are associated with on (0,1] and [1, ∞). On [1, ∞), Mf = λf is in the strong-limit-point case and the inequality is of the type attributed to Everitt (HELP). The equation Mf = λf is limit-circle and non-oscillatory (for λ ∊ R) on (0,1] and the inequality is of the type studied by Evans and Everitt. Estimates of the best constants and cases of equality are determined in each case.

Weighted interpolation inequalities are derived which bound a weighted p norm of the jth derivative of a function u by a product of a weighted q norm of u and a weighted r norm of the nth derivative of u. The weights are expressed in terms of a monotone function M which satisfies mild conditions on its regularity of growth. Examples are given which apply the inequality to the spectral theory of differential equations.

In this paper we determine that the best constant of the inequality in the title is . Our approach consists of reducing the problem to various equivalent inequalities on a finite interval and determining necessary conditions on the extremals. The best constant is shown to satisfy an algebraic equation which is solved exactly with the help of MAPLE. The best constant for several other similar inequalities are also determined.

In this paper we deal with some sufficient conditions for a function to be convex with respect to an n-parameter family of functions and we present an application of the conditions to n-th order ordinary differential inequalities.

Simple methods from elementary calculus are used to obtain new proofs of some classical inequalities. This will be done by looking on weighted means as functions of their weights.

Initial value problems for nonlinear wave equations in normal form may not have classical solutions if the nonlinearity is only continuous in the lower order derivatives. Then a natural solution concept is the class of absolutely continuous functions (w.r. to Lebesgue measure). We try to find such solutions in the case of two independent variables and state some interesting open problems.

Assuming a certain two term nth order differential equation is right disfocal we show under certain assumptions the existence of a smallest positive eigenvalue for a related problem. Under further assumptions we prove the existence of a corresponding eigenfunction in a certain cone. Finally we prove a comparison theorem for smallest positive eigenvalues for two eigenvalue problems.

We establish the following lower bound

Similar inequality holds for the kth positive zero y_νk of the Bessel function Y_ν(x) of the second kind. Additionally some comparisons of inequalities in question are given.

The theory of u₀-positive operators with respect to a cone in a Banach space is applied to a class of two-point boundary value problems for a system of linear ordinary differential equations. The existence of a smallest positive eigenvalue is established, and then a comparison theorem for smallest positive eigenvalues is obtained.

A boundary value problem for a singular ordinary differential equation, defined on a bounded domain, is studied. A uniquely determined and absolutely integrable Green's function is constructed such that standard fixed point theorems can be applied. In this paper, sign properties of the Green's function are determined in order that monotone methods and the Schauder fixed point theorem can be applied. Several examples are given and in one example, a delay equation is studied.

We study the equation −Δu = λg(|x|)f(u) R₁ < |x| < R₂, x ∊ ℝ^N, N ≥ 1 subject to linear boundary conditions at R₁ and R₂. Under assumptions concerning sub- or superlinearity of f, we establish existence, non-existence, and multiplicity results for positive solutions.

The present paper is devoted to some aspects, concerned with the τ-modulus, a measure of smoothness for integrable functions on compact intervals. While at first structural properties such as continuity and saturation are worked out, the paper continues with considerations on an improper extension to unbounded intervals. Finally an estimation between the τ-modulus and the ordinary L^∞-modulus is discussed for convex functions, which illustrates some effects, observed in numerical analysis.

Let be analytic in the unit disc U. If α ∊ [1,2] it given, we discuss inequalities of the type

We find connections between Hardy, Littlewood, and Polya's majorization inequalities, Farkas' Lemma, and some interpolation problems. We also consider what “majorization” should look like for functions with higher monoticities.

The notion of delta-convexity (yielding a generalization of functions which are representable as a difference of two convex functions) has been extended to the case of higher order convexity. Examples are given and various characterizations presented. Finally, some stability type results (in the spirit of [3]) are established, including a corollary on supporting polynomial functionals.

The purpose of this paper is to prove a multivariate Nikolskii-type inequality in a generic particular case and to show that the best exponent of the estimate depends on the fact that the set of zeros of the weight-function is transversal or tangential to the boundary of the domain.

Using Chebyshev expansions and coefficient estimates, we obtain inequalities concerning functions, derivatives and integrals. In particular these inequalities are useful for the treatment of differential equations (cf. Breuer–Everson [1] and Plum [7]).

In this paper we shall derive existence and comparison results for extremal solutions of a first order ordinary differential equation in an ordered Banach space whose order cone is regular and has a nonempty interior. We shall assume that the dependent variable in the equation is decomposed into two parts, one being Lipschits continuous and quasi monotone nondecreasing, while the other is nondecreasing, but not necessarily continuous. No kind of compactness conditions are assumed.

Boundary value problems of the first kind for infinite systems of second order ordinary differential equations of the form

The purpose of this paper is to study the weighted relative p–capacity and to apply it to examine solutions for certain degenerate elliptic equations. First, we shall establish metric properties of the p–capacity in terms of weighted Hausdorff measure using the relativity of E and F. Secondly, we shall apply these to degenerate elliptic equations and describe fundamental theorems for weak solutions.

In this paper we examine the null controllability set and the minimal time function for a class of finite dimensional systems with control constraints and in which the state variable enters linearly. In particular, we study how those items respond to changes in the data and we use the minimal time function to completely describe the null-controllability set. Finally we prove a “bang-bang” principle for the time-optimal controls for a class of nonlinear systems.

The problem of rotating plane Couette flow is investigated using singular evolution equations. It is shown that Hopf bifurcation occurs leading to a continuum of disturbance flows in the form of periodic waves that are supercritical and asymptotically stable.

The paper is concerned with the problem of reconstructing inaccessible parts of the state x of a finite - dimensional system from observed output y: Find an estimate for x(t) given y{τ), τ ≤ t. By an observer we mean a control system with input y whose state (or parts of it) provide an estimate which is asymptotically accurate. Under the assumption that x evolves according to a differential equation and that y is part of x concrete proposals for observers will be made in two situations. The results are new at least in the general non-linear and non-autonomous case. They will be obtained with the help of the advanced invariant manifold technique which has been developed in [1] for the purposes of control theory.

This work is dedicated, in admiration, to Wolfgang Walter. It was inspired by discussions the second named author had with Professor Walter at a meeting in Gregynog, Wales in the summer of 1993. One of Walter's keen interests is in providing elementary proofs. It is very much in this spirit that this paper is written.

The present paper wants to relate two classes of convex subsets of real vector spaces: On the one hand these are the superconvex sets, studied after Simons [1972] by the author [1986] and some of his former students, and in König-Wittstock [1990]. The other class consists of the convex sets which are complete in the internal metric (or part metric), defined in Bear-Weiss [1967] as an abstraction of the famous Gleason-Harnack metric in the theory of complex function algebras and studied in Bauer-Bear [1969], Bauer [1970], Bear [1970], and Bauer [1972]. Both notions have the flavour of Interior completeness; for example, the open unit ball of a Banach space is both superconvex and internally complete. But it seems that the two concepts have not been related so far. We shall prove that superconvexity implies internal completeness. In the opposite direction one cannot expect natural results but for those convex sets which coincide with their interior, that means which consist of one part. For these sets superconvexity will be proved to be equivalent to internal completeness plus a remarkable property in superconvex analysis called internal boundedness.

One-dimensional Hardy inequalities can be interpreted as continuous embeddings between (homogeneous) weighted Sobolev spaces and weighted L_p-spaces. Inspired of an inequality of Grisvard⁴ (see also Kufner&Triebel⁸) we want to create a scale of fractional order Hardy inequalities, where a Hardy type inequality is one endpoint estimate and where some trivial or well-known embeddings give the other endpoint estimate. We present an interpolation technique to obtain such scales of inequalities. Some examples of inequalites obtained in this way are proved and discussed. The relations to other similar results are pointed out.

The following sections are included:

• Introduction

• Preliminaries

• Method of upper and lower solutions

• Main result

• References

The inequality of Gabushin concerned relates to Lebesgue L^p-, L^q- and L^r-norms of a function f and its kth and lth derivatives. The two inequalities of the title have this same general character, but k and l are not restricted to be integers, and there is also a weight function in the norms. To pay for this extra generality it is required that f and the derivatives concerned be non-negative.

An important item in Gabushin's hypothesis is an inequality which may for the moment be expressed as A ≥ B, where A and B are functions of k, l, p, q, r. The very same inequality occurs in the hypothesis of one of the inequalities presented here, except that k and l may have non-integral values; while in the other inequality the hypothesis includes the reversed inequality, A ≤ B.

In this paper we extend the method of generalized quasilinearization to second order nonlinear boundary value problem to obtain two sided bounds for the solution. Further we prove the iterates converge uniformly and monotonically to the unique solution of the boundary value problem and the convergence is quadratic.

Using Hölder inequality and degree arguments, this paper provides a new simple proof and some consequences of a recent existence theorem of Srzednicki for periodic solutions of some planar non-autonomous polynomial differential equations when the coefficient of the highest order term are time-periodic.

Discrete inequalities of Bihari-type with several nonlinear terms are considered. The stable properties of the estimations obtained allow us to get the stability of very general nonlinear difference systems.

For linear m-th order difference equations (inequalities) sufficient conditions for positivity and monotonicity of solutions satisfying nonnegative initial conditions are provided. A comparison theorem of Olver type is also obtained.

The main result of the paper obtains a Cauchy-type mean value theorem for the ratio of functional determinants. It generalizes Cauchy's and Taylor's Mean Value Theorems as well as other mean value theorems known for divided differences.

Some inequalities involving functions and their integrals and derivatives are considered. In fact, we give a generalization of some inequalities which are due to G. Pólya.

For nonlinear second-order elliptic boundary value problems, a numerical method for proving the existence of a weak solution in an explicit H₁⁰-neighborhood of some approximate solution ω is presented. The method is based on a H₋₁-bound for the defect of ω, and on a suitable norm bound for the inverse of the linearization of the given problem at ω, which is obtained via eigenvalue estimates. All kinds of monotonicity or inverse-positivity assumptions are avoided.

We discuss the use of the theory of nonassociative algebras in the study of quadratic systems of ordinary differential equations and provide an application to a system of differential equations which arises in the discretization of Boltzmann equations.

For some real-valued and absolutely continuous functions f(x) on x > a > 0, an integral inequality of exponential type is derived. Furthermore, the functions satisfying equality in the inequality are determined.

For a continuous right-hand side (of an ordinary differential equation) we show that a well known theorem on differential inequalities does not only follow from but also implies its quasimonotonicity.

In this paper a semilinear elliptic boundary value problem in the exterior of a finite domain is considered. An important example in applications is the Poisson-Boltzmann problem. Isoperimetric inequalities for a functional of the solution are proven using optimal sub- or supersolutions.

Real-valued functions u defined in euclidean n-dimensional space ℝⁿ are detected that render the Lorentz L(p, 1) norm a maximum and obey the following constraints: the support of u has a prescribed measure and |∇u|, the length of the gradient of u, has a prescribed rearrangement.

The following sections are included:

• Introduction

• Qualitative properties

• The increasing problem

• The decreasing case

• Example

• References