Zeitschrift für Analysis und ihre Anwendungen
Journal for Analysis and its Applications
Volume 19 (2000)
Abstracts
Th. Chryssakis: Geometry of Numerical Ranges in Locally m-Convex *-Algebras,
19 (2000) 003--012
- We examine the symmetry of numerical ranges in a unital locally m-convex
C*-algebra of a given element and its adjoint, with respect to a
rotated real-axis, where the rotation angle depends on the value of the positive
linear forms of the algebra (states) at the unit element of the algebra.
- We present some results about the characterization of modular conjugations
of von Neumann algebras. Further, we show that hyperfinite factors of type II,
III1, and IIIl have algebraic
conjugations which are not modular conjugations.
- We consider the one-parameter family of linear operators that A. Belleni
Morante recently introduced and called B-bounded semigroups. Such a family was
studied by A. Belleni Morante himself and by J. Banasiak. Here we give a necessary
and sufficient condition that a pair (A,B) of linear operators be the generator of
a B-bounded semigroup. Our procedure is constructive and is equivalent to the Yosida
procedure for the construction of a C0-semigroup when B = I. We also show
that our result represents a generalization of Banasiak's result.
- Acyclicity of solution sets to asymptotic problems, when the value is
prescribed either at the origin or at infinity, is proved for differential
inclusions and discontinuous autonomous differential inclusions. Existence
criteria showing that such sets are non-empty are obtained as well.
- In models for infectious diseases, the basic reproduction number is the crucial
parameter which determines the possibility of an outbreak. In simple situations
it depends in a monotone way on the infectivity. Non-monotone behavior may occur in
diseases where infectivity depends on time since infection and where transmission
depends on social structure, as is shown by an example. A typical application is
the HIV infection where transmission rates depend on existing pair bonds and
infectivity changes drastically over time.
For a class of epidemic models with pair formation models and infectivity depending
on time since infection it is shown that the basic reproduction number is a monotone
function of infectivity. This observation is a consequence of a general result on a
class of cyclic linear reaction chains with tridiagonal structure for which it is
shown that the number of passages depends in a monotone way on the rates.
- A nonlinear parabolic partial differential equation model describing the
behaviour of a distributed parameter fixed-bed bioreactor is studied here.
Exponential stability around the steady state solution for exponentially
decaying deviations in the input and disturbance are proved via abstract
formulation of the model as an evolution equation and by utilizing semigroup
theory and asymptotic stability of the corresponding evolution operator.
- We show in a novel way that periodic pseudodifferential operators are
pseudodifferential operators in Hörmander's definition. In our approach,
commutators a la Beals, Dunau, Coifman and Meyer on Rn
and on closed manifolds are involved.
- We consider the problem of identifying the subset domain
O of R2 of a semilinear elliptic
equation subject to given Cauchy data on part of the known outer boundary
G and to the zero flux condition on the unknown
inner boundary g, where it is assumed that
G is a piecewise C1 curve and that
g is the boundary of a finite disjoint union of
simply connected domains, each bounded by a piecewise C1 Jordan
curve. It is shown that, under appropriate smoothness conditions, the domain
O is uniquely determined. The problem of existence
of solution for given data is not considered since it is usually of lesser
importance in view of measurement errors giving data for which no solution exists.
- We prove a theorem about existence, uniqueness and regularity of the solution
to an initial-boundary value problem for a nonlinear coupled parabolic system
consisting of two equations. Such a system appears in the thermodiffusion in a
solid body. In our proof we use an energy method, methods of Sobolev spaces,
semigroup theory and the Banach fixed point theorem.
- We continue our investigation of a slowly rotating Timoshenko beam in a
horizontal plane whose movement is controlled by the angular acceleration of
the disk of a driving motor into which the beam is clamped. We show how to
choose a feedback control allowing to stabilize our system (the beam plus the
disk) in a preassigned position of rest.
- The coupled Maxwell-Lorentz system describes feed-back action of electromagnetic
fields in classical electrodynamics. When applied to point-charge sources (viewed as
limiting cases of charged fluids) the resulting nonlinear weakly hyperbolic system
lies beyond the scope of classical distribution theory. Using regularized derivatives
in the framework of Colombeau algebras of generalized functions we analyze a two-
dimensional analogue of the Maxwell-Lorentz system. After establishing existence and
uniqueness of solutions in this setting we derive some results on distributional limits
of solutions with delta-like initial values.
- Existence and uniqueness results for the integro-differential equation
ut(x,t) - auxx(x,t) = c(x,t)u(x,t) + Integral0
1 k(s,x)h(s,t,u(s,t)) ds + f(x,t), where (x,t) in Q,
subject to the boundary condition u(x,t) = F(x,t),
where (x,t) in R, and, especially, for the linear case h(s,t,u) = u are given.
To this end, this equation is written as operator equation in a suitable Hölder
space. The main tools are the calculation of the spectral radius in the linear
case, and fixed point principles in the nonlinear case.
- For a fixed Dirichlet form, we study the space of positive Borel measures
(possibly infinite) which do not charge polar sets. We prove the density in
this space of the set of the measures which represent varying domains. Our
method is constructive. For the Laplace operator, the proof was based on a
pavage of the space. Here, we substitute this notion by that of homogeneous
covering in the sense of Coiffman and Weiss.
- For de Rham's singular function we derive new properties, in particular some
formulas which express its self-similarity. Inversions and compositions of de
Rham's function are considered as well as generalizations of de Rham's functional
equations which have a connection to the (3n+1)-iteration of Collatz.
- By linearization of nonlinear semi-eigenvalue problems, especially for analytic
maps with completely continuous Frechet derivative, we develop the (primary) branching
solutions and the corresponding semi-eigenvalues into power series with respect to
a real parameter. We consider the algebraically simple case of the semi-eigenvalue of
the linearization using the implicit function theorem and the Fredholm alternative.
- Asymptotic expansions for second-order moments of integral functionals
of a family of random processes are considered. The random processes are
assumed to be wide-sense stationary and e-correlated, i.e. the values are
not correlated excluding an e-neighbourhood of each point. The asymptotic
expansions are derived for e approaching 0. Using a special weak assumption
there are found easier expansions as in the case of general weakly correlated
random processes. Expansions are given for integral functionals of real-valued
as well as of complex vector-valued processes.
- We consider infinite-horizon optimal control problems. First, by a suitable
change of variable, we transform the problem to a finite-horizon nonlinear optimal
control problem. Then the problem is modified into one consisting of the minimization
of a linear functional over a set of positive Radon measure. The optimal measure is
approximated by a finite combination of atomic measures and the approximate solution
of the fist problem is found by the optimal solution of a finite-dimensional linear
programming problem. The solution of this problem is used to find a piecewise constant
control for the original one, and finally by using the approximate control signals we
obtain the approximate trajectories.
- The Bernstein condition of boundedness of the derivatives of an a priori
bounded solution of a 2nd order ordinary differential equation is extended to
systems in which each equation has its own order.
- This paper deals with a generalization of the classical Choquet theorem. We
consider metric spaces which are endowed with an abstract notion of convexity.
Convex combinations are obtained by the solutions of variational inequalities.
A generalized Krein-Milman theorem is derived from our Choquet theorem. We end
with an example based on hyperbolic geometry.
- We describe some connections between three different fields: combinatorics
(umbral calculus), functional analysis (linear functionals and operators) and
harmonic analysis (convolutions on group-like structures). Systematic usage of
cancellative semigroups, their convolution algebras, and tokens between
them provides a common language for description of objects from these three
fields.
- By means of a certain conformal covariant differentiation process explicit
formulae are derived for
(i) a conformally invariant generalized Bach tensor in dimension 6;
(ii) conformally invariant differential operators acting on weighted functions,
especially one with a leading term with exponent 4;
(iii) conformal covariants on symmetric, trace-free p-tensor bundles,
especially one with a leading term with exponent 2;
(iv) conformal covariants on differential forms.
Furthermore, theorems for uniqueness, existence and non-existence of conformal
covariants, in particular in dimension 4, are given.
- We consider convolution operators generated by L1 functions in
Lp spaces and various spaces of almost periodic functions. It turns
out to be that if such an operator is invertible in one of these spaces, then
it is invertible in all the spaces we consider. Further, we prove that any
convolution has identical norms in many natural couples of function spaces.
- We derive (weighted) Sobolev-Poincare inequalities for s-John domains
and s-cusp domains, both with optimal exponents. These results are obtained
as consequences of a more comprehensive criterion.
- We present a new approximate joint selection theorem which unifies Michael's
theorem (1956) on continuous selections and Cellina's theorem (1969) on continuous
e-approximate selections. More precisely, we show that,
given a convex-valued H-upper semicontinuous multifunction F and a convex-closed-valued
lower semicontinuous multifunction G such that F(x) and G(x) have nonempty intersection,
one can find a continuous function f which is both a selection of G and an
e-approximate selection of F. We also prove a parametric
version of this theorem for multifunctions F and G of two variables (s,u) in M times
X where M is a measure space. Using this selection theorem, we obtain an existence
result for elliptic systems involving a vector Laplacian and a strongly nonlinear
multi-valued right-hand side, subject to Dirichlet boundary conditions.
- We consider the non-characteristic Cauchy problem for the degenerate nonlinear
parabolic equation |u|a ut -
D u - g|u|-bu = 0 under some assumptions on a, b
and g. The problem is improperly posed in the sense of
Hadamard. We derive for such solutions an estimate in terms of the Cauchy data
and a prescribed bound of the solution.
- We derive necessary and sufficient optimality conditions for a relaxed (in
terms of Young measures) variational problem governing steady states of
ferromagnetic materials. Such conditions here stated in the form of a
generalized Weierstrass maximum principle enable us to establish uniqueness
of a solution in some specific situations and can also be used in efficient
numerical algorithms solving the relaxed problems, for instance.
- We prove a theorem about local existence (in time) of the solution to the
first initial-boundary value problem for a nonlinear hyperbolic-parabolic
system of eight coupled partial differential equations of second order
describing the process of thermodiffusion in a three-dimensional micropolar
medium. At first, we prove existence, uniqueness and regularity of the
solution to this problem for the associated linearized system by using the
Faedo-Galerkin method and semi-group theory. Next, we prove (basing on this
theorem) an energy estimate for the solution to the linearized system by
applying the method of Sobolev spaces. At last, by using the Banach fixed
point theorem we prove that the solution of our nonlinear problem exists and
is unique.
- We consider a problem which models the evolution of sound in a compressible
fluid with reflection of sound at the surface of the material. Different methods
such as the concavity method of Levine, the potential well method and an argument
due to Tsutsumi are used to derive global non-existence theorems.
- We present some sufficient conditions for the asymptotic stability of forced
almost-periodic oscillations in nonlinear systems subject to small hysteresis
perturbations. The main technical restriction on hysteresis nonlinearity
comes to a contraction-type property, which holds for some classical models
of hysteresis. Also we require a special stability property of the
unperturbed system in the sense of Lyapunov and the bounded input - bounded
output.
- Basic inverse problems for identification of memory kernels in linear heat
conduction and viscoelasticity in the infinite time interval (0,infinity) are
treated by Laplace transform method in coupling with Fourier's method for the
direct initial-boundary value problem of the corresponding integro-differential
equation. Under suitable assumptions on the data existence and uniqueness of the
memory kernel are shown.
- We consider the system of boundary value problems
ui(ni)(t) + fi(t, u1(t)
,
... , um(t)) = 0, ui(j)(0) = 0,
ui(pi)(1) = 0,
for 0 <= t <= 1, i = 1, ... , m and 0 <= j <= ni - 2 where ni
>= 2
and 1 <= pi <= ni - 1.
- Several criteria are offered for
the existence of single and twin solutions of the system that are of fixed
signs.
- It is well-known that the zeros of holomorphic functions in more than one
complex variable are not isolated. Nevertheless, there exist so-called
identity surfaces such that a holomorphic function vanishes identically
everywhere if only it equals zero on an identity surface. In the paper
identity surfaces will be constructed using the technique of completely
integrable overdetermined systems of partial differential equations. Moreover,
identity surfaces will be constructed not only for holomorphic functions but
also for solutions of more general first order systems of partial differential
equations.
The present paper deals only with systems with real-analytic coefficients
and, therefore, the classical Cauchy-Kovalevskaya and Holmgren theorems are
applicable (while many recent papers deal with infinitely differentiable
coefficients or they solve initial value problems with generalized analytic
initial functions). Using the compatibility conditions of an overdetermined
system, in the present paper the construction of identity surfaces (of minimal
dimension) is carried out as some kind of inverse problem to an initial value
problem.
- The classical Neumann problem for the inhomogeneous pluriholomorphic system
in a polydisc is considered. Its solvability conditions and its solution are
given. It is shown that the problem is not well-posed. To fix the solution
the boundary condition is modified. For the modified problem the solvability
conditions and the solution which is unique up to an arbitrary constant are
explicitly given.
- We study certain correlations between lower and upper bounds
of exponential Riesz sequences, in particular between sharp lower
and upper bounds, where we show that the product of the sharp bounds
of an exponential Riesz sequence is bounded from above by a universal
constant. The result is applied to the norms of coefficient and frame
operators and their inverses.
- The existence of almost periodic piecewise continuous functions of
Lyapunov's type for impulsive differential equations is considered. The
impulses take place at fixed moments of time.
- Strong solutions to the class of semilinear elliptic equations
-Du=f(x,u) on the entire space and with
possibly supercritical growth for f(x, . ) are obtained by mainly
using fixed points arguments. The case of discontinuous non-linearities
is then examined.
- The paper deals with the asymptotics of zeros of the Wright
function j (r,
b; z) in the case the parameter
b is a real number. The exact formulae
for the order, the type and the indicator function of the entire
function j (r,
b;z) are given for r
> -1. On the basis of these results and using the obtained distribution
of the zeros of the Wright function it is shown to be a function of
completely regular growth.
- A generalization of the GNS-representation is investigated that
represents partial *-algebras as systems of operators acting on a
partial inner product space (i.e., PIP-space). It is based on possibly
indefinite B-weights which are closely related
to the positive B-weights introduced by J.-P. Antoine,
Y. Soulet and C. Trapani. Some additional assumptions had to be made in order
to guarantee the GNS-construction. Different partial products of operators on a
PIP-space are considered which allow the GNS-construction under suitable conditions.
Several examples illustrate the argumentation and indicate inherent problems.
- The order of differentiability of the inversion operator J
between certain spaces or manifolds of distributionally differentiable functions is shown
to be sharp in the following sense. Up to a certain order k guaranted by inverse function
arguments, the operator J is everywhere differentiable and
J(k) is continuous. On the other hand, J is nowhere k+1 times differentiable.
- The Dirac operator with pseudoscalar, scalar or electric potential and the
Schrödinger operator are considered. For any potential depending on an arbitrary
function x satisfying the equation
(*) Dx - g(x) (dg(x) /
dx) = 0
where g(x) = |grad x| there
are constructed special solutions of the Dirac and the Schrödinger equations, and
in some cases the fundamental solutions are obtained also. The class of solutions
of equation (*) is sufficiently ample. For example, if (1) x
is harmonic and (2) the gradient squared of x is constant,
then x satisfies (*). That is, in particular, any complex
linear combination of three variables x = ax1 +
bx2 + cx3 + d satisfies equation (*), and the solutions may be
obtained for any potential depending on such x. All results
are obtained using some special biquaternionic projection operators constructed after
having solved an eikonal equation corresponding to x.
- It is shown that best approximation by trigonometric polynomials is achieved
in average by families of linear polynomial operators in the Lp-metric
for all p, 0 < p <= infty. This is compared with approximation by
Fourier means and interpolation means which is restricted to 1 <= p
<= infty and p = infty, respectively.
- A characterization of weighted L2(I) spaces in terms of their images
under various integral transformations is derived, where I is a finite interval. The
class of integral transformations considered is related to certain singular Dirac
systems on a half line.
- We study asymptotic properties of the fundamental solution to an Oseen-type
system coming from fluid mechanics. We show that the solution has similar
anisotropic structure near infinity as the fundamental solution to the
(classical) Oseen problem. We also study integral operators with kernels
representing the second gradient of the fundamental solution.
- A nonlinear perturbation problem for steady two-dimensional inviscid
transonic flow in a nozzle is studied. The existence of a smooth solution
to the problem is proved under the condition of positive acceleration of the
given flow. The proof involves the method of singular perturbations for solving
a linear problem associated with the nonlinear one. The technique for obtaining
a priori estimates is simpler than that used in previous papers.
- This paper is concerned with the capillary problem in a class of non-cylindrical
domains in a subset K of Rn+1 obtained by scaling a bounded cross-section
W, where W is a subset of Rn
along the vertical axis. The capillary surfaces are described in two different ways.
In the first model, they are described as the boundary of a Caccioppoli set and in a
second model, after transforming K to a cylinder, they are described as graphs of
functions on W. The volume of the fluid is prescribed.
For both models, the energy functional is derived and declared on the appropriate
function space consisting of BV-functions. Main results are existence and a priori
bounds of minimizers, using the direct methods in the calculus of variations. For
the special case of a cone over the domain W, a criterion
is given to assure that the tip is not filled with liquid. Another point of examination
concerns modelling the volume restriction by means of a Lagrange multiplier.
- For the trace of Besov spaces Bsp,q onto a hyperplane,
the borderline case with s = (n/p) - n + 1 and 0 < p < 1 is analysed and
a new dependence on the sum-exponent q is found. Through examples the restriction
operator defined for s down to 1/p, and valued in Lp, is shown to be
distinctly different and, moreover, unsuitable for elliptic boundary problems.
All boundedness properties (both new and previously known) are found to be easy
consequences of a simple mixed-norm estimate, which also yields continuity with
respect to the normal coordinate. The surjectivity for the classical borderline
s = 1/p (1 <= p < infty) is given a simpler proof for all q with
0 < q <= 1, using only basic functional analysis. The new borderline results
are based on corresponding convergence criteria for series with spectral conditions.
- The present paper deals with (logarithmic) Lipschitz spaces of type
Lipp,q(1, -a) (1 <= p <=
infty, 0 < q <= infty, a > 1/q).
We study their properties and derive some (sharp) embedding results. In that
sense this paper can be regarded as some continuation and extension of our previuos
papers, but there are also connections with some recent work of Triebel concerning
Hardy inequalities and sharp embeddings. Recall that the nowadays almost 'classical'
forerunner of investigations of this type is the Brezis-Wainger result about the
'almost' Lipschitz continuity of elements of the Sobolev spaces Hp1+n/p
(Rn) when 1 < p < infty.
- Using fractional derivatives we show that the drift form "integral from
-infty to infty over u(x) (dv(x) / dx) dx" can be approximated by non-symmetric
Dirichlet forms. A similar result holds for the drift form in Rn with
variable coefficients if the coefficient functions satisfy certain regularity and
commutator conditions. Since time-dependent Dirichlet forms (in the sense of Y.
Oshima) can be interpreted as sums of a drift form (in t
-direction) and a mixture of t-parametrized Dirichlet forms
over Rn, our results show that time-dependent Dirichlet forms arise as
limits of ordinary non-symmetric Dirichlet forms in R ´
Rn-space. An abstract result on fractional powers of Markov generators
allows to extend this observation to generalized Dirichlet forms. Another consequence
is that the bilinear form induced by an arbitrary Levy process is the limit of
non-symmetric Dirichlet forms.
- The authors aim at deriving a family of series representations for
z(2n+1) (n a natural number) by evaluating
certain trigonometric integrals in several different ways. They also
show how the results presented in this paper relate to those that were
obtained in other works. Finally, some illustrative computational examples,
using Mathematica (Version 4.0) for Linux, are considered.
- Given a closed subset M of [0, 1] of Lebesgue measure zero, we construct
a C1 function f with the property that f-1({y}) is a
perfect set for every y in M.
- A new multiplicity result is presented for maps between Frechet spaces.
Our argument relies on fixed point results in Banach spaces together with a
result on hemicompact maps. An application is also given to illustrate how
the theory can be applied in practice.
- Sufficient conditions are established for the oscillation of all solutions
of the fourth order difference equation
D (an D
(bn D (cn
D yn))) + qnf(yn+1)
= hn (n >=0)
- where D is the forward difference operator
D yn = yn+1 - yn,
{an}, {bn}, {cn}, {qn}, {hn}
are real sequences, and f is a real-valued continuous function. Also, sufficient
conditions are provided which ensure that all non-oscillatory solutions of
the equation approach zero as n approaches infinity. Examples are inserted to
illustrate the results.
- A class of ergodicity coefficients for infinite stochastic matrices is
introduced and investigated with respect to connections to the well-known
d-coefficient. The theory yields results on the
behaviour of infinite products of stochastic matrices, in particular on
inhomogeneous Markov chains and Markov systems.
- A theorem in our previous paper, although correct, was formulated incompletely. A complete
correct and complete version is stated here.
- We discuss a generalization of Fueter's theorem which states
that whenever f(x0, \ul{x})$ is holomorphic in $x_0 + \ul{x}$, then it
satisfies $D\Box\!f = 0$, $D = \p_{x_0} + i\p_{x_1} + j\p_{x_2} + k\p_{x_3}$
being the Fueter operator.
- We propose a new approach for obtaining approximate solutions of Maxwell's
equations in inhomogeneous media. This work is based on the application of
quaternionic analysis technique and consists of some approximate
diagonalization of Maxwell's equations. They are reduced to a pair of
quaternionic equations which under some additional conditions can be solved
exactly.
- This paper is devoted to the study of nonlinear geometric optics in Colombeau
algebras of generalized functions in the case of Cauchy problems for
semilinear hyperbolic systems in one space variable. Extending classical
results, we establish a generalized variant of nonlinear geometric optics.
As an application, a nonlinear superposition principle is obtained when
distributional initial data are perturbed by rapid oscillations.
- We discuss the viscosity solutions of a Dirichlet problem for
weakly coupled systems of fully nonlinear second order degenerated elliptic
equations. We prove the existence, uniqueness and continuity of solutions
by Perron's method combined with the technique of coupled solutions. Our
results generalize those of H. Ishii and S. Koike [Communications in Partial
Differential Equations 16 (1991) 1095--1128] for the case of general quasi-monotonic
systems.
- We consider a conserved phase-field model with memory in which the Fourier
heat conduction law is replaced by a constitutive assumption of Gurtin-Pipkin
type; the system is conserved in the sense that the initial mass of the order
parameter is preserved during the evolution. We investigate a Cauchy-Neumann
problem for this model which couples an integro-differential equation with a
nonlinear fourth-order equation for the phase field. Here we assume that the
heat flux memory kernel has a decreasing exponential as principal part and
we study the behaviour of solutions when this kernel converges to a Dirac
mass. We show that the solution to the conserved phase-field model with
memory converges to a solution to the phase-field problem without memory
under suitable assumptions on the data.
- We study an evolution problem for filtration through porous media, accounting
for hysteresis in the saturation versus pressure constitutive relation. We
provide a weak formulation of the problem, assuming that the memory effect
in the constitutive relation consists not only of a rate-independent component
but also of a rate-dependent one. We prove an existence result, which also
applies to the case where the hysteresis operator is of Preisach-type.
- A special system of two discrete two-scale difference equations with
polynomial solutions is investigated. For the solutions, addition and
subtraction theorems are established showing in particular the behaviour
of the solutions for a great argument, as well as further relations and
inequalities. Also, corresponding generating functions are constructed
which imply explicit representations for the solutions.
- We study the system of functional equations
$$
f_i(x) = \sum_{j=1}^n \sum_{k=1}^m a_{ijk}[x,f_j(S_{ijk}(x))] + g_i(x) \qquad
(1 \le i \le n)
$$
for $x \in \Omega_i$ where $\Omega_i$ are compact or non-compact domains of
$\R^p$, \ $g_i: \, \Omega_i \to R$, $S_{ijk}: \, \Omega_i \to \Omega_j$,
$a_{ijk}: \, \Omega_i \times \R \to \R$ are given continuous functions and
$f_i: \, \Omega_i \to \R$ are unknown functions. The paper consists of two
mains parts. In the first part we give some results on existence, uniqueness
and stability of the solutions of such systems and study sufficient conditions
to obtain quadratic convergence. In the second part we obtain the Maclaurin
expansion and approximation of solution in the case that $a_{ijk}$ are linear
and $S_{ijk}$ are affine functions.
- We use the Galerkin and compactness method in appropriate weighted Sobolev
spaces to prove the existence of a unique weak solution of the nonlinear
boundary valued problem
- (1 / xg) (d/dx) M (x, u'(x)) + f (x, u(x)) =
F(x) (0 < x < 1)
| lim xg/p u'(x) | < +infty for x approaching 0
M(1, u'(1)) + h(u(1)) = 0,
where g > 0, p >= 2 are given constants and f, F, h, M
are given functions.
- Let M(n, R) be the set of real positive definite symmetric (n x n)-matrices equipped with the Euclidean norm, and let
A be an element from M(n, R).
Let L(n, R) be the set of all real non-degenerate lower-triangular (n x
n)-matrices equipped with the Euclidean norm, and let L from M(n, R) to L(n,
R) be a (differentiable) map assigning
to a positive definite symmetric matrix its lower-triangular factor in the
LU-decomposition. We give an effective upper estimate for the norm of L'(A).
- Let S be the class of functions f(z) = z + a0 +
a-1 z-1 + ... analytic and univalent in |z| > 1. We investigate
the problem to maximize Re a-1 in two subclasses of S:
(i) the class of all functions f in S which omit two given values
+/- w1 (0 < |w1| < 2) and (ii) the class of all functions f in
S with a0 = 0 which map onto regions of prescribed width
bf = b (0 < b < 4) in the direction of the imaginary axis. We solve
these problems by applying a variational method to a coefficient problem in two subclasses
of univalent Bieberbach-Eilenberg functions which are equivalent to these problems.
- Some properties of Legendre functions in an asymmetric interval (with respect
to zero) with zero boundary values are obtained through variational methods.
There are given some applications to the monotonicity and estimates of the
first Dirichlet eigenvalue for moving bands on S2.
- The paper establishes a general coerciveness property for a class of
non-smooth functionals satisfying an appropriate Palais-Smale condition.
This result is obtained by applying an abstract principle supplying
qualitative information concerning the asymptotic behaviour of a non-smooth
functional. Comparison with other results in this field is provided.