Functional Analysis and Evolution Equations

We are interested in evolution phenomena on star-shaped networks composed of

n

semi-infinite branches which are connected at their origins. Using spectral theory we construct the equivalent of the Fourier transform, which diagonalizes the weighted Laplacian on the

n

-star. It is designed for the construction of explicit solution formulas to various evolution equations such as the heat, wave or the Klein-Gordon equation with different leading coefficients on the branches.

In this paper we summarize some of our recent results on diffusion equations with finite speed of propagation. These equations have been introduced to correct the infinite speed of propagation predicted by the classical linear diffusion theory.

In this article we explore properties of subordinated

d

-parameter groups. We show that they are semi-groups, inheriting the properties of the subordinator via a transference principle. Applications range from infinitely divisible processes on a torus to the definition of inhomogeneous

d

-dimensional fractional derivative operators.

The integral equation that plays a key role in AeroElasticity is known as the Possio Integral Equation, named after its discoverer. From its inception in 1938, this equation was formulated in the Fourier Transform domain using divergent integrals, until 2002 when a more precise formulation valid in a right half-plane was given. In this paper we express it in the time-domain, which requires the language of Functional Analysis,

L

p

spaces, 1 <

p

< 2, and Semigroup Theory. A key role is played by the Finite Hilbert Transform and the Tricomi-Sohngen airfoil equation, which may actually be considered a special case.

The asymptotic behavior of the eigenvalue sequence of the eigenvalue problem

$$ - \Delta \phi + q\left( x \right)\phi = \lambda \phi $$

in a bounded Lipschitz domain

D

⊂ ℝ

N

under the eigenvalue dependent boundary condition

$$ \varphi n = \sigma \lambda \varphi $$

with a continuous function

Σ

is investigated in the case

Σ

−

≢ 0, the dissipative one

Σ

≥ 0 having been settled in [

6

]. For

N

= 1 the eigenvalues grow like

k

2

with leading asymptotic coefficient equal to the Weyl constant. For

N

≥ 2 the positive eigenvalues grow like

k

2/

N

, while the negative eigenvalues grow in absolute value like |

k

|

1/(

N−1

)

. Moreover, asymptotic bounds in dependence on the dynamical coefficient function

Σ

are derived, firstly in the constant case, secondly for

Σ

of constant sign, and finally for a function

Σ

changing sign.

In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates.

Let

a

∈

W

1,∞

(0,1),

a

(

x

) ≥ α > 0,

b, c

∈

L

∞

(0,1) and consider the differential operator

A

given by

Au

=

au

″ +

bu

′ +

cu

. Let α

j

, β

j

(

j

= 0, 1) be complex numbers satisfying α

j

, β

j

≠ (0,0) for

j

= 0, 1. We prove that a realization of

A

with the boundary conditions

$$ \alpha _j u\prime \left( j \right) + \beta _j u\left( j \right) = 0,{\text{ }}j = 0,1, $$

generates a cosine family on

L

p

(0, 1) for every

p

∈ [1, ∞]. This result is obtained by an explicit calculation, using simply d’Alembert’s formula, of the solutions in the case of the Laplace operator.

We study a quasilinear fractional evolution equation, which is of order four in space and 1 + β in time, where β ∈ (0, 1). Under the restriction β < 3/5 we are able to prove existence and uniqueness of global smooth solutions. This result can be seen as the analogue of a result obtained by Engler for a related problem of second order.

Let

f

: ℝ → ℝ be a continuous function. We prove that under some additional assumptions on

f

and

A

: ℝ → ℝ

+

, weak ℓ

1

solutions of the differential inequality — div(

A

(⌝∇

u

⌝)∇

u

) ≥

f

(

u

) on ℝ

N

are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot Groups are considered.

In this article we analyze the stochastic parabolic integral equation

$$ u\left( {t,x,\omega } \right) = c_\alpha t^{ - 1 + \alpha } *\Delta u + \sum\limits_{k = 1}^\infty {\smallint _0^t g^k \left( {s,x,\omega } \right)} dw_s^k , $$

where

t

≥ 0,

x

∈ ℝ

d

,

α

∈ (1/2, 1) and

ω

∈ Ω. We take

w

k

t

⌝ k

= 1, 2, . . . to be a family of independent

$$ \mathcal{F}_t $$

-adapted Wiener processes defined on a probability space

$$ \left( {\Omega ,\mathcal{F},P} \right) $$

. Here

$$ \mathcal{F}_t \subset \mathcal{F}{\text{and }}\mathcal{F}_t $$

is an increasing filtration.

By applying and modifying the method of Krylov we obtain existence and regularity results in

L

p

-spaces,

p

≥ 2.

We consider a resolvent equation arising from a stability problem for exterior Navier-Stokes flows with nonzero velocity at infinity.

In this short note we show that delay equations can be reformulated as abstract weak*-integral equations (AIE) involving dual semigroups, even in the case of infinite delay and/or when the solution takes values in a non-reflexive Banach space. The advantage is that for such (AIE) the standard local stability and bifurcation results are already available, see [

8

]. Our motivation derives from models of physiologically structured populations, as explained in more detail in [

12

].

A result of Huang and van Neerven [

12

] establishes weak individual stability for orbits of

C

0

-semigroups under boundedness assumptions on the local resolvent of the generator. We present an elementary proof for this using only the inverse Fourier-transform representation of the orbits of the semigroup in terms of the local resolvent.

If

X

is a non-degenerate derivation on

R

and

H

=

−X

2

we examine conditions for the closure of

H

to generate a weakly* continuous semigroup on

L

∞

which extends to the

L

p

-spaces. We give an example which cannot be extended and an example which extends but for which the real part of the generator on

L

2

is not lower semibounded.

In this paper we show that the curve shorting flow with contact angle and triple junction in a mirror symmetric configuration is locally well posed in suitable Hölder spaces.

The aim of this paper is to prove

a priori

error estimates for the semi-discrete solution of the dual mixed method for the heat diffusion equation in a polygonal domain. Due to the geometric singularities of the domain, the solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces. In order to recapture the optimal order of convergence, the meshes are refined in an appropriate fashion near the reentrant corners of the domain.

It is well known that the Helmholtz decomposition of

L

q

-spaces fails to exist for certain unbounded smooth domains unless

q

≠ 2. Hence also the Stokes operator and the Stokes semigroup are not well defined for these domains when

q

≠ 2. In this note, we generalize a new approach to the Stokes operator in general unbounded domains from the three-dimensional case, see [

6

], to the

n

-dimensional one,

n

≥ 2, by replacing the space

L

q

, 1 <

q

< ∞, by

$$ \tilde L^q {\text{ where }}\tilde L^q $$

=

L

q

∩

L

2

for

q

≥ 2 and

$$ \tilde L_q $$

=

L

q

+

L

2

for 1 <

q

< 2. As a main result we show that the nonstationary Stokes equation has maximal regularity in

L

8

(0,

T

;

$$ \tilde L_q $$

), 1 <

s, q

< ∞,

T

> 0, for every unbounded domain of uniform

C

1,1

-type in ℝ

n

.

Pontryagin’s maximum principle in its infinite-dimensional version provides (separate) necessary and sufficient conditions for both time and norm optimality for the system

y

′ =

Ay

+

u

(

A

an infinitesimal generator). The question whether targets in

D

(

A

) guarantee a smooth costate has been open. We show the answer is “no” by means of a counterexample involving an analytic semigroup. Another analytic semigroup sheds some light on other subjects such as the existence of hypersingular time optimal controls (thus answering another open question) and the characterization of the reachable space and of singular functionals in its dual.

The mathematical theory of viscous multipolar fluids, based on the general ideas of Green and Rivlin [

8

], was proposed by Nečas and Šilhavý [

17

] (see also Nečas et al. [

15

], [

16

] for relevant existence theory) in order to develop a general framework for studying viscous fluids and to present a suitable alternative to the boundary layer theory (see Bellout et al. [

1

]). The theory is compatible with the basic principles of thermodynamics as well as with the principle of material frame indifference. The present paper is concerned with the mathematical description of the motion of one or several rigid bodies immersed in a viscous multipolar fluid. The principal and very natural idea behind the analysis presented below is the fact that the dissipation of mechanical energy, being much stronger than for classical newtonian fluids, yields better estimates on the gradient of the velocity field, in particular, the streamlines are well defined, which seems crucial for this class of problems partially formulated in terms of the Lagrangian coordinate system.

The Stokes resolvent equations are studied in locally uniform

L

p

spaces where the domain is an exterior of a bounded domain. The unique existence of a solution of the Stokes resolvent equations is proved with a resolvent estimate. In particular, the analyticity of the Stokes semigroup is established. An interesting aspect of locally uniform

L

p

spaces is that these spaces contain non-decaying functions.

This paper is devoted to study the generation of analytic semigroup for a family of degenerate elliptic operators (with unbounded coefficients) which includes well-known operators arising in mathematical finance. The generation property is proved by assuming some compensation conditions among the coefficients and applying a suitable modification of the techniques developed in [

16

]. Using the results proved in [

11

] concerning the generation in the space

L

2

(ℝ

d

), we prove the generation results in

L

p

(ℝ

d

) for

p

∈ [1,+

∞

]. These results have several consequences in connection with the financial applications [

3

,

11

].

A general recipe for high-order approximation of generalized functions is introduced which is based on the use of L

2

-orthonormal bases consisting of C

∞

-functions and the appropriate choice of a discrete quadrature rule. Particular attention is paid to maintaining the distinction between point-wise functions (that is, which can be evaluated point-wise) and linear functionals defined on spaces of smooth functions (that is, distributions). It turns out that “best” point-wise approximation and “best” distributional approximation cannot be achieved simultaneously. This entails the validity of a kind of “numerical uncertainty principle”: The local value of a function and its action as a linear functional on test functions cannot be known at the same time with high accuracy, in general.

In spite of this, high-order accurate point-wise approximations can be obtained in special cases from a high accuracy distributional approximation when more information is available concerning the function which is to be approximated. A few special cases with application to PDEs are considered in detail.

Nonlinear evolution equations in the theory of exothermic chemical reactions lead to semilinear parabolic and elliptic boundary value problems with exponential nonlinearities. In contrast to a commonly employed (Frank-Kamenetskii) approximation, which permits similarity variables for the asymptotic analysis of solution behavior near thermal runaway, we show that the more correct (Arrhenius-Semenov) equation permits no radial symmetries. We also establish that a more general class of thermal nonlinearities also possess no symmetries.

We obtain spectral conditions that characterize mild well-posed inhomogeneous differential equations in a general Banach space

X. L

p

periodic solutions of first and second-order equations are considered. The results are expressed in terms of operator-valued Fourier multipliers. Our approach provides a unified framework for various notions of strong and mild solutions. Applications to semilinear equations of second order in Hilbert spaces are given.

Backward uniqueness for thermoelastic plates and thermoelastic waves with time- and space-dependent coefficients is established. While this result has been proved recently, in the case of

time-independent

coefficients, it is new for the case of time-dependent coefficients. The proof relies on a combination of energy and Carleman’s estimates, hence it is very different from the one given in [LRT], which is based on complex analysis methods. These latter methods are not applicable to nonlinear models and to models with time-dependent coefficients. Our results have consequences for several nonlinear models of thermoelasticity.

The present article wants to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence.

We prove a Post-Widder inversion formula for the Laplace transform of hyperfunctions with compact support in [0,∞). We observe that any hyperfunction with support in [0,∞) has Laplace transforms which are analytic on the right half-plane ℂ

+

, and we extend the Post-Widder inversion formula to suitably bounded representatives of arbitrary hyperfunctions with support in [0,∞).

We prove optimal Schauder estimates for classical solutions of the nonhomogeneous Cauchy problem associated with a class of elliptic operators with unbounded coefficients depending both on time and space variables. We deal both with the case when the coefficients of the elliptic operator are continuous and the case when they are merely measurable in the pair (

t, x

). In both the cases we assume that they are Hölder continuous in

x

, uniformly with respect to

t

.

This paper is concerned with time-dependent relatively continuous perturbations of analytic semigroups and applications to convective reaction-diffusion systems. A general class of time-dependent semilinear evolution equations of the form

u

t

= (

A

+

B

(

t

))

u

(

t

),

t

∈ (

s, τ

);

u

(

s

) =

v

∈

D

(

s

) is introduced in a general Banach space

X

. Here

A

is the generator of an analytic semigroup in

X

and

B

(

t

) is a possibly nonlinear operator from a subset of the domain of a fractional power (−

A

)

α

into

X

and

D

(

t

) =

D

(

B

(

t

)) ⊂

D

((−

A

)

α

). This type of semilinear evolution equations admit only local and mild solutions in general. In order to restrict the growth of mild solutions and formulate a Lipschitz conditions in a local sense for

B

(

t

), a lower semicontinuous functional

ϕ

:

D

((−

A

)

α

) → [0,+

∞

] is introduced and the growth condition of

u

(

·

) is formulated in terms of the nonnegative function

ϕ

(

u

(

·

)) and the nonlinear operator

B

(

t

) is assumed to be Lipschitz continuous on

D

ρ

(

t

) ≡ {

v

∈

D

(

t

):

ϕ

(

v

) ≤

ρ

for

ρ

> 0. The main objective is to establish a generation theorem for a nonlinear evolution operator which provides mild solutions to the semilinear evolution equation under the assumption that a consistent discrete scheme exists under a growth condition with respect to

ϕ

as well as closedness condition for the noncylindrical domain ∪({

t

}×

D

ρ

(

t

)). Moreover, a characterization theorem for the existence of such evolution operator is established in terms of the existence of

ϕ

-bounded discrete schemes. Our generation theorem can be applied to a variety of semilinear convective reaction-diffusion systems. We here make an attempt to apply our result to a mathematical model which describes a complex physiological phenomena of bone remodeling.

We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix-Haase, thus extending several known results and obtaining optimal analyticity angle.

We give exponential and polynomial stability results for the wave equation with variable coefficients in a bounded domain of ℝ

n

, subject to a Dirichlet boundary condition on one part of the boundary and boundary conditions of memory type on the other part of the boundary. Moreover, analogous stability results are given for a system of Maxwell’s equations in heterogeneous media subject to dissipative boundary conditions with memory.

In this contribution we consider degenerate evolution equations on the real line that have the distinguished feature that they contain an exponential weight function in front of the time derivative.

The objective of this paper is to provide an analytic theory for pricing of Asian options of European type. We present a partial differential equation describing the fair price process of an Asian option. This appears as

$$ \left( {\partial _t - A - x \cdot \nabla _y } \right)u = 0 $$

and the associated payoff function as the end value. Here the operator

A

is the

d

-dimensional Black-Scholes operator, and

B

=

x

·∇

y

represents the path dependence in terms of the price averaging in Asian options. The main result will be to prove, that a solution of this partial differential equation exists, is unique, and depends continuously on the data in appropriate function spaces, i.e., that the problem is well posed. On our way we are going to employ semigroup methods, in particular the

Lumer-Phillips theorem

.

The paper is concerned with the general theory of nonlinear agedependent population dynamics. We present (a) a principle of linearized stability and (b) a result on regularity of solutions to the general nonlinear model.

We consider reaction diffusion equations of the form (*)

∂

t

u

=

ν

Δ

u

+

ζ

u

+

$$ \varsigma u + \mathcal{P}\left( u \right),\mathcal{P}\left( u \right) = \sum _z^m a_k u^k $$

and seek solutions on ℝ

n

which are almost periodic in the space variables

x

. Such solutions are constructed in the space

H

0

(ℝ

n

) of almost periodic functions

f

(

x

) subject to (**)

$$ f\left( x \right) = \sum f_k e^{i\nabla _k x} ,\sum \left| {fk} \right| < \infty $$

, provided that the coefficients

a

k

in (*) are also in this class. Such solutions are obtained via an instable manifold construction, which yields solutions on

t

∈ (− ∞, 0] of slow exponential decay. An extension of the method to Fourier transforms of complex measures is outlined.

This paper is concerned with the generation of

C

0

semigroup associated with the Oseen equation with rotating effect and its

L

p

-

L

q

decay estimate. The theorems presented in this paper give us one of the key steps in order to show a globally in time existence of solutions to the Navier-Stokes equations describing the motion of viscous incompressible fluid flow past a rotating rigid body.

We consider the Schrödinger equation, with

H

1

-level terms having variable coefficients in time and space, as defined on an open bounded connected set Ω of an

n

-dimensional complete Riemannian manifold. We show that it is exactly controllable on the state space

L

2

(Ω) on an arbitrarily small interval [0,

T

], by means of Neumann boundary controls in the class

L

2

(0,

T

;

L

2

(Г

1

)), where Г

1

=

∂

Ω SHIELA Г

0

, and the equation is homogeneous on Г

0

, either in the Dirichlet or in the Neumann B.C. Different geometric conditions apply in the two cases. This result is a vast generalization over the literature.