Estimates for matrix operators
Introduction
Description of results
Outline of the proof of the main result
Estimates for systems of ordinary differential operators on a semi-axis
Some assumptions and notation
Transformation of the basic inequality
The simplest lower bound of the constant
On solutions of the system P+(-id/dt)I=0
Properties of the matrix T()
Integral representation for (-id/dt)
Properties of the matrix G()
A quadratic functional
Necessary and sufficient conditions for the validity of inequality (1.1.1)
On condition 4 of Theorem 1.1.19
Matrix G() for estimates with a large" number of boundary operators
Explicit representations of the matrix G()
Estimates for vector functions satisfying homogeneous boundary conditions
Estimates for vector functions without boundary conditions
Estimates in a half-space. Necessary and sufficient conditions
Basic assumptions and notation
Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space
Matrix G(; ) and its properties
The case of a single boundary operator
The case of a polynomial P(; ) with roots in the half-plane Im 0
Estimates of the types (1.2.1), (1.2.12), (1.2.13) in the norms "026B30D "026B30D bold0mu mumu dotted and "426830A to."426830A to."526930B to."526930B to.bold0mu mumu dotted
The case, where the lower-order terms have no influence
Estimates in a half-space. Sufficient conditions
Sufficient condition for the validity of the estimate (1.3.1)
The case M()=T+-1/2()
The case of the diagonal matrix M()
Sufficient conditions for the validity of the estimate (1.3.21)
Examples
Generalized-homogeneous quasielliptic systems
The Lamรฉ system of the static elasticity theory
The CauchyโRiemann system
The stationary linearized NavierโStokes system
Hyperbolic systems
Operators of first order in the variable t. Scalar case
An example of a second-order operator w.r.t. t
On well-posed boundary value problems in a half-space
Notes
Boundary estimates for differential operators
Introduction
Description of results
Outline of the proof of the main result
Estimates for ordinary differential operators on the semi-axis
A lemma on polynomials
A variational problem in finite-dimensional space
Reduction of the estimate for ordinary differential operators on the semi-axis to a variational problem in a finite-dimensional space
Two properties of the matrix B
An estimate without boundary operators in the right-hand side
Necessary and sufficient conditions for the validity of inequality (2.1.1)
Estimates for functions satisfying homogeneous boundary conditions
Estimates in a half-space. Necessary and sufficient conditions
Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space
Corollaries
The case when the lower-order terms play no role
An example of estimate for operators of first order with respect to t
Description of the trace space
Preliminary results
Embedding and extensions theorems
On the extension of functions from H(Rn) to H(Rn+)
Notes
Dominance of differential operators
Introduction
Description of results
Remarks on the method of proving the main result
Estimates for ordinary differential operators on the semi-axis
A variational problem in a finite-dimensional space
The simplest lower bound for the constant
Reduction of the estimates for ordinary differential operators on the semi-axis to variational problems in a finite-dimensional space
Necessary and sufficient conditions for the validity of inequalities (3.1.1) and (3.1.1')
Inequalities for functions without boundary conditions
Estimates in a half-space. Necessary and sufficient conditions
Necessary and sufficient conditions for the validity of the estimates (3.0.1), (3.0.2), and (3.0.1')
On the minimal number and algebraic properties of the boundary operators; formulas for (;)
Estimates for polynomials whose -roots lie in the lower complex half-plane
Examples
The theorem of N. Aronszajn on necessary and sufficient conditions for the coercivity of a system of operators
The case m=1, N=N() in Theorems 3.2.2, 3.2.3, and 3.2.4
Examples of estimates for operators of first order with respect to t
Notes
Estimates for a maximal operator
Introduction
Preliminary results
Results concerning the estimate (4.0.1)
Results concerning the estimate (4.0.2)
Quasielliptic polynomials
Polynomials with a generalized-homogeneous principal part
The estimate (4.2.16) for quasielliptic polynomials of type l 1
The estimate (4.2.19) for quasielliptic polynomials of type l 1
Homogeneous polynomials with simple roots
Asymptotic representations of the -roots of the polynomial H+(;) as ||
Necessary and sufficient conditions for the validity of the estimate (4.2.16)
Necessary and sufficient conditions for the validity of the estimate (4.2.19)
Some classes of nonhomogeneous polynomials with simple roots
A formula for the function ()
Asymptotic representations as N for the -roots j () of the polynomial H+(;)
An asymptotic representation of the function () as N for polynomials P with the real -roots
Necessary and sufficient conditions for the validity of the estimates (4.0.1), (4.0.2) for a polynomial P with real -roots
An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im <0
Necessary and sufficient conditions for the validity of the estimates (4.0.1), (4.0.2) for a polynomial P with the -roots lying in the half-plane Im <0
An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im >0
Necessary and sufficient conditions for the validity of the estimates (4.0.1), (4.0.2) for a polynomial P with the -roots lying in the half-plane Im >0
Second-order polynomials of
Preliminary results
The case p1() 0
The case Im pk() 0 (k=0,1,2)
The estimate (4.2.16) in the case Re p1()0, Im pk() 0 (k=0,2)
Space of traces of functions for the maximal operator
The maximal operator as closure of its restriction on the set of functions infinitely differentiable up to the boundary
Description of thetrace space''
Notes
Notation
Bibliography
Index