Estimates for Differential Operators in Half-space

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 Lamé system of the static elasticity theory
		The Cauchy–Riemann system
		The stationary linearized Navier–Stokes 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 the ``trace space\'\'
	Notes
Notation
Bibliography
Index