Estimates for Differential Operators in Half-space (EMS Tracts in Mathematics)

Description

Translated from the German by Darya Apushkinskaya. Originally published in 1981 by Akademie-Verlag as Abschätzungen für Differentialoperatoren im Halbraum. A publication of the European Mathematical Society Inequalities for differential operators play a fundamental role in the modern theory of partial differential equations. Among the numerous applications of such inequalities are existence and uniqueness theorems, error estimates for numerical approximations of solutions and for residual terms in asymptotic formulas, as well as results on the structure of the spectrum. The inequalities cover a wide range of differential operators, boundary conditions and norms of the corresponding function spaces. The book focuses on estimates up to the boundary of a domain. It contains a great variety of inequalities for differential and pseudodifferential operators with constant coefficients. Results of final character are obtained, without any restrictions on the type of differential operators. Algebraic necessary and sufficient conditions for the validity of the corresponding a priori estimates are presented. General criteria are systematically applied to particular types of operators found in classical equations and systems of mathematical physics (such as Lamé's system of static elasticity theory or the the linearized Navier Stokes system), Cauchy Riemann's operators, Schrödinger operators, among others. The well-known results of Aronszajn, Agmon Douglis Nirenberg and Schechter fall into the general scheme and sometimes are strengthened. The book will be interesting and useful to a wide audience, including graduate students and specialists interested in the theory of differential equations. 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 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