Estimates for Differential Operators in Half-space

✍ Scribed by Igor V. Gel’man, Vladimir G. Maz’ya

• Publisher: European Mathematical Society
• Year: 2019
• Tongue: English
• Leaves: 264
• Series: EMS Tracts in Mathematics Vol. 31
• Category: Library

✦ Synopsis

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 Lame’s system of static elasticity theory or 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 in the theory of differential equations.

Keywords: Differential operators with constant coefficients, differential operators in a half-space, pseudo-differential operators, domination of differential operators, boundary traces, maximal operator, estimates for Lame system, estimates for Stokes system

✦ Table of Contents

Description of results......Page 18
Outline of the proof of the main result......Page 22
Some assumptions and notation......Page 29
Transformation of the basic inequality......Page 30
The simplest lower bound of the constant......Page 31
On solutions of the system P+(-id/dt)I=0......Page 32
Properties of the matrix T()......Page 36
Integral representation for (-id/dt)......Page 38
Properties of the matrix G()......Page 41
A quadratic functional......Page 44
Necessary and sufficient conditions for the validity of inequality (1.1.1)......Page 46
On condition 4 of Theorem 1.1.19......Page 49
Matrix G() for estimates with a large" number of boundary operators......Page 52<br>Explicit representations of the matrix G()......Page 54<br>Estimates for vector functions satisfying homogeneous boundary conditions......Page 55<br>Estimates for vector functions without boundary conditions......Page 56<br>Estimates in a half-space. Necessary and sufficient conditions......Page 57<br>Basic assumptions and notation......Page 58<br>Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space......Page 60<br>Matrix G(; ) and its properties......Page 62<br>The case of a single boundary operator......Page 65<br>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......Page 67<br>The case, where the lower-order terms have no influence......Page 70<br>Estimates in a half-space. Sufficient conditions......Page 74<br>Sufficient condition for the validity of the estimate (1.3.1)......Page 75<br>The case M()=T+-1/2()......Page 77<br>The case of the diagonal matrix M()......Page 82<br>Sufficient conditions for the validity of the estimate (1.3.21)......Page 83<br>Generalized-homogeneous quasielliptic systems......Page 86<br>The Lamé system of the static elasticity theory......Page 89<br>The Cauchy–Riemann system......Page 91<br>The stationary linearized Navier–Stokes system......Page 93<br>Hyperbolic systems......Page 94<br>Operators of first order in the variable t. Scalar case......Page 97<br>An example of a second-order operator w.r.t. t......Page 99<br>On well-posed boundary value problems in a half-space......Page 101<br>Notes......Page 106<br>Description of results......Page 110<br>Outline of the proof of the main result......Page 112<br>Estimates for ordinary differential operators on the semi-axis......Page 116<br>A lemma on polynomials......Page 117<br>A variational problem in finite-dimensional space......Page 123<br>Reduction of the estimate for ordinary differential operators on the semi-axis to a variational problem in a finite-dimensional space......Page 127<br>Two properties of the matrix B......Page 131<br>An estimate without boundary operators in the right-hand side......Page 132<br>Necessary and sufficient conditions for the validity of inequality (2.1.1)......Page 134<br>Estimates for functions satisfying homogeneous boundary conditions......Page 136<br>Estimates in a half-space. Necessary and sufficient conditions......Page 139<br>Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space......Page 140<br>Corollaries......Page 142<br>The case when the lower-order terms play no role......Page 145<br>An example of estimate for operators of first order with respect to t......Page 147<br>Preliminary results......Page 150<br>Embedding and extensions theorems......Page 154<br>On the extension of functions from H(Rn) to H(Rn+)......Page 158<br>Notes......Page 161<br>Description of results......Page 162<br>Remarks on the method of proving the main result......Page 164<br>Estimates for ordinary differential operators on the semi-axis......Page 166<br>A variational problem in a finite-dimensional space......Page 167<br>The simplest lower bound for the constant......Page 170<br>Reduction of the estimates for ordinary differential operators on the semi-axis to variational problems in a finite-dimensional space......Page 171<br>Necessary and sufficient conditions for the validity of inequalities (3.1.1) and (3.1.1')......Page 175<br>Inequalities for functions without boundary conditions......Page 178<br>Necessary and sufficient conditions for the validity of the estimates (3.0.1), (3.0.2), and (3.0.1')......Page 179<br>On the minimal number and algebraic properties of the boundary operators; formulas for (;)......Page 182<br>Estimates for polynomials whose -roots lie in the lower complex half-plane......Page 185<br>The theorem of N. Aronszajn on necessary and sufficient conditions for the coercivity of a system of operators......Page 186<br>The case m=1, N=N() in Theorems 3.2.2, 3.2.3, and 3.2.4......Page 187<br>Examples of estimates for operators of first order with respect to t......Page 191<br>Notes......Page 194<br>Introduction......Page 196<br>Results concerning the estimate (4.0.1)......Page 198<br>Results concerning the estimate (4.0.2)......Page 202<br>Polynomials with a generalized-homogeneous principal part......Page 204<br>The estimate (4.2.16) for quasielliptic polynomials of type l 1......Page 207<br>The estimate (4.2.19) for quasielliptic polynomials of type l 1......Page 209<br>Homogeneous polynomials with simple roots......Page 212<br>Asymptotic representations of the -roots of the polynomial H+(;) as ||......Page 213<br>Necessary and sufficient conditions for the validity of the estimate (4.2.16)......Page 214<br>Necessary and sufficient conditions for the validity of the estimate (4.2.19)......Page 218<br>Some classes of nonhomogeneous polynomials with simple roots......Page 221<br>Asymptotic representations as N for the -roots j () of the polynomial H+(;)......Page 222<br>An asymptotic representation of the function () as N for polynomials P with the real -roots......Page 225<br>Necessary and sufficient conditions for the validity of the estimates (4.0.1), (4.0.2) for a polynomial P with real -roots......Page 227<br>An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im <0......Page 228<br>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......Page 230<br>An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im >0......Page 231<br>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......Page 232<br>Preliminary results......Page 234<br>The case p1() 0......Page 236<br>The case Im pk() 0 (k=0,1,2)......Page 238<br>The estimate (4.2.16) in the case Re p1()0, Im pk() 0 (k=0,2)......Page 239<br>Space of traces of functions for the maximal operator......Page 243<br>The maximal operator as closure of its restriction on the set of functions infinitely differentiable up to the boundary......Page 244<br>Description of thetrace space''......Page 246
Notes......Page 250
Notation......Page 252
Bibliography......Page 256
Index......Page 260