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
Author(s): Igor V. Gel’man, Vladimir G. Maz’ya
Series: EMS Tracts in Mathematics Vol. 31
Publisher: European Mathematical Society
Year: 2019
Language: English
Pages: 264
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
Explicit representations of the matrix G()......Page 54
Estimates for vector functions satisfying homogeneous boundary conditions......Page 55
Estimates for vector functions without boundary conditions......Page 56
Estimates in a half-space. Necessary and sufficient conditions......Page 57
Basic assumptions and notation......Page 58
Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space......Page 60
Matrix G(; ) and its properties......Page 62
The case of a single boundary operator......Page 65
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
The case, where the lower-order terms have no influence......Page 70
Estimates in a half-space. Sufficient conditions......Page 74
Sufficient condition for the validity of the estimate (1.3.1)......Page 75
The case M()=T+-1/2()......Page 77
The case of the diagonal matrix M()......Page 82
Sufficient conditions for the validity of the estimate (1.3.21)......Page 83
Generalized-homogeneous quasielliptic systems......Page 86
The Lamé system of the static elasticity theory......Page 89
The Cauchy–Riemann system......Page 91
The stationary linearized Navier–Stokes system......Page 93
Hyperbolic systems......Page 94
Operators of first order in the variable t. Scalar case......Page 97
An example of a second-order operator w.r.t. t......Page 99
On well-posed boundary value problems in a half-space......Page 101
Notes......Page 106
Description of results......Page 110
Outline of the proof of the main result......Page 112
Estimates for ordinary differential operators on the semi-axis......Page 116
A lemma on polynomials......Page 117
A variational problem in finite-dimensional space......Page 123
Reduction of the estimate for ordinary differential operators on the semi-axis to a variational problem in a finite-dimensional space......Page 127
Two properties of the matrix B......Page 131
An estimate without boundary operators in the right-hand side......Page 132
Necessary and sufficient conditions for the validity of inequality (2.1.1)......Page 134
Estimates for functions satisfying homogeneous boundary conditions......Page 136
Estimates in a half-space. Necessary and sufficient conditions......Page 139
Theorems on necessary and sufficient conditions for the validity of the estimates in a half-space......Page 140
Corollaries......Page 142
The case when the lower-order terms play no role......Page 145
An example of estimate for operators of first order with respect to t......Page 147
Preliminary results......Page 150
Embedding and extensions theorems......Page 154
On the extension of functions from H(Rn) to H(Rn+)......Page 158
Notes......Page 161
Description of results......Page 162
Remarks on the method of proving the main result......Page 164
Estimates for ordinary differential operators on the semi-axis......Page 166
A variational problem in a finite-dimensional space......Page 167
The simplest lower bound for the constant......Page 170
Reduction of the estimates for ordinary differential operators on the semi-axis to variational problems in a finite-dimensional space......Page 171
Necessary and sufficient conditions for the validity of inequalities (3.1.1) and (3.1.1')......Page 175
Inequalities for functions without boundary conditions......Page 178
Necessary and sufficient conditions for the validity of the estimates (3.0.1), (3.0.2), and (3.0.1')......Page 179
On the minimal number and algebraic properties of the boundary operators; formulas for (;)......Page 182
Estimates for polynomials whose -roots lie in the lower complex half-plane......Page 185
The theorem of N. Aronszajn on necessary and sufficient conditions for the coercivity of a system of operators......Page 186
The case m=1, N=N() in Theorems 3.2.2, 3.2.3, and 3.2.4......Page 187
Examples of estimates for operators of first order with respect to t......Page 191
Notes......Page 194
Introduction......Page 196
Results concerning the estimate (4.0.1)......Page 198
Results concerning the estimate (4.0.2)......Page 202
Polynomials with a generalized-homogeneous principal part......Page 204
The estimate (4.2.16) for quasielliptic polynomials of type l 1......Page 207
The estimate (4.2.19) for quasielliptic polynomials of type l 1......Page 209
Homogeneous polynomials with simple roots......Page 212
Asymptotic representations of the -roots of the polynomial H+(;) as ||......Page 213
Necessary and sufficient conditions for the validity of the estimate (4.2.16)......Page 214
Necessary and sufficient conditions for the validity of the estimate (4.2.19)......Page 218
Some classes of nonhomogeneous polynomials with simple roots......Page 221
Asymptotic representations as N for the -roots j () of the polynomial H+(;)......Page 222
An asymptotic representation of the function () as N for polynomials P with the real -roots......Page 225
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
An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im <0......Page 228
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
An asymptotic representation of the function () as N for a polynomial P with the -roots lying in the half-plane Im >0......Page 231
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
Preliminary results......Page 234
The case p1() 0......Page 236
The case Im pk() 0 (k=0,1,2)......Page 238
The estimate (4.2.16) in the case Re p1()0, Im pk() 0 (k=0,2)......Page 239
Space of traces of functions for the maximal operator......Page 243
The maximal operator as closure of its restriction on the set of functions infinitely differentiable up to the boundary......Page 244
Description of the ``trace space''......Page 246
Notes......Page 250
Notation......Page 252
Bibliography......Page 256
Index......Page 260
