## Degenerate Elliptic EquationsThis volume is the first to be devoted to the study of various properties of wide classes of degenerate elliptic operators of arbitrary order and pseudo-differential operators with multiple characteristics. Conditions for operators to be Fredholm in appropriate weighted Sobolev spaces are given, a priori estimates of solutions are derived, inequalities of the Gårding type are proved, and the principal term of the spectral asymptotics for self-adjoint operators is computed. A generalization of the classical Weyl formula is proposed. Some results are new, even for operators of the second order. In addition, an analogue of the Boutet de Monvel calculus is developed and the index is computed. For postgraduate and research mathematicians, physicists and engineers whose work involves the solution of partial differential equations. |

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### Contents

II | 9 |

III | 12 |

IV | 19 |

V | 24 |

VI | 33 |

VII | 37 |

VIII | 40 |

IX | 43 |

LV | 190 |

LVI | 193 |

LVII | 199 |

LVIII | 202 |

LIX | 203 |

LX | 207 |

LXI | 210 |

LXII | 213 |

X | 45 |

XII | 50 |

XIII | 56 |

XIV | 59 |

XV | 63 |

XVI | 68 |

XVII | 69 |

XVIII | 70 |

XIX | 75 |

XX | 78 |

XXI | 79 |

XXII | 81 |

XXIII | 85 |

XXIV | 87 |

XXV | 89 |

XXVI | 95 |

XXVII | 98 |

XXVIII | 103 |

XXX | 105 |

XXXI | 108 |

XXXII | 112 |

XXXIII | 123 |

XXXIV | 129 |

XXXV | 131 |

XXXVI | 135 |

XXXVII | 137 |

XXXVIII | 143 |

XXXIX | 145 |

XL | 146 |

XLI | 156 |

XLII | 158 |

XLIII | 162 |

XLIV | 163 |

XLV | 164 |

XLVII | 165 |

XLVIII | 166 |

XLIX | 171 |

L | 174 |

LI | 176 |

LII | 180 |

LIII | 182 |

LIV | 187 |

LXIV | 216 |

LXV | 225 |

LXVI | 226 |

LXVII | 231 |

LXVIII | 232 |

LXIX | 239 |

LXX | 240 |

LXXI | 245 |

LXXII | 247 |

LXXIII | 249 |

LXXIV | 251 |

LXXV | 255 |

LXXVI | 262 |

LXXVII | 265 |

LXXVIII | 270 |

LXXIX | 272 |

275 | |

LXXXI | 279 |

LXXXII | 285 |

LXXXIII | 291 |

LXXXIV | 296 |

LXXXV | 301 |

LXXXVI | 304 |

LXXXVII | 311 |

LXXXVIII | 317 |

LXXXIX | 320 |

XC | 329 |

XCI | 330 |

XCII | 335 |

XCIII | 338 |

XCIV | 344 |

XCV | 355 |

XCVI | 359 |

XCVII | 363 |

XCIX | 366 |

C | 377 |

CI | 389 |

CII | 399 |

423 | |

429 | |

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### Common terms and phrases

admits a unique algebra analogue boundary value problems bounded operator calculus Chapter closed manifold coefficients compact conditions of Theorem construct coordinates defined Definition degenerate elliptic operators denote differential operators eigenvalues elliptic differential operators elliptic equations finite norm following conditions hold following estimate forms of Type Fredholm function Hardy inequality Hence Hilbert space Hormander hypoelliptic operators implies integral invertible involutory ISBN kernel Lemma Let A G Levendorskii Math metric g model classes Monvel neighbourhood obtain open set operators of Type parametrix partition of unity principal symbol priori estimate proof of Theorem pseudodifferential operators R+;C Remark resp right hand side Rn_1 satisfy conditions Section smooth Sobolev spaces spectral asymptotics spectrum Subsection subspace suffices to prove sufficiently small symplectic temperate metric Theorem 1 Let topological uniformly with respect unique bounded extension V'do valued symbol Weyl formula Weyl symbol