[Bar] R.G. Bartle, The Elements of Real Analysis, 2nd ed., Wiley, New York, 1976.

Transcription

1 AppendixA References A.1 Elementary Texts [Bar] R.G. Bartle, The Elements of Real Analysis, 2nd ed., Wiley, New York, [BC] D. Bleecker and G. Csordas, Basic Partial Differential Equations, Van Nostrand Reinhold, New York, [BD] W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 4th ed., Wiley, New York, [Bu] R.C. Buck, Advanced Calculus, 3rd ed., McGraw-Hill, New York, [Kr] E. Kreysig, Introductory Functional Analysis with Applications, Wiley, New York, [MH] J.E. Marsden and M.J. Hoffman, Basic Complex Analysis, W.H. Freeman, New York, 3rd ed., [Rud] W. Rudin, Principles of Mathematical Analysis, 3rd ed. McGraw Hill, New York, [Stak] I. Stakgold, Boundary Value Problems of Mathematical Physics, Vol. 1/2, Macmillan, New York, [ZT] E.C. Zachmanoglou and D.W. Thoe, Introduction to Partial Differential Equations with Applications, Dover, New York, 1986.

2 A.2. Basic Graduate Texts 427 A.2 Basic Graduate Texts [CH1] R. Courant and D. Hilbert, Methods of Mathematical Physics I, Wiley, New York, [CH2] R. Courant and D. Hilbert, Methods of Mathematical Physics II, Wiley, New York, [DiB] E. DiBenedetto, Partial Differential Equations, Birkhäuser, Boston, [[Eva] L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, [GS] I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 1, Academic Press, New York, [Ha] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, [In] E.L. Ince, Ordinary Differential Equations, Dover, New York, [Jo] F. John, Partial Differential Equations, 4th ed., Springer-Verlag, New York, [La] O.A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (English Edition), Springer-Verlag, New York, [Rau] J. Rauch, Partial Differential Equations, Springer-Verlag, New York, [RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, [Sc] L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, Reading, MA, [Wlok] J. Wloka, Partial Differential Equations, Cambridge University Press, New York, 1987 A.3 Specialized or Advanced Texts [Adam] R.A. Adams, Sobolev Spaces, Academic Press, New York, [Dac] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, [DS] N. Dunford and J.T. Schwartz, Linear Operators I, Wiley, New York, [ET] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.

3 428 AppendixA. References [EN] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, [Fri1] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, [Fri2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, [Go] J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, [GR] I.S. Gradshteyn and I.M. Ryshik, Table of Integrals, Series and Products, Academic Press, New York, [He] G. Hellwig, Differential Operators of Mathematical Physics, Addison- Wesley, Reading, MA, [Ka] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer- Verlag, New York, [Ke] O.D. Kellogg, Foundations of Potential Theory, Dover, NewYork, [KJF] A. Kufner, O. John, and S. Fucik, Function Spaces, Noordhoff International Publishers, Leyden, [LSU] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, [LU] O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, [LM] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, New York, [Li] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, [Mor] C.B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, [PW] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, [Sm] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer- Verlag, New York, 1983.

4 A.4. Multivolume or Encyclopedic Works 429 [Ze] E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, A.4 Multivolume or Encyclopedic Works [DL] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 6 vol., Springer-Verlag, Berlin, [ESFA] Y.V. Egorov, M.A. Shubin, M.V. Fedoryuk, M.S. Agranovich (eds.), Partial Differential Equations I-IX, in: Encyclopedia of Mathematical Sciences, Vols , 63-65, 79, Springer-Verlag, New York, from [Hor] L. Hörmander, The Analysis of Linear Partial Differential Operators, 4 vol., Springer-Verlag, Berlin, [Tay] M.E. Taylor, Partial Differential Equations, 3 vol. Springer-Verlag, New York, A.5 Other References [Ab] E.A. Abbott, Flatland, Harper & Row, New York, [ADN1] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), [ADN2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), and 17 (1964), [Ba] J. Ball, Convexity conditions and existence theorems in nonlinear elasiticy, Arch. Rational Mechan. Anal., 63 (1977), [Fra] L.E. Fraenkel, On regularity of the boundary in the theory of Sobolev spaces, Proc. London Math. Soc. 39 (1979), No. 3, [Fri] K.O. Friedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), [GNN] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1980), [La] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957),

5 430 AppendixA. References [Max] J.C. Maxwell, Science and free will, in: L. Campbell and W. Garnett (eds.), The Life of James Clerk Maxwell, Macmillan, London, [Mas] W.S. Massey, Singular Homology Theory, Springer-Verlag, New York, 1980, p. 218ff. [Mo] T. Morley, A simple proof that the world is three-dimensional, SIAM Rev. 27 (1985), [Se] M. Sever, Uniqueness failure for entropy solutions of hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 42 (1989), [Vo] L.R. Volevich, A problem of linear programming arising in differential equations, Uspekhi Mat. Nauk 18 (1963), No. 3, (Russian).

6 Index C 0-semigroup, 397 L p spaces, 177 p system, 68 Abel s integral equation, 161 Adjoint, 311 adjoint, 61, 251 adjoint, boundary-value problem, 166 adjoint, formal, 163 adjoint, Hilbert, 253 admissibility conditions, 83, 94 Agmon s Condition, 315 Alaoglu s theorem, 200 analytic, 248 Analytic Fredholm theorem, 266 Analytic Functions, 46 analytic semigroup, 413 analytic, weakly, 250 Arzela-Ascoli theorem, 110 backwards heat equation, 26 Banach contraction principle, 336 Banach space, 175 Banach space valued functions, 380 barrier, 113 basis, 186 bifurcation, 5, 340 Boundary Integral Methods, 170 Boundary Regularity, 324 bounded below, 240 Bounded inverse theorem, 241 bounded linear operator, 194, 230 bounded, relative, 241 Brouwer fixed point theorem, 361 Browder-Minty theorem, 364 Burgers equation, 68 calculus of variations type, 371 Carathéodory conditions, 370 Cauchy problem, 31 Cauchy s integral formula, 10 Cauchy-Kovalevskaya Theorem, 46 Cauchy-Schwarz inequality, 180 characteristic, 40 classical solution, 287 closable, 237 closed, 237 Closed graph theorem, 241 coercive, 291, 360, 363 Coercive Problems, 315 compact, 259 compact imbedding, 211 compact, relative, 270 comparison principle, 103

7 432 Index Complementing Condition, 306 completion, 175 compression spectrum, 245 continuous imbedding, 209 continuous spectrum, 245 contraction semigroup, 406 convergence, distribution, 130 convergence, strong, 232 convergence, test functions, 124 convergence, weak, 199 convergence, weak-, 199 convex, 347 convolution, 143 corners, 325 D Alembert s solution, 31 deficiency, 245, 280 deficiency indices, 256 delta convergent sequences, 139 diffeomorphism, 221 Difference Quotients, 321 Dirac delta function, 127 direct product, 143 Dirichlet conditions, 15 Dirichlet system, 311 discrete spectrum, 245 dissipative opertor, 407 distribution, 126 distribution, approximation by test functions, 146 distribution, convergence, 130 distribution, derivative, 135 distribution, finite order, 128 distribution, primitive, 141 distribution, sequential completeness, 130 div-curl lemma, 352 divergence form, 284 domain, 229 domain of determinacy, 64 dual space, 195 dual spaces, Sobolev, 218 DuBois-Reymond lemma, 20 Duhamel s principle, 29 Ehrling s lemma, 212 Eigenfunction expansions, 300 eigenfunction expansions, 268, 273 eigenvalues, 245 eigenvectors, 245 elasticity, 342 elliptic, 39, 284 Energy estimate, 11, 33 energy estimate, 28 Entropy Condition, 94 entropy/entropy-flux pair, 95 equicontinuous, 110 essentially self-adjoint, 256 Euler equations, 45 Euler-Lagrange equations, 344 exponential matrix, 395 extension, 231 extension property, 208 finite rank, 261 Fourier series, 17, 188 Fourier transform, 38, 151, 208 Fréchet derivative, 336 Fréchet derivative, Fréchet, 336 Fractional Powers, 416 Fredholm alternative theorem, 267 Fredholm index, 280 Fredholm operator, 279 Friedrichs lemma, 409 functions, Banach space valued, 380 fundamental lemma of the calculus of variations, 20 fundamental solution, 147 fundamental solution, heat equation, 148 fundamental solution, Laplace s equation, 148 fundamental solution, ODE, 147 fundamental solution, wave equation, 150, 156 Galerkin s method, 365, 383 Gas dynamics, 69 generalized function, 126 genuinely nonlinear, 72 graph, 237 graph norm, 240 Green s function, 167, 274 Green s Functions, 163 Gronwall s inequality, 10 Gårding s inequality, 292 Hölder s inequality, 177

8 Index 433 Hahn-Banach Theorem, 197 heat equation, 24, 408 hemi-continuous, 378 Hilbert adjoint, 253 Hilbert space, 181 Hilbert-Schmidt kernel, 235, 262 Hilbert-Schmidt theorem, 268 Hille-Yosida theorem, 403 Holmgren s Uniqueness Theorem, 61 hyperbolic, 39 imbedding, compact, 211 imbedding, continuous, 209 Implicit function theorem, 3 implicit function theorem, 50 index, Fredholm, 280 infinitesimal generator, 399 inner product, 180 integral operator, 235 Inverse function theorem, 3, 337 isometric, 175 Jordan curve theorem, 105 jump condition, 79 Laplace transform, 397 Laplace transforms, 159 Laplace s Equation, 15 Lax Shock Condition, 83 Lax-Milgram lemma, 290 Legendre-Hadamard condition, 286 linear functional, 195 linear operator, 229 linearly degenerate, 73 Lipschitz continuous, 207 lower convex envelope, 356 lower semicontinuous, 347 Lumer-Phillips theorem, 407 Majorization, 50 Maximum modulus principle, 12 maximum principle, strong, 103, 118 maximum principle, weak, 102, 117 Mazur s lemma, 350 method of descent, 157 mild solution, 402 monotone, 360, 363 negative Sobolev spaces, 218 Nemytskii Operators, 370 Neumann conditions, 15 Neumann series, 246 norm, 174 norm, equivalent, 175 norm, operator, 195, 230 null Lagrangian, 358 null Lagrangians, 352 null space, 229 nullity, 280 ODE, continuity with respect to initial conditions, 7 ODE, eigenvalues, 5 ODE, existence, 2 ODE, uniqueness, 4 Open mapping theorem, 241 operator norm, 230 operator, Fredholm, 279 operator, norm, 195 operator, quasi-dissipative, 407 operators, strong convergence, 232 orthogonal, 182 Orthogonal polynomials, 190 orthonormal, 185 parabolic, 39 partition of unity, 125, 222 Perturbation, 246, 335 perturbation, 241, 270 perturbations, analytic semigroups, 419 phase transitions, 355 Picard-Lindelöf theorem, 2 Poincaré s inequality, 213 point spectrum, 245 Poisson s formula, 108 Poisson s integral formula, 19 polyconvex, 353 principal part, 38 principal value, 130 Projection theorem, 182 Pseudo-monotone Operators, 371 quasi-dissipative operator, 407 quasi-m-dissipative operator, 407 quasicontraction semigroup, 406 quasiconvex, 356 quasilinear, 45

9 434 Index radial symmetry, 114 range, 229 rank one convex, 357 Rankine-Hugoniot condition, 79 rarefaction wave, 81, 85 Rarefaction waves, 88 reflexive, 197 regular values, 244 regularization, singular integrals, 130 residual spectrum, 245 resolvent set, 244 Riemann invariants, 70 Riemann Problems, 84 Riesz representation theorem, 196 Schrödinger Equation, 411 Schwartz reflection principle, 60 self-adjoint, 254 self-adjoint, essentially, 256 semi-fredholm, 279 semigroup, 397 semigroup, analytic, 413 semigroup, contraction, 406 semigroup, type, 399 semigroups, perturbations, 419 semilinear, 45 separable, 182 separation of variables, 15 shock wave, 67 Shock waves, 86 Sobolev imbedding theorem, 209 Sobolev Spaces, 203 spectral radius, 247 spectrum, 244 stability, 6 Stokes system, 45, 56 strictly hyperbolic, 42 strong solution, 287 strongly continuous semigroup, 397 strongly convex, 72 Sturm-Liouville problem, 271 subharmonic, 103, 109 subsolution, 103, 107 surfaces, smoothness, 53 symbol, 37 symmetric, 254 Symmetric Hyperbolic Systems, 408 test function, 124 test functions, convergence, 124 Tonelli s theorem, 347 Trace Theorem, 214 type, semigroup, 399 types, 38 ultrahyperbolic, 40 Uniform Boundedness Theorem, 198 uniformly elliptic, 284 unit ball, surface area, 114 unit ball, volume, 114 variation of parameters, 9 Variational problems, 19 variational problems, nonconvex, 355 Variational problems, nonexistence, 14 Variational problems, nonlinear, 342 vector valued functions, 380 Viscosity Solutions, 97 Wave Equation, 410 wave equation, 30 Weak compactness theorem, 200 weak convergence, 199 weak solution, 21, 35, 67, 78, 289, 366 weakly analytic, 250 Weierstraß Approximation Theorem, 64 weighted L 2 -spaces, 191 well-posed problems, 8 tempered distribution, 133