Bibliography [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1965. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3) 1973, 637-654. [3] H. Brezis, Analyse fonctionelle: Théorie et Applications, Masson, Paris, 1987. [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer Verlag, 2011. [5] A.P. Calderón, Lebesgue spaces of differentiable functions and distributions, in Proc. Symp. Pure Math., IV, 1961, 33-49. [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture in Mathematics and its Applications n. 13, Clarendon Press, Oxford, 1998. [7] D. Chenais, On the existence of a solution in a domain identification problem, Journal Math. Anal. and Appl. (52) 1975, 189-219. [8] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, Oxford University Press, 1999. [9] D. Cioranescu, V. Girault and K. Rajagopal Mechanics and Mathematics of Fluids of the Differential Type, Springer, 2016. [10] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955. [11] E. De Giorgi, Sulla differenziabilità e l analiticità delle estremali degli Integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3) 25-43, 1957. [12] J. Diestel and J.J. Uhl, Vector Measures, Mathematical Surveys, 15, American Mathematical Society, Providence, 1977. [13] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967. [14] N. Dunford and J.T. Schwartz, Linear Operators, Interscience Publishers, New York, 1958. [15] Yu. V. Egorov and M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations, Springer, Berlin Heidelberg, 1998. [16] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathe- 273
274 An Introduction to Second Order Partial Differential Equations matics Vol. 19, American Mathematical Society, 1998. [17] T. Gallouët and A. Monier, On the regularity of Solutions to Elliptic Equations, Rendiconti di Matematica, Serie VII, Volume 19, Roma, 1999, 471-488. [18] P. R. Garabedian, Partial Differential Equations, Wiley, New York, 1964. [19] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer Verlag, Berlin Heidelberg New York 1983. [20] E. Kreyszig, Advanced Engineering Mathematics, 7th edition, John Wiley and Sons, New York, 1993. [21] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, Inc., New York, 1989. [22] O. A. Ladyzenskaya and N.N. Ural tseva, Linear and Quasilinear Elliptic Equations, English Translation: Academic Press, New York, 1968. [23] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [24] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1 and Vol. 2, Dunod, Paris, 1968. [25] R. Merton, Theory of rational option pricing, Bell J. Economics and Management Sci. 4 (1) 1973, 141-183. doi:10.2307/3003143. [26] N. G. Meyers, An L p -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, 17 1963, 189-206. [27] V. P. Mikhailov, Partial Differential Equations, Mir, Moscow, 1976. [28] J. Nash, Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 1958(80), 931-954. [29] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967. [30] J. Ockendon, S. Howinson, A. Lacey and A. Movchan, Applied Partial Differential Equations, Oxford University Press, 2006. [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Lecture Notes, University of Maryland, 1974. [32] R. Precup, Ecuaţii cu derivate parţiale (in romanian), Transilvania Press, Cluj, 1997. [33] H. L. Royden, Real Analysis, Macmillan, 1963. [34] W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1966. [35] L. Schwartz, Théorie des distributions, Hermann, Paris, 1965. [36] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus, Séminaire sur les équations aux dérivées partielles, Collège de France, 1964. [37] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) (15) fasc. 1, 1965, 189-258. [38] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. [39] M. E. Taylor, Partial Differential Equations: Basic Theory, Springer, New York, 1996.
Bibliography 275 [40] G. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics, Plenum Press, New York, 1987. [41] J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. [42] Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, 2006. [43] K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1964. [44] E. Zeidler, Nonlinear Functional Analysis and its Applications (Part I and Part II), Springer-Verlag, Berlin, 1980.
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Arzela Ascoli Compactness Theorem, 202 Banach Fixed Point Theorem, 219 Banach space, 116 reflexive, 120, 248 separable, 131, 248 bidual, 119 bilinear form, 217 bounded, 217 coercive, 217 H -elliptic, 217 positive, 217 symmetric, 217 Black Scholes equation, 13 boundary conditions, 9 Dirichlet, 9, 217 Neumann, 9, 217 Robin, 9, 217 C k boundary, 165 canonical form, 31 elliptic PDE, 38 hyperbolic PDE, 36, 37 nonconstant coefficients, 40 parabolic PDE, 39 Cauchy Kovalevskaya Theorem, 26 Cauchy Schwarz inequality, 120, 126 Chain Rule in W 1,p (Ω), 167 characteristic curve, 27, 29 compact inclusion, 202 Index compact mapping, 202 continuous embedding, 189 convergence in D (O), 152 convergence in D(O), 149 convolution of f and g, 135 d Alembert solution, 107 density of D(R N ) in L p (R N ), 137 D(R N ) in W 1,p (R N ), 162 D( Ω) in W 1,p (Ω), 166 Dirac function, 150 Dirichlet problem, 55 for R 2, 57 for unit disk, 55 distribution, 149 convergence in the sense of, 152 derivative, 152 regular, 151 domain of dependence, 103 dual space, 119 Eberlein Šmuljan Theorem, 143 eigenvalue problems, 235 Extension Theorem in W 1,p (Ω), 166 Fatou s Lemma, 130 Fourier transform, 185, 186 Fubini s Theorem, 131 gradient of u, 8 277
278 An Introduction to Second Order Partial Differential Equations Green formulas, 61 Green function, 69 Green Representation Theorem, 132 Hölder continuous function, 73 Hölder s inequality, 126 harmonic function, 49 Average Theorem for, 65 radial, 50 heat equation, 11 one-dimensional, 78 three-dimensional, 88 two-dimensional, 88 Helmholtz equation, 17, 111 Hilbert space, 120 Holmgren s Theorem, 31 ill-posed problem, 21 Hadamard s example, 23 inner product space, 120 Interpolation Inequality Lemma, 129 inverse Fourier transform, 185, 186 Kirchhoff formula, 97 L p -spaces, 125 norm, 126 vector-valued functions, 248 L p (0, T ; X), 248 dual, 248 norm, 248 Laplace equation, 11, 49 polar coordinates, 52 spherical coordinates, 54 Laplace operator polar coordinates, 49 spherical coordinates, 50 Lax Milgram Theorem, 219 Lebesgue s Dominated Convergence Theorem, 131 linear operator, 117 bounded, 117 norm, 117 Lipschitz-continuous boundary, 165 M(α, β, Ω), 216 Maximum Principle general parabolic equations, 89 heat equation, 78 Laplace equation, 66 Maxwell equations, 18 mean value of a function, 196, 211 n-mollifier, 136 Navier Stokes equations, 20 norm, 115 equivalence, 116 Euclidean, 116 sup norm, 116 normal derivative, 8 normed space, 116 orthogonal basis, 121 orthogonal complement, 121 orthogonal projection, 122 orthonormal basis, 122 Ostrogradski Theorem, 61 parabolic equations general case, 88 parallelogram rule, 102 partial differential equation, 4 elliptic, 32 hyperbolic, 32 linear, 4 parabolic, 32 quasilinear, 5 semilinear, 5 Plancherel s Theorem, 186 Poincaré inequality, 182 Poincaré Wirtinger Inequality, 210 Poisson equation, 11 Poisson formula, 99 Poisson kernel, 56, 71 Principle of Superposition, 4 quotient space W (Ω), 211 Rademacher s Theorem, 174 range of influence, 103 regular, 28 regularity of weak solutions, 241
Index 279 regularizing sequence, 136 Rellich Kondrachov Theorem, 204 Representation Theorem, 63 Riesz Representation Theorem, 125 Schauder s estimates, 75 separation of variables, 6 heat equation, 82 Laplace equation, 52, 54 Sobolev Compact Embedding p = N, 208 p > N, 203 p [1, N), 204 Sobolev Continuous Embedding p = N, 195 p > N, 196 p [1, N), 191 Sobolev spaces, 154 norm, 155 support of a function in C 0 (O), 132 in L p (O), 134 telegraph equations, 19 tensor product, 250 test function, 148 trace of a function, 176 Trace Theorem, 175 traveling waves, 92 backward, 92 forward, 92 uniformly elliptic operator, 74, 216 Urysohn s Lemma, 138 variational elliptic problems, 215 with Neumann boundary condition, 226 with homogeneous Dirichlet boundary condition, 222 with nonhomogeneus Dirichlet boundary condition, 224 with Robin boundary conditions, 233 variational evolution problems hyperbolic, 264 parabolic, 257 variational formulation, 148 wave equation, 14 Cauchy problem, 23 one-dimensional, 101 weak convergence, 141 in L p -space, 142 weak convergence, 144 well-posedness, 21 Young s inequality, 129 zero extension, 139