Functional Analysis, Sobolev Spaces and Partial Differential Equations

Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.

[1]  Leopoldo Nachbin,et al.  A theorem of the Hahn-Banach type for linear transformations , 1950 .

[2]  R. C. James.  A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[3]  J. L. Kelley.  Banach spaces with the extension property , 1952 .

[4]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[5]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[6]  J. Neveu,et al.  Mathematical foundations of the calculus of probability , 1965 .

[7]  J. Lions,et al.  Problèmes aux limites dans les équations aux dérivées partielles , 1965 .

[8]  H. Weinberger,et al.  A first course in partial differential equations , 1965 .

[9]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[10]  L. Carleson.  On convergence and growth of partial sums of Fourier series , 1966 .

[11]  L. Karlovitz,et al.  On the radial projection in normed spaces , 1967 .

[12]  Joseph J. Kohn,et al.  Degenerate elliptic-parabolic equations of second order , 1967 .

[13]  F. Trèves.  Topological vector spaces, distributions and kernels , 1967 .

[14]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[15]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[16]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[17]  Achi Brandt,et al.  Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle , 1969 .

[18]  M. S. Baouendi,et al.  Régularité et théorie spectrale pour une classe d'opérateurs elliptiques dégénérés , 1969 .

[19]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[20]  V. A. Kondrat'ev,et al.  On Positive Solutions of Elliptic Equations , 1971 .

[21]  Joram Lindenstrauss,et al.  On the complemented subspaces problem , 1971 .

[22]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[23]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[24]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[25]  D. Varberg.  Convex Functions , 1973 .

[26]  H. Weinberger.  Variational Methods for Eigenvalue Approximation , 1974 .

[27]  James R. Munkres,et al.  Topology; a first course , 1974 .

[28]  Peter B. Gilkey,et al.  The index theorem and the heat equation , 1974 .

[29]  J. Moreau.  Application of convex analysis to the treatment of elastoplastic systems , 1976 .

[30]  Felix E. Browder,et al.  Nonlinear Operators and Nonlinear Equations of Evolution , 1976 .

[31]  E. Magenes.  Topics in Parabolic Equations: Some Typical Free Boundary Problems , 1977 .

[32]  C. Pearcy.  Some recent developments in operator theory , 1978 .

[33]  J. Toland.  Duality in nonconvex optimization , 1978 .

[34]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[35]  Ivar Ekeland,et al.  Hamiltonian trajectories having prescribed minimal period , 1980 .

[36]  Barry F. Knerr,et al.  Parabolic interior Schauder estimates by the maximum principle , 1980 .

[37]  R. Rockafellar.  The theory of subgradients and its applications to problems of optimization : convex and nonconvex functions , 1981 .

[38]  Louis Nirenberg,et al.  Variational and topological methods in nonlinear problems , 1981 .

[39]  Roger D. Nussbaum,et al.  Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem , 1981 .

[40]  B(H) does not have the approximation propertydoes not have the approximation property , 1981 .

[41]  I. Ekeland,et al.  Infinite-Dimensional Optimization And Convexity , 1983 .

[42]  H. Triebel.  Theory Of Function Spaces , 1983 .

[43]  R. Varga,et al.  Proof of Theorem 2 , 1983 .

[44]  Elliott H. Lieb,et al.  A Relation Between Pointwise Convergence of Functions and Convergence of Functionals , 1983 .

[45]  Elliott H. Lieb,et al.  Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities , 1983 .

[46]  R. Noyé,et al.  Numerical Solutions of Partial Differential Equations , 1983 .

[47]  Giles Auchmuty.  Duality for non-convex variational principles , 1983 .

[48]  B. Beauzamy.  Introduction to Banach spaces and their geometry , 1985 .

[49]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[50]  Leonard M. Adleman,et al.  Proof of proposition 3 , 1992 .

[51]  Y. Meyer.  Wavelets and Operators , 1993 .

[52]  C(X) in the weak topology , 1995 .

[53]  N. Krylov,et al.  Lectures on Elliptic and Parabolic Equations in Holder Spaces , 1996 .

[54]  J. Toland.  Self-Adjoint Operators and Cones , 1996 .

[55]  P. Wojtaszczyk,et al.  A Mathematical Introduction to Wavelets: Wavelets and smoothness of functions , 1997 .

[56]  Y. Meyer,et al.  Wavelets: Calderón-Zygmund and Multilinear Operators , 1997 .

[57]  Ya-Zhe Chen,et al.  Second Order Elliptic Equations and Elliptic Systems , 1998 .

[58]  F. Browder,et al.  Partial Differential Equations in the 20th Century , 1998 .

[59]  Olivier Druet,et al.  Best constants in Sobolev inequalities , 1998 .

[60]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[61]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[62]  K. Taira.  Proof of Theorem 1.3 , 2004 .

[63]  Rafael Ortega,et al.  Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications , 2005 .

[64]  L. Caffarelli,et al.  A Geometric Approach to Free Boundary Problems , 2005 .

[65]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

[66]  E. Davies,et al.  Linear Operators and their Spectra , 2007 .

[67]  H. Bui.  Duality and symmetry lost in solid mechanics , 2008 .

[68]  Claudio Perez Tamargo.  Can one hear the shape of a drum , 2008 .

[69]  P. Rabinowitz.  Variational Methods for Nonlinear Eigenvalue Problems , 2009 .

[70]  Isoperimetric Inequalities and Eigenvalues of the Laplacian Robert Osserman , 2010 .

[71]  S. Yau.  The Role of Partial Differential Equations in Differential Geometry , 2010 .

[72]  F. Lin,et al.  Geometric Measure Theory: An Introduction , 2010 .

[73]  I. Singer.  Eigenvalues of the Laplacian and Invariants of Manifolds , 2010 .