CONTENTS 11

33.4 Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113633.5 Helmholtz Decompositions . . . . . . . . . . . . . . . . . . . . . . . . .1146

34 Gelfand Triples And Related Stuff 115534.1 An Unnatural Example . . . . . . . . . . . . . . . . . . . . . . . . . . .115734.2 Standard Techniques In Evolution Equations . . . . . . . . . . . . . . . .116134.3 An Important Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .117134.4 The Implicit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117934.5 The Implicit Case, B = B(t) . . . . . . . . . . . . . . . . . . . . . . . . .119634.6 Another Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120834.7 Some Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . .121534.8 Some Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . .1224

III Sobolev Spaces 1229

35 Weak Derivatives 123135.1 Weak ∗ Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . .123135.2 Test Functions And Weak Derivatives . . . . . . . . . . . . . . . . . . . .123235.3 Weak Derivatives In Lp

loc . . . . . . . . . . . . . . . . . . . . . . . . . . .123535.4 Morrey’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123835.5 Rademacher’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .124135.6 Change Of Variables Formula Lipschitz Maps . . . . . . . . . . . . . . .1244

36 Integration On Manifolds 125336.1 Partitions Of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125336.2 Integration On Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .125636.3 Comparison With H n . . . . . . . . . . . . . . . . . . . . . . . . . . . .1262

37 Basic Theory Of Sobolev Spaces 126537.1 Embedding Theorems For W m,p (Rn) . . . . . . . . . . . . . . . . . . . .127337.2 An Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .128737.3 General Embedding Theorems . . . . . . . . . . . . . . . . . . . . . . .129437.4 More Extension Theorems . . . . . . . . . . . . . . . . . . . . . . . . . .1296

38 Sobolev Spaces Based On L2 130138.1 Fourier Transform Techniques . . . . . . . . . . . . . . . . . . . . . . . .130138.2 Fractional Order Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .130538.3 An Intrinsic Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130738.4 Embedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . .131438.5 The Trace On The Boundary Of A Half Space . . . . . . . . . . . . . . .131538.6 Sobolev Spaces On Manifolds . . . . . . . . . . . . . . . . . . . . . . . .1323

38.6.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . .132338.6.2 The Trace On The Boundary . . . . . . . . . . . . . . . . . . .1327

39 Weak Solutions 1331