12 CONTENTS

39.1 The Lax Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . .133139.2 An Application Of The Mountain Pass Theorem . . . . . . . . . . . . . .1335

40 Korn’s Inequality 134140.1 A Fundamental Inequality . . . . . . . . . . . . . . . . . . . . . . . . . .134140.2 Korn’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1346

41 Elliptic Regularity 134941.1 The Case Of A Half Space . . . . . . . . . . . . . . . . . . . . . . . . . .134941.2 The Case Of Bounded Open Sets . . . . . . . . . . . . . . . . . . . . . .1358

42 Maximal Monotone Operators, Hilbert Space 136742.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136742.2 Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137342.3 Subgradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1379

42.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . .137942.3.2 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . .1393

42.4 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139542.5 Moreau’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139642.6 A Perturbation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .139942.7 An Evolution Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .140142.8 A More Complicated Perturbation Theorem . . . . . . . . . . . . . . . .140442.9 An Evolution Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .1405

43 Interpolation In Banach Space 141343.1 Some Standard Techniques In Evolution Equations . . . . . . . . . . . . .1413

43.1.1 Weak Vector Valued Derivatives . . . . . . . . . . . . . . . . . .141343.2 An Important Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .142343.3 The Implicit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143043.4 Some Implicit Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . .144043.5 Some Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . .144343.6 The K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144843.7 The J Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145343.8 Duality And Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . .1459

44 Trace Spaces 146944.1 Definition And Basic Theory Of Trace Spaces . . . . . . . . . . . . . . .146944.2 Trace And Interpolation Spaces . . . . . . . . . . . . . . . . . . . . . . .1476

45 Traces Of Sobolev Spaces 148345.1 Traces Of Sobolev Spaces, Half Space . . . . . . . . . . . . . . . . . . .148345.2 A Right Inverse For The Trace For A Half Space . . . . . . . . . . . . . .148545.3 Intrinsic Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148745.4 Fractional Order Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . .1508

46 Sobolev Spaces On Manifolds 1513