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Gradient Systems Ralph Chill, Eva Fasangovˇ a´ Gradient Systems – 13th International Internet Seminar – February 10, 2010 c by R. Chill, E. Faˇsangov´a Contents 0 Introduction ................................................... 1 1 Gradient systems in euclidean space .............................. 5 1.1 Thespace Rd .............................................. 5 1.2 Fr´echetderivative .............................. ............ 7 1.3 Euclideangradient............................... ........... 8 1.4 Euclideangradientsystems ........................ .......... 9 1.5 Algebraicequationsand steepest descent ............ ........... 12 1.6 Exercises ....................................... .......... 13 2 Gradient systems in finite dimensional space ...................... 15 2.1 Definitionofgradient ............................. .......... 17 2.2 Definitionofgradientsystem....................... .......... 18 2.3 Local existence for ordinary differential equations . ............. 18 2.4 Local and global existence for gradientsystems .. ..... .......... 21 2.5 Newton’smethod.................................. ......... 23 2.6 Exercises ....................................... .......... 25 3 Gradients in infinite dimensional space: the Laplace operator on a bounded interval ....................... 27 3.1 Definitionofgradient ............................. .......... 28 3.2 Gradientsofquadraticforms ....................... .......... 30 3.3 Sobolevspacesinonespacedimension ................ ........ 31 3.4 The Dirichlet-Laplace operator on a bounded interval . .......... 33 3.5 The Dirichlet-Laplace operator with multiplicative coefficient ..... 35 3.6 Exercises ....................................... .......... 37 4 Gradients in infinite dimensional space: the Laplace operator and the p-Laplace operator .................. 39 4.1 Sobolevspacesinhigherdimensions ................. ......... 39 4.2 TheDirichlet-Laplaceoperator .................... ........... 42 v vi Contents 4.3 The Dirichlet-p-Laplaceoperator ............................. 44 4.4 Anauxiliaryresult............................... ........... 46 4.5 OriginsoftheLaplaceoperator..................... .......... 49 4.6 Exercises ....................................... .......... 52 5 Bochner-Lebesgue and Bochner-Sobolev spaces ................... 55 5.1 TheBochnerintegral.............................. .......... 56 5.2 Bochner-Lebesguespaces .......................... ......... 58 5.3 Bochner-Sobolevspaces in one space dimension . ..... .. ........ 59 5.4 Exercises ....................................... .......... 63 6 Gradient systems in infinite dimensional spaces: existence and uniqueness of solutions ......................................... 65 6.1 Gradient systems in infinite dimensional spaces .. ..... .......... 65 6.2 Global existence and uniqueness of solutions for gradient systems withconvexenergy................................... ...... 66 6.3 Exercises ....................................... .......... 75 7 Diffusion equations ............................................. 77 7.1 Heatconduction.................................. .......... 77 7.2 Thereaction-diffusionequation................... ............ 80 7.3 The Perona-Malikmodelin imageprocessing........... ........ 81 7.4 Global existenceand uniquenessof solutions ......... .......... 82 7.5 Exercises ....................................... .......... 84 8 Gradient systems in infinite dimensional spaces: existence and uniqueness of solutions II ....................................... 87 8.1 Global existence and uniqueness of solutions for gradient systems withellipticenergy ................................. ........ 88 8.2 Exercises ....................................... .......... 97 9 Regularity of solutions .......................................... 99 9.1 Interpolation................................... ............ 99 9.2 Timeregularity–Angenent’strick .................. ..........101 9.3 Exercises ....................................... ..........104 10 Existence of local solutions ......................................107 10.1 The local inverse theorem and existence of local solutions of abstractdifferentialequations ...................... ..........108 10.2 Gradients of quadratic forms and identification of the trace space.. 109 10.3 Asemilineardiffusionequation................... ............112 10.4 Exercises ...................................... ...........117 Contents vii 11 Asymptotic behaviour of solutions of gradient systems .............119 11.1 The ω-limit set of a continuous function on R+ .................120 11.2 A stabilisation result for global solutions of gradientsystems ......121 11.3 Nonstabilisation results for global solutions of gradient systems. 123 11.4 Exercises ...................................... ...........126 12 Asymptotic behaviour of solutions of gradient systems II ...........129 12.1 The Łojasiewicz-Simon inequality and stabilisation of global solutionsofgradientsystems......................... ........130 12.2 The Łojasiewicz-Simon inequality in Hilbert spaces . ...........133 12.3 Exercises ...................................... ...........138 13 The universe, soap bubbles and curve shortening ..................141 References ................................................... ......151 A Primerontopology .............................................1001 A.1 Metricspaces.................................... ..........1001 A.2 Sequences,convergence........................... ..........1004 A.3 Compactness ..................................... .........1006 A.4 Continuity ...................................... ..........1007 A.5 Completionofametricspace........................ .........1008 B Banach spaces and bounded linear operators ......................1011 B.1 Normedspaces.................................... .........1011 B.2 Productspacesandquotientspaces.................. ..........1017 B.3 Boundedlinearoperators .......................... ..........1020 B.4 TheArzela-Ascolitheorem ......................... .........1025 C Calculus on Banach spaces ......................................1029 C.1 Differentiable functions between Banach spaces . ...........1029 C.2 Local inverse theorem and implicit function theorem . ..........1030 C.3 *Newton’smethod................................. ........1034 D Hilbert spaces .................................................1035 D.1 Innerproductspaces .............................. ..........1035 D.2 Orthogonaldecomposition......................... ..........1039 D.3 *Fourierseries .................................. ..........1044 D.4 LinearfunctionalsonHilbertspaces................ ...........1048 D.5 WeakconvergenceinHilbertspaces .................. .........1049 E Dual spaces and weak convergence ...............................1053 E.1 ThetheoremofHahn-Banach ......................... .......1053 E.2 Weak∗ convergenceand the theorem of Banach-Alaoglu . .1059 E.3 Weakconvergenceandreflexivity .................... .........1060 E.4 *Minimizationofconvexfunctionals................ ..........1065 viii Contents E.5 *ThevonNeumannminimaxtheorem.................... .....1068 F The divergence theorem (BY WOLFGANG ARENDT) ................1071 F.1 Open sets of class C1 .......................................1071 F.2 Thedivergencetheorem ............................ .........1074 F.3 Proofofthedivergencetheorem ..................... .........1077 G Sobolev spaces .................................................1083 G.1 Test functions, convolution and regularization . .............1083 G.2 Sobolevspacesinonedimension ..................... ........1086 G.3 Sobolevspacesinseveraldimensions................ ..........1092 G.4 * Elliptic partial differentialequations ..... ..... ..............1094 H Bochner-Lebesgue and Bochner-Sobolev spaces ...................1097 H.1 TheBochnerintegral.............................. ..........1097 H.2 Bochner-Lebesguespaces .......................... .........1100 H.3 Bochner-Sobolevspaces in onedimension............. .........1102 Index ................................................... ..........1105 Lecture 0 Introduction Philosophy is written in that gigantic book which is perpetually open in front of our eyes (I allude to the universe), but no one can understand it who does not strive beforehand to learn the language and recognize the letters in which it is written. It is written in mathematical language, and its letters are triangles, circles and other geometrical figures, and without these means it is humanly impossible to understand any of it; without them, all we can do is to wander aimlessly in an obscure labyrinth. Galileo Galilei The evolution of many systems from physics, biology, chemistry, engineering, economics, social sciences can be rewritten in the mathematical language which uses ordinary or partial differential equations. Due to the analysis of the differen- tial equations we have the hope that we need not wander aimlessly in an obscure labyrinth but that we understand the corresponding evolution problems. In order to illustrate this statement, think of the motion of a pendulum, for example the oscillations of the great lantern in the nave of Pisa cathedral. At the end of the 16th century, after mere observation of this lantern and probably other pendula, Galilei conjectured that on the one hand the period of one swing is proportional to the length of the pendulum and on the other hand this period is independent of the amplitude of the swing. The conjecture about the precise period of one swing would have
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