Second order parabolic differential equations(¶þ½×Å×Îï΢·Ö·½³Ì)

PREFACEPREFACE TO REVISED EDITIONChapter ¢ñ INTRODUCTION 1.Outline of this book 2.Further remarks 3.NotationChapter ¢ò MAXIMUM PRINCIPLES Introduction I.The weak maximum principle 2.The strong maximum principle 3.A priori estimates Notes ExercisesChapter ¢ó INTRODUCTION TO THE THEORY OF WEAK SOLUTIONS Introduction 1.The theory of weak derivatives 2.The method of continuity 3.Problems in small balls 4.Global existence and the Perron process Notes ExercisesChapter ¢ô HOLDER ESTIMATES Introduction 1.Ho1der continuity 2.Campanato spaces 3.Interior estimates 4.Estimates near a flat boundary 5.Regularized distance 6.Intermediate Schauder estimates 7.Curved boundaries and nonzero boundary data 8.Two special mixed problems Notes ExercisesChapter ¢õ EXISTENCE, UNIQUENESS AND REGULARITY OF SOLUTIONS Introduction 1.Uniqueness of solutions 2.The Cauchy-Dirichlet problem with bounded coefficients 3.The Cauchy-Dirichlet problem with unbounded coefficients 4.The oblique derivative problem Notes ExercisesChapter ¢ö FURTHER THEORY OF WEAK SOLUTIONS Introduction 1.Notation and basic results 2.Differentiability of weak solutions 3.Sobolev inequalities 4.Poincarf's inequality 5.Global boundedness 6.Local estimates 7.Consequences of the local estimates 8.Boundary estimates 9.More Sobolev-type inequalities 10.Conormal problems 11.A special mixed problem 12.Solvability in H61der spaces 13.The parabolic DeGiorgi classes Notes ExercisesChapter ¢÷ STRONG SOLUTIONS Introduction 1.Maximum principles 2.Basic results from harmonic analysis 3.Lp estimates for constant coefficient divergence structure equations 4.Interior Lp estimates for solutions of nondivergence form constant coefficient equations 5.An interpolation inequality 6.Interior Lp estimates 7.Boundary and global estimates 8.Wp2,1 estimates for the oblique derivative problem 9.The local maximum principle 10.The weak Harnack inequality 11.Boundary estimates Notes ExercisesChapter ¢ø FIXED POINT THEOREMS AND THEIR APPLICATIONS Introduction 1.The Schauder fixed point theorem 2.Applications of the Schauder theorem 3.A theorem of Caristi and its applications Notes ExercisesChapter ¢ù COMPARISON AND MAXIMUM PRINCIPLES Introduction I.Comparison principles 2.Maximum estimates 3.Comparison principles for divergence form operators 4.The maximum principle for divergence form operators Notes ExercisesChapter ¢ú BOUNDARY GRADIENT ESTIMATES Introduction 1.The boundary gradient estimate in general domains 2.Convex-increasing domains 3.The spatial distance function 4.Curvature conditions 5.Nonexistence results 6.The case of one space dimension 7.Continuity estimates Notes ExercisesChapter ¢û GLOBAL AND LOCAL GRADIENT BOUNDS Introduction 1.Global gradient bounds for general equations 2.Examples 3.Local gradient bounds 4.The Sobolev theorem of Michael and Simon 5.Estimates for equations in divergence form 6.The case of one space dimension 7.A gradient bound for an intermediate situation Notes ExercisesChapter ¢ü HOLDER GRADIENT ESTIMATES AND EXISTENCE THEOREMS Introduction 1.Interior estimates for equations in divergence form 2.Equations in one space dimension 3.Interior estimates for equations in general form 4.Boundary estimates 5.Improved results for nondivergence equations 6.Selected existence results Notes ExercisesChapter ¢ú¢ó THE OBLIQUE DERIVATIVE PROBLEM FOR QUASILINEAR PARABOLIC EQUATIONS Introduction 1.Maximum estimates 2.Gradient estimates for the conormal problem 3.Gradient bounds for uniformly parabolic problems in general form 4.The H61der gradient estimate for the conormal problem 5.Nonlinear boundary conditions with linear equations 6.The H61der gradient estimate for quasilinear equations 7.Existence theorems Notes ExercisesChapter ¢ú¢ô FULLY NONLINEAR EQUATIONS ¢ñ. INTRODUCTION Introduction 1.Comparison and maximum principles 2.Simple uniformly parabolic equations 3.Higher regularity of solutions 4.The Cauchy-Dirichlet problem 5.Boundary second derivative estimates 6.The oblique derivative problem 7.The case of one space dimension Notes ExercisesChapter ¢ú¢õ FULLY NONLINEAR EQUATIONS ¢ò. HESSIAN EQUATIONS Introduction 1.General results for Hessian equations 2.Estimates on solutions 3.Existence of solutions 4.Properties of symmetric polynomials 5.The parabolic analog of the Monge-Ampere equation Notes ExercisesBibliographyIndex