THE MAXIMUM PRINCIPLE

By ELIZABETH THOMAS

CONTENTS INTRODUCTION NOTATION THE CONTRIBUTIONS OF EBERHARD HOPF TANGENCY AND COMPARISON PRINCIPLES FOR QUASILINEAR INEQUALITIES MAXIMUM AND SWEEPING PRINCIPLES FOR QUASILINEAR INEQUALITIES COMPARISON THEOREMS FOR DIVERGENCE STRUCTURE INEQUALITIES TANGENCY THEOREMS VIA HARNACK’S INEQUALITY UNIQUENESS OF THE DIRICHLET PROBLEM THE BOUNDARY POINT LEMMA APPENDIX: PROOF OF EBERHARD HOPF’S MAXIMUM PRINCIPLE NOTES PROBLEMS DISTRIBUTION SOLUTIONS MAXIMUM PRINCIPLES FOR HOMOGENEOUS INEQUALITIES A MAXIMUM PRINCIPLE FOR THIN SETS A COMPARISON THEOREM IN W 1,P(?) COMPARISON THEOREMS FOR SINGULAR ELLIPTIC INEQUALITIES STRONGLY DEGENERATE OPERATORS MAXIMUM PRINCIPLES FOR NON-HOMOGENEOUS ELLIPTIC INEQUALITIES UNIQUENESS OF THE SINGULAR DIRICHLET PROBLEM APPENDIX: SOBOLEV’S INEQUALITY PRELIMINARY LEMMAS EXISTENCE THEOREMS EXISTENCE THEOREMS ON A HALF-LINE THE END POINT LEMMA APPENDIX: PROOF OF PROPOSITION 4 2 1 THE STRONG MAXIMUM PRINCIPLE THE COMPACT SUPPORT PRINCIPLE A SPECIAL CASE STRONG MAXIMUM PRINCIPLE: GENERALIZED VERSION A BOUNDARY POINT LEMMA COMPACT SUPPORT PRINCIPLE: GENERALIZED VERSION MAXIMUM PRINCIPLES FOR STRUCTURED INEQUALITIES PROOF OF THEOREM 6 1 3 AND THE ?RST PART OF THEOREM 6 1 5 PROOF OF THEOREM 6 1 4 AND THE SECOND PART OF THEOREM 6 1 5 THE CASE P = 1 AND THE MEAN CURVATURE EQUATION LOCAL BOUNDEDNESS AND THE WEAK HARNACK INEQUALITY THE HARNACK INEQUALITY H¨OLDER CONTINUITY THE CASE P ?N APPENDIX THE JOHN–NIRENBERG THEOREM CAUCHY–LIOUVILLE THEOREMS RADIAL SYMMETRY SYMMETRY FOR OVERDETERMINED BOUNDARY VALUE PROBLEMS THE PHENOMENON OF DEAD CORES THE STRONG MAXIMUM PRINCIPLE FOR RIEMANNIAN MANIFOLDS