Regularity of the One-phase Free Boundaries

Description

This open access book is an introduction to the regularity theory for free boundary problems. The focus is on the one-phase Bernoulli problem, which is of particular interest as it deeply influenced the development of the modern free boundary regularity theory and is still an object of intensive research.  The exposition is organized around four main theorems, which are dedicated to the one-phase functional in its simplest form. Many of the methods and the techniques presented here are very recent and were developed in the context of different free boundary problems. We also give the detailed proofs of several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories. This book is aimed at graduate students and researches and is accessible to anyone with a moderate level of knowledge of elliptical PDEs. Preface Acknowledgment Contents 1 Introduction and Main Results 1.1 Free Boundary Problems: Classical and Variational Formulations 1.2 Regularity of the Free Boundary 1.3 The Regularity Theorem of Alt and Caffarelli 1.4 The Dimension of the Singular Set 1.5 Regularity of the Free Boundary for Measure Constrained Minimizers 1.6 An Epiperimetric Inequality Approach to the Regularity of the Free Boundary in Dimension Two 1.7 Further Results 2 Existence of Solutions, Qualitative Properties and Examples 2.1 Properties of the Functional F 2.2 Proof of Proposition 2.1 2.3 Half-Plane Solutions 2.4 Radial Solutions 3 Lipschitz Continuity of the Minimizers 3.1 The Alt-Caffarelli's Proof of the Lipschitz Continuity 3.2 The Laplacian Estimate 3.3 The Danielli-Petrosyan Approach 4 Non-degeneracy of the Local Minimizers 5 Measure and Dimension of the Free Boundary 5.1 Density Estimates for the Domain Ωu 5.2 The Positivity set Ωu Has Finite Perimeter 5.3 Hausdorff Measure of the Free Boundary 6 Blow-Up Sequences and Blow-Up Limits 6.1 Convergence of Local Minimizers 6.2 Convergence of the Free Boundary 6.3 Proof of Proposition 6.2 6.4 Regular and Singular Parts of the Free Boundary 7 Improvement of Flatness 7.1 The Optimality Condition on the Free Boundary 7.2 Partial Harnack Inequality 7.2.1 Interior Harnack Inequality 7.2.2 Partial Harnack Inequality at the Free Boundary 7.3 Convergence of Flat Solutions 7.4 Improvement of Flatness: Proof of Theorem 7.4 8 Regularity of the Flat Free Boundaries 8.1 Improvement of Flatness, Uniqueness of the Blow-Up Limit and Rate of Convergence of the Blow-Up Sequence 8.2 Regularity of the One-Phase Free Boundaries 9 The Weiss Monotonicity Formula and Its Consequences 9.1 The Weiss Boundary Adjusted Energy 9.2 Stationary Free Boundaries 9.3 Homogeneity of the Blow-Up Limits 9.4 Regularity of the Free Boundaries in Dimension Two 9.5 The Optimality Condition on the Free Boundary: A Monotonicity Formula Approach 9.6 Energy and Lebesgue Densities 10 Dimension of the Singular Set 10.1 Hausdorff Measure and Hausdorff Dimension 10.2 Convergence of the Singular Sets 10.3 Dimension Reduction 10.4 Proof of Theorem 1.4 11 Regularity of the Free Boundary for Measure Constrained Minimizers 11.1 Existence of Minimizers 11.2 Euler-Lagrange Equation 11.3 Strict Positivity of the Lagrange Multiplier 11.4 Convergence of the Lagrange Multipliers 11.5 Almost Optimality of u at Small Scales 12 An Epiperimetric Inequality Approach to the Regularity of the One-Phase Free Boundaries 12.1 Preliminary Results 12.2 Homogeneity Improvement of the Higher Modes: Proof of Lemma 12.6 12.3 Epiperimetric Inequality for the Principal Modes: Proof of Lemma 12.7 12.3.1 Reduction to the Case c1=1 12.3.2 An Estimate on the Energy Gain 12.3.3 Computation of f 12.3.4 Conclusion of the Proof of Lemma 12.7 12.4 Proof of Theorem 12.1 12.5 Epiperimetric Inequality and Regularity of the Free Boundary 12.6 Comparison with Half-Plane Solutions A The Epiperimetric Inequality in Dimension Two A.1 Proof of Theorem 12.3 A.2 Proof of Lemma A.2 A.3 Epiperimetric Inequality for Large Cones: Proof of Lemma A.3 B Notations and Definitions References Index