Preface......Page 6
Acknowledgments......Page 8
Contents......Page 10
Introduction......Page 12
1 A Probabilistic Motivation......Page 14
1.1 The Random Walk with Arbitrarily Long Jumps......Page 15
1.2 A Payoff Model......Page 17
2.1 Preliminary Notions......Page 19
2.2 Fractional Sobolev Inequality and Generalized Coarea Formula......Page 28
2.3 Maximum Principle and Harnack Inequality......Page 31
2.4 An s-Harmonic Function......Page 36
2.5 All Functions Are Locally s-Harmonic Up to a Small Error......Page 41
2.6 A Function with Constant Fractional Laplacian on the Ball......Page 45
3 Extension Problems......Page 50
3.1 Water Wave Model......Page 51
3.1.1 Application to the Water Waves......Page 53
3.2 Crystal Dislocation......Page 54
3.3 An Approach to the Extension Problem via the Fourier Transform......Page 67
4 Nonlocal Phase Transitions......Page 77
4.1 The Fractional Allen-Cahn Equation......Page 80
4.2 A Nonlocal Version of a Conjecture by De Giorgi......Page 95
5 Nonlocal Minimal Surfaces......Page 106
5.1 Graphs and s-Minimal Surfaces......Page 110
5.2 Non-existence of Singular Cones in Dimension 2......Page 120
5.3 Boundary Regularity......Page 128
6 A Nonlocal Nonlinear Stationary SchrΓΆdinger Type Equation......Page 136
6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality......Page 145
A.1 Another Proof of Theorem 2.4.1......Page 148
A.2 Another Proof of Lemma 2.3......Page 152
References......Page 158