The stickiness property of nonlocal Limits Boundary behavior - PowerPoint PPT Presentation

The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction The stickiness property of nonlocal Limits Boundary behavior minimal surfaces of nonlocal minimal surfaces Enrico Valdinoci University of Western Australia 2020 Fields Medal Symposium 1 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 2 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 3 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 4 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 5 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 6 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 7 / 91

The stickiness Outline property of nonlocal minimal surfaces Nonlocal minimal surfaces Enrico Valdinoci Energy functional dealing with “pointwise interactions” Introduction between a given set and its complement Limits Boundary behavior Main idea: the “surface tension” is the byproduct of long-range of nonlocal minimal surfaces interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence on the local structures of these new objects STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary” 8 / 91

The stickiness The fractional perimeter functional property of nonlocal minimal surfaces Given s ∈ ( 0 , 1 ) and a bounded open set Ω ⊂ R n with Enrico Valdinoci C 1 ,γ -boundary, the s -perimeter of a (measurable) set E ⊆ R n in Introduction Ω is defined as Limits Boundary behavior Per s ( E ; Ω) := L ( E ∩ Ω , ( C E ) ∩ Ω) of nonlocal minimal surfaces + L ( E ∩ Ω , ( C E ) ∩ ( C Ω)) + L ( E ∩ ( C Ω) , ( C E ) ∩ Ω) , where C E = R n \ E denotes the complement of E , and L ( A , B ) denotes the following nonlocal interaction term � � 1 ∀ A , B ⊆ R n , L ( A , B ) := | x − y | n + s dx dy A B This notion of s -perimeter and the corresponding minimization problem were introduced in [Caffarelli-Roquejoffre-Savin, 2010]. 9 / 91

The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior of nonlocal minimal surfaces Side 1. Perimeter 4. √ Approximate Perimeter 4 2. √ Error 4 ( 2 − 1 ) . 10 / 91

The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior of nonlocal minimal surfaces Error in each pixel O ( ǫ 2 − s ) . Number of pixels O ( ǫ − 1 ) Error O ( ǫ 1 − s ) . 11 / 91

The stickiness [Caffarelli-Roquejoffre-Savin, 2010] property of nonlocal minimal surfaces Enrico Valdinoci 1) Existence theorem: there exists E s -minimizer for Per s in Ω with Introduction E \ Ω = E 0 \ Ω . Limits Boundary behavior 2) Maximum principle: of nonlocal minimal E s -minimizer and ( ∂ E ) \ Ω ⊂ {| x n | � a } ⇒ surfaces ∂ E ⊂ {| x n | � a } . 3) If ∂ E is an hyperplane, then E is s -minimizer. 4) If E is s -minimizer in B 1 , then ∂ E is C 1 ,α in B 1 / 2 except in a closed set Σ , with Hausdorff dimension less or equal than n − 2. 5) If E is s -minimizer and 0 ∈ ∂ E , then χ E ( y ) − χ E c ( y ) � dy = 0 . | y | n + s R n 12 / 91

The stickiness [Caffarelli-Roquejoffre-Savin, 2010] property of nonlocal minimal surfaces Enrico Valdinoci 1) Existence theorem: there exists E s -minimizer for Per s in Ω with Introduction E \ Ω = E 0 \ Ω . Limits Boundary behavior 2) Maximum principle: of nonlocal minimal E s -minimizer and ( ∂ E ) \ Ω ⊂ {| x n | � a } ⇒ surfaces ∂ E ⊂ {| x n | � a } . 3) If ∂ E is an hyperplane, then E is s -minimizer. 4) If E is s -minimizer in B 1 , then ∂ E is C 1 ,α in B 1 / 2 except in a closed set Σ , with Hausdorff dimension less or equal than n − 2. 5) If E is s -minimizer and 0 ∈ ∂ E , then χ E ( y ) − χ E c ( y ) � dy = 0 . | y | n + s R n 13 / 91

The stickiness [Caffarelli-Roquejoffre-Savin, 2010] property of nonlocal minimal surfaces Enrico Valdinoci 1) Existence theorem: there exists E s -minimizer for Per s in Ω with Introduction E \ Ω = E 0 \ Ω . Limits Boundary behavior 2) Maximum principle: of nonlocal minimal E s -minimizer and ( ∂ E ) \ Ω ⊂ {| x n | � a } ⇒ surfaces ∂ E ⊂ {| x n | � a } . 3) If ∂ E is an hyperplane, then E is s -minimizer. 4) If E is s -minimizer in B 1 , then ∂ E is C 1 ,α in B 1 / 2 except in a closed set Σ , with Hausdorff dimension less or equal than n − 2. 5) If E is s -minimizer and 0 ∈ ∂ E , then χ E ( y ) − χ E c ( y ) � dy = 0 . | y | n + s R n 14 / 91

The stickiness [Caffarelli-Roquejoffre-Savin, 2010] property of nonlocal minimal surfaces Enrico Valdinoci 1) Existence theorem: there exists E s -minimizer for Per s in Ω with Introduction E \ Ω = E 0 \ Ω . Limits Boundary behavior 2) Maximum principle: of nonlocal minimal E s -minimizer and ( ∂ E ) \ Ω ⊂ {| x n | � a } ⇒ surfaces ∂ E ⊂ {| x n | � a } . 3) If ∂ E is an hyperplane, then E is s -minimizer. 4) If E is s -minimizer in B 1 , then ∂ E is C 1 ,α in B 1 / 2 except in a closed set Σ , with Hausdorff dimension less or equal than n − 2. 5) If E is s -minimizer and 0 ∈ ∂ E , then χ E ( y ) − χ E c ( y ) � dy = 0 . | y | n + s R n 15 / 91