On the regularity of the two-phase free boundaries u = 0 u < 0 u - PowerPoint PPT Presentation

On the regularity of the two-phase free boundaries u = 0 u < 0 u > 0 Bozhidar Velichkov Università degli Studi di Napoli Federico II V A R EG

T HE TWO - PHASE B ERNOULLI PROBLEM Given: g = 0 − a domain D ⊆ R d (we assume D = B 1 ), g > 0 g 0 0 0 < − positive constants λ + , λ − , and λ 0 , > = < u u u 0 − a boundary datum g : ∂ D → R , minimize the TWO - PHASE functional λ − λ 0 λ + � |∇ u | 2 dx + λ 2 � � � { u > 0 } ∩ D J TP ( u , D ) = + � D + λ 2 � � { u < 0 } ∩ D � g = 0 − � + λ 2 � � � { u = 0 } ∩ D � , 0 among all functions u : D → R such that u = g on ∂ D .

T HE TWO - PHASE B ERNOULLI PROBLEM g = 0 First considerations: 1. The solutions u are harmonic in the set { u � = 0 } . g > 0 g 0 0 0 < > = < 2. Is there an equation for u in the entire D = B 1 ? u u u 0 � 0 < ∆ u , ϕ > := ∇ u · ∇ ϕ 0 = = u ∆ u D ∆ � � ∂ u � ∂ u = ϕ ∆ u + ∂ n ϕ = ∂ n ϕ { u � = 0 } ∂ { u � = 0 } ∂ { u � = 0 } Then g = 0 � � � � H d − 1 H d − 1 ∂ { u > 0 } ∂ { u < 0 } in ∆ u = |∇ u + | − |∇ u − | D Solution u ⇆ Free boundary ∂ { u > 0 } ∪ ∂ { u < 0 }

R EGULARITY OF THE MINIMIZING FUNCTION u 1981 Alt-Caffarelli ( J. Reine Agnew. Math. ) − u ∈ C 0 , 1 loc ( D ) − for u ≥ 0 , 1984 Alt-Caffarelli-Friedman ( Trans. Amer. Math. Soc. ) − u ∈ C 0 , 1 loc ( D ) − for any u. The Lipschitz continuity of u is optimal. In fact, for every λ + > λ 0 = 0 and e ∈ ∂ B 1 , ∆ h = 0 the function q λ 2 + − λ 2 |r h | = 2 max { 0 , x · e } 0 � 1 / λ 2 + − λ 2 � h ( x ) = − e 0 is a local minimizer in R d . h = 0 Corollary: • Ω + u = { u > 0 } and Ω − u = { u < 0 } are open sets ; • ∆ u = 0 in Ω + ∆ u = 0 in Ω − and u . u Question: What is the regularity of the free boundary ∂ Ω + u ∪ ∂ Ω − u ∩ D ?

On the regularity of the two-phase free boundaries P ART I u = 0 Known results, u < 0 u > 0 main theorem, and applications

S TRUCTURE OF THE TWO - PHASE FREE BOUNDARIES Suppose that dist � { g > 0 } , { g < 0 } � > 0 on the sphere . ...and consider the following three cases. u = 0 u > 0 u > 0 ∆ u = 0 u > 0 u = 0 u < 0 u < 0 u < 0 ∆ u = 0 branching point u = 0 λ + > > λ 0 and λ − > > λ 0 λ + > λ 0 and λ − > λ 0 λ + ≥ λ 0 and λ − = λ 0

S TRUCTURE OF THE TWO - PHASE FREE BOUNDARIES Suppose that dist � { g > 0 } , { g < 0 } � > 0 on the sphere . ...and consider the following three cases. u = 0 u > 0 u > 0 ∆ u = 0 u > 0 u = 0 u < 0 u < 0 u < 0 ∆ u = 0 branching point u = 0 λ + > λ + > > λ 0 and λ − > > λ 0 and λ − > > λ 0 > λ 0 λ + > λ 0 and λ − > λ 0 λ + ≥ λ 0 and λ − = λ 0

R EGULARITY OF THE ONE - PHASE FREE BOUNDARIES Recall that dist � { g > 0 } , { g < 0 } � > 0 on the sphere . and > λ 0 . Suppose that λ + > > λ 0 λ − > g = 0 g = 0 In the small ball B ′ B 0 � |∇ u | 2 dx J TP ( u , B 1 ) = the positive part u + minimizes g < 0 g < 0 B 1 u > 0 u > 0 u = 0 u = 0 + λ 2 the one-phase functional � � { u > 0 } ∩ B 1 � + � u < 0 u < 0 + λ 2 � � � { u < 0 } ∩ B 1 � |∇ u | 2 dx g > 0 g > 0 − � J OP ( u , B 1 ) = + λ 2 � { u = 0 } ∩ B 1 � � B 1 0 � λ 2 + − λ 2 � � � � { u > 0 }∩ B 1 � + 0 � g = 0 g = 0

R EGULARITY OF THE ONE - PHASE FREE BOUNDARIES the graph of u over B 0 Definition: We say that u is the graph of u over ∂ B 0 a local minimizer of J OP in B ′ if J OP ( u , B ′ ) ≤ J OP ( v , B ′ ) u > 0 u = 0 for every v : B ′ → R such that u = v on ∂ B ′ . ∂ B 0 ∂ Ω u \ B 0 Theorem (Alt-Caffarelli’81, Weiss’00) . There is d ∗ ∈ { 5 , 6 , 7 } such that: If u is a (nonnegative) local minimizer of J OP in B ′ ⊆ R d , u ∩ B ′ = Reg ( ∂ Ω + ∂ Ω + u ) ∪ Sing ( ∂ Ω + then the free boundary decomposes as: u ) u ) is a C 1 ,α -regular manifold; • Reg ( ∂ Ω + u ) is empty ( d < d ∗ ), discrete ( d = d ∗ ), of dimension d − d ∗ ( d > d ∗ ). • Sing ( ∂ Ω +

S TRUCTURE OF THE TWO - PHASE FREE BOUNDARIES Suppose that dist � { g > 0 } , { g < 0 } � > 0 on the sphere . ...and consider the following three cases. u = 0 u > 0 u > 0 ∆ u = 0 u > 0 u = 0 u < 0 u < 0 u < 0 ∆ u = 0 branching point u = 0 λ + > > λ 0 and λ − > > λ 0 λ + > λ 0 and λ − > λ 0 λ + ≥ λ 0 and λ − = λ 0

R EGULARITY OF THE TWO - PHASE FREE BOUNDARIES − THE EXTREMAL CASE | r u + | 2 − | r u − | 2 = λ 2 + − λ 2 − Theorem (Alt-Caffarelli-Friedman’84) . Let d = 2 , and λ − = λ 0 . λ + ≥ λ 0 u > 0 ∆ u = 0 If u minimizes J TP in B 1 , then: ∆ u = 0 u < 0 • ∂ Ω + u = ∂ Ω − u ; u is C 1 ,α -regular curve ; • ∂ Ω + • |∇ u + | 2 −|∇ u − | 2 = λ 2 + − λ 2 ∂ Ω + on u ∩ B 1 . − Regularity of free boundaries satisfying a transmission condition: 1987 - 1989 Caffarelli ( Comm. Pure Appl. Math.,... ) − Harnack inequality approach; 2005 Caffarelli-Salsa − A geometric approach to free boundary problems (book); 2014-2018 De Silva-Ferrari-Salsa − Partial Boundary Harnack approach ( 2010 De Silva − Partial Boundary Harnack for the one-phase pb).

S TRUCTURE OF THE TWO - PHASE FREE BOUNDARIES Suppose that dist � { g > 0 } , { g < 0 } � > 0 on the sphere . ...and consider the following three cases. u = 0 u > 0 u > 0 ∆ u = 0 u > 0 u = 0 u < 0 u < 0 u < 0 ∆ u = 0 branching point u = 0 λ + > > λ 0 and λ − > > λ 0 λ + > λ 0 and λ − > λ 0 λ + ≥ λ 0 and λ − = λ 0

R EGULARITY OF THE TWO - PHASE FREE BOUNDARIES − THE MAIN RESULT u = 0 Assume λ + > λ 0 and λ − > λ 0 . W.l.o.g. λ 0 = 0. u > 0 Decomposition of the free boundary: u < 0 • one-phase points: branching point Γ + Γ − OP = ∂ Ω + u \ ∂ Ω − and OP = ∂ Ω − u \ ∂ Ω + u u u = 0 • two-phase points: Γ TP = ∂ Ω + u ∩ ∂ Ω − u Theorem. Let d ≥ 2. Let u be a minimizer of J TP with λ 0 = 0, λ + > 0 and λ − > 0. Then, in a neighborhood of any x 0 ∈ ∂ Ω + u ∩ ∂ Ω − u ∩ D , u are C 1 ,α -regular manifolds. the free boundaries ∂ Ω + u and ∂ Ω − 2017 Spolaor-Velichkov ( Comm. Pure Appl. Math. ) − the case d = 2 ; 2018 Spolaor-Trey-Velichkov ( Comm. PDE ) − almost-minimizers in R 2 ; 2019 De Philippis-Spolaor-Velichkov ( to appear ) − any d ≥ 2.

R EGULARITY OF THE TWO - PHASE FREE BOUNDARIES - THE COMPLETE RESULT Corollary (Alt-Caffarelli; Weiss; Spolaor-Trey-V.; De Philippis-Spolaor-V.) . Let d ≥ 2 and D ⊆ R d be an open set. Let u be a local minimizer of J TP in D with λ 0 = 0, λ + > 0 and λ − > 0. Then, for each of the sets Ω + u and Ω − u , the free boundary ∂ Ω ± u ∩ D can be decomposed as ∂ Ω ± u ∩ D = Reg ( ∂ Ω ± u ) ∪ Sing ( ∂ Ω ± u ) , where: u ) is a C 1 ,α manifold; • the regular part Reg ( ∂ Ω ± • Sing ( ∂ Ω ± u ) is a (possibly empty) closed set of one-phase singularities , and − Sing ( ∂ Ω ± u ) is empty, if d < d ∗ ; − Sing ( ∂ Ω ± u ) is discrete, if d = d ∗ ; − Sing ( ∂ Ω ± u ) has Hausdorff dimension d − d ∗ , if d > d ∗ ; − Sing ( ∂ Ω ± u ) ∩ ∂ Ω + u ∩ ∂ Ω − u = ∅ .

A PPLICATION : A MULTIPHASE SHAPE OPTIMIZATION PROBLEM n � � � Minimize among λ 1 (Ω k ) + m k | Ω k | k = 1 all n -uples of sets (Ω 1 , . . . , Ω n ) such that: • Ω k ⊆ D , where D is a C 1 ,α -regular box; • Ω k ∩ Ω j = ∅ , whenever k � = j . * Numerical simulations by Beniamin Bogosel ( http://www.cmap.polytechnique.fr/ beniamin.bogosel/ ) Theorem. Let d = 2. Let (Ω 1 ,..., Ω n ) be a solution of the multiphase problem in a C 1 ,α -regular box D . Then each of the sets Ω k is C 1 ,α -regular. 2015 Bogosel-Velichkov ( SIAM Numer. Anal. ) 2019 Spolaor-Trey-Velichkov ( Comm. PDE )

A MULTIPHASE SHAPE OPTIMIZATION PROBLEM - REGULARITY IN R d Theorem (regularity in higher dimension - part I). Let d ≥ 2. Let (Ω 1 ,..., Ω n ) be a solution of the multiphase problem in the C 1 ,α -regular box D ⊆ R d . Then: • each of the sets Ω k is open and has finite perimeter 2014 Bucur-Velichkov ( SIAM Contr. Optim. ) ; • there are no triple points: ∂ Ω i ∩ ∂ Ω j ∩ ∂ Ω k = ∅ 2014 Bucur-Velichkov, 2014 Velichkov − three-phase monotonicity formula, 2015 Bogosel-Velichkov ( SIAM Numer. Anal. ) - new proof in 2D; • there are no two-phase points on the boundary of the box: ∂ Ω i ∩ ∂ Ω j ∩ ∂ D = ∅ 2015 Bogosel-Velichkov;