Regularity for almost minimizers with free boundary Tatiana Toro - PowerPoint PPT Presentation

Regularity for almost minimizers with free boundary Tatiana Toro University of Washington Harmonic Analysis & Partial Differential Equations September 19, 2014 Joint work with G. David Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 1 / 22

Minimizers with free boundary Let Ω ⊂ R n be a bounded connected Lipschitz domain, q ± ∈ L ∞ (Ω) and u ∈ L 1 loc (Ω) ; ∇ u ∈ L 2 (Ω) � � K (Ω) = . Minimizing problem with free boundary: Given u 0 ∈ K (Ω) minimize ˆ |∇ u ( x ) | 2 + q 2 + ( x ) χ { u > 0 } ( x ) + q 2 J ( u ) = − ( x ) χ { u < 0 } ( x ) Ω among all u = u 0 on ∂ Ω. One phase problem arises when q − ≡ 0 and u 0 ≥ 0. The general problem is know as the two phase problem. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 2 / 22

Alt-Caffarelli Minimizers for the one phase problem exist. If u is a minimizer of the one phase problem, then u ≥ 0, u is subharmonic in Ω and ∆ u = 0 in { u > 0 } u is locally Lipschitz in Ω. If q + is bounded below away from 0, that is there exists c 0 > 0, such that q + ≥ c 0 , then: ◮ for x ∈ { u > 0 } u ( x ) δ ( x ) ∼ 1 where δ ( x ) = dist( x , ∂ { u > 0 } ) ◮ { u > 0 } ∩ Ω is a set of locally finite perimeter, thus ∂ { u > 0 } ∩ Ω is (n-1)-rectifiable. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 3 / 22

Alt-Caffarelli-Friedman Minimizers for the two phase problem exist. If u is a minimizer of the two phase problem, then u ± are subharmonic and ∆ u = 0 in { u > 0 } ∪ { u < 0 } u is locally Lipschitz in Ω. If q ± are bounded below away from 0, then ◮ for x ∈ { u ± > 0 } u ± ( x ) δ ( x ) ∼ 1 where δ ( x ) = dist( x , ∂ { u ± > 0 } ) ◮ { u ± > 0 } ∩ Ω are sets of locally finite perimeter. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 4 / 22

Regularity of the free boundary Γ( u ) If u is a minimizer for the one phase problem Γ( u ) = ∂ { u > 0 } If q + is H¨ older continuous and q + ≥ c 0 > 0 then ◮ if n = 2 , 3, Γ( u ) is a C 1 ,β (n-1)-dimensional submanifold. ◮ if n ≥ 4, Γ( u ) = R ( u ) ∪ S ( u ) where R ( u ) is a C 1 ,β (n-1)-dimensional submanifold and S ( u ) is a closed set of Hausdorff dimension less than n-3. If u is a minimizer for the two phase problem Γ( u ) = ∂ { u > 0 } ∪ ∂ { u < 0 } If q ± are H¨ older continuous q + > q − ≥ 0 and q + ≥ c 0 > 0 then ◮ if n = 2 , 3, Γ( u ) is a C 1 ,β (n-1)-dimensional submanifold. ◮ if n ≥ 4, Γ( u ) = R ( u ) ∪ S ( u ) where R ( u ) is a C 1 ,β (n-1)-dimensional submanifold and S ( u ) is a closed set of Hausdorff dimension less than n-3. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 5 / 22

Contributions One phase problem: ◮ n=2, Alt-Caffarelli ◮ n ≥ 3, Alt-Caffarelli, Caffarelli-Jerison-Kenig / Weiss Two phase problem: ◮ n=2 , Alt-Caffarelli-Friedman ◮ n ≥ 3, Alt-Caffarelli-Friedman, Caffarelli-Jerison-Kenig / Weiss DeSilva-Jerison: There exists a non-smooth minimizer for J in R 7 such that Γ( u ) is a cone. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 6 / 22

Almost minimizers for the one phase problem Let Ω ⊂ R n be a bounded connected Lipschitz domain, q + ∈ L ∞ (Ω) and u ∈ L 1 loc (Ω) ; u ≥ 0 a . e . in Ω and ∇ u ∈ L 2 � � K + (Ω) = loc (Ω) u ∈ K + (Ω) is a ( κ, α )-almost minimizers for J + in Ω if for any ball B ( x , r ) ⊂ Ω J + x , r ( u ) ≤ (1 + κ r α ) J + x , r ( v ) for all v ∈ K + (Ω) with u = v on ∂ B ( x , r ), where ˆ |∇ v | 2 + q 2 J + x , r ( v ) = + χ { v > 0 } . B ( x , r ) Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 7 / 22

Almost minimizers for the two phase problem Let Ω ⊂ R n be a bounded connected Lipschitz domain, q ± ∈ L ∞ (Ω) and u ∈ L 1 loc (Ω) ; ∇ u ∈ L 2 � � K (Ω) = loc (Ω) . u ∈ K (Ω) is a ( κ, α )-almost minimizers for J in Ω if for any ball B ( x , r ) ⊂ Ω J x , r ( u ) ≤ (1 + κ r α ) J x , r ( v ) for all v ∈ K (Ω) with u = v on ∂ B ( x , r ), where ˆ |∇ v | 2 + q 2 + χ { v > 0 } + q 2 J x , r ( v ) = − χ { v > 0 } . B ( x , r ) Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 8 / 22

Almost minimizers are continuous Theorem: Almost minimizers of J are continuous in Ω. Moreover if u is an almost minimizer for J there exists a constant C > 0 such that if B ( x 0 , 2 r 0 ) ⊂ Ω then for x , y ∈ B ( x 0 , r 0 ) 2 r 0 � � | u ( x ) − u ( y ) | ≤ C | x − y | 1 + log . | x − y | Remark: Since almost-minimizers do not satisfy an equation, good comparison functions are needed. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 9 / 22

Sketch of the proof To prove regularity of u , an almost minimizer for J , we need to control the quantity � 1 / 2 � |∇ u | 2 ω ( x , s ) = B ( x , s ) for s ∈ (0 , r ) and B ( x , r ) ⊂ Ω. Consider u ∗ r satisfying ∆ u ∗ r = 0 in B ( x , r ) and u ∗ r = u on ∂ B ( x , r ). Then r | 2 is subharmonic since |∇ u ∗ � 1 / 2 � 1 / 2 � � |∇ u − ∇ u ∗ r | 2 |∇ u ∗ r | 2 ω ( x , s ) ≤ + B ( x , s ) B ( x , s ) � 1 / 2 � 1 / 2 � � � n / 2 � r |∇ u − ∇ u ∗ r | 2 |∇ u ∗ r | 2 ≤ + s B ( x , r ) B ( x , r ) Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 10 / 22

The almost minimizing property comes in Since ∆ u ∗ r = 0 in B ( x , r ) and u ∗ r = u on ∂ B ( x , r ) and q ± ∈ L ∞ (Ω) ˆ ˆ ˆ |∇ u | 2 − |∇ u − ∇ u ∗ r | 2 |∇ u ∗ r | 2 = B ( x , r ) B ( x , r ) B ( x , r ) ˆ ˆ r | 2 − r | 2 + Cr n (1 + κ r α ) |∇ u ∗ |∇ u ∗ ≤ B ( x , r ) B ( x , r ) ˆ r | 2 + Cr n κ r α |∇ u ∗ ≤ B ( x , r ) ˆ |∇ u | 2 + Cr n . κ r α ≤ B ( x , r ) Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 11 / 22

Iteration scheme � � n / 2 � � n / 2 � r � r r α/ 2 ω ( x , s ) ≤ 1 + C ω ( x , r ) + C . s s Set r j = 2 − j r for j ≥ 0, iteration yields ω ( x , r j ) ≤ C ω ( x , r ) + Cj , which for s ∈ (0 , r ) ensures ω ( x , r ) + log r � � ω ( x , s ) ≤ C . s Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 12 / 22

Local regularity on each phase Theorem: Let u be an almost minimizer for J in Ω. Then u is locally Lipschitz in { u > 0 } and in { u < 0 } . Theorem: Let u be an almost minimizer for J in Ω. Then there exists β ∈ (0 , 1) such that u is C 1 ,β locally in { u > 0 } and { u < 0 } . Proof: Refine the argument above. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 13 / 22

Local regularity for minimizers Theorem [AC], [ACF]: Let u be a minimizer for J in Ω. Then u is locally Lipschitz. Elements of the proof: u ± are subharmonic in Ω, u harmonic on { u ± > 0 } , the 2-phase case requires a monotonicity formula introduced by Alt-Caffarelli-Friedman [ACF], that is � ˆ � � ˆ � |∇ u + | 2 |∇ u − | 2 Φ( r ) = 1 | x − y | n − 2 dy | x − y | n − 2 dy r 4 B ( x , r ) B ( x , r ) is an increasing function of r > 0. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 14 / 22

Local regularity for almost minimizers Theorem: Let u be an almost minimizer for J in Ω. Then u is locally Lipschitz. Elements of the proof: analysis of the interplay between � 1 / 2 � m ( x , r ) = 1 u , 1 |∇ u | 2 | u | and ω ( x , r ) = r r ∂ B ( x , r ) ∂ B ( x , r ) B ( x , s ) the 2-phase case requires an almost [ACF]-monotonicity formula, i.e. we need to control the oscillation of Φ( r ) on small intervals. Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 15 / 22

Sketch of the proof For 1 ≪ K and 0 < γ ≪ 1 if B ( x , 2 r ) ⊂ Ω consider: Case 1: � ω ( x , r ) ≥ K | m ( x , r ) | ≥ γ (1 + ω ( x , r )) Case 2: � ω ( x , r ) ≥ K | m ( x , r ) | < γ (1 + ω ( x , r )) Case 3: ω ( x , r ) ≤ K Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 16 / 22

Case 1 If u is an almost minimizer for J in Ω, B ( x , 2 r ) ⊂ Ω and � ω ( x , r ) ≥ K | m ( x , r ) | ≥ γ (1 + ω ( x , r )) then there exists θ ∈ (0 , 1) such that u ∈ C 1 ,β ( B ( x , θ r )) and sup |∇ u | � ω ( x , r ) . B ( x ,θ r ) Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 17 / 22

Cases 2 & 3 If u is an almost minimizer for J + in Ω, B ( x , 2 r ) ⊂ Ω and � ω ( x , r ) ≥ K m ( x , r ) γ (1 + ω ( x , r )) < then for θ ∈ (0 , 1) there exists β ∈ (0 , 1) such that ω ( x , θ r ) ≤ βω ( x , r ) . If only cases 2 and 3 occur then lim sup ω ( x , s ) � K s → 0 and if x is a Lebesgue point of ∇ u then |∇ u ( x ) | � K . Tatiana Toro (University of Washington) Almost minimizers with free boundary September 19, 2014 18 / 22