Regularity of the free boundary for the two phase Bernoulli problem - PowerPoint PPT Presentation

Regularity of the free boundary for the two phase Bernoulli problem G. De Philippis (j/w L. Spolaor, B. Velichkov)

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The Bernoulli Free Boundary Problem Let λ 0 , λ + , λ − ≥ 0 be given and for D ⊂ R d let us consider � |∇ u | 2 + λ + |{ u > 0 }| + λ − |{ u < 0 }| + λ 0 |{ u = 0 }| . J ( u , D ) = D and the minimization problem (TPBP) u | ∂ D = g J ( u , D ). min where g is a given function. G. De Philippis (CIMS): Two phase Bernoulli problem

The Bernoulli Free Boundary Problem: some remarks A few simple properties. - Minimizers are easily seen to exist. - Uniqueness in general fails. - A minimizers would like to be harmonic where it is � = 0, but the functional might penalize to be always non zero and/or might impose a “balance” between the negative and positive phase G. De Philippis (CIMS): Two phase Bernoulli problem

The Bernoulli Free Boundary Problem: some remarks When λ 0 , λ − = 0 and g ≥ 0, the problem reduces to the one phase free boundary problem : � min J ( u , D ) u = g , u ≥ 0 � (OPBP) |∇ u | 2 + λ + |{ u > 0 }| � J ( u , D ) := D G. De Philippis (CIMS): Two phase Bernoulli problem

Motivations These problems have been introduced in the 80’s by Alt-Caffarelli (OPBP) and by Alt-Caffarelli-Friedmann (TPBP) motivated by some problems in flows with jets and cavities. Since then they have been the model problems for a huge class of free boundary problems. More recently these types of problems turned out to have applications in the study of shape optimization problems. G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems Let us consider the following minimization problem: U ⊂ D Cap( U , D ) − λ | U | min where �� � |∇ u | 2 u ∈ W 1,2 Cap( U , D ) = min ( D ), u = 1 on U 0 D is the Newtonian capacity of U relative to D . The problem is equivalent to � |∇ v | 2 − λ |{ v = 1 }| min v ∈ W 1,2 ( D ) D 0 � |∇ v | 2 + λ |{ 0 < v < 1 }| − λ | D | . = min v ∈ W 1,2 ( D ) D 0 u = 1 − v solves a one phase problem. G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems Let us consider the following minimal partition problem : �� � min λ ( D i ) + m i | D i | D i ⊂ D , D i ∩ D j = ∅ if i � = j . i Here λ ( D i ) is the first eigenvalue of the Dirichlet Laplacian on D i , i.e. �� � D i |∇ u | 2 : u ∈ W 1,2 � λ ( D i ) = inf ( D i ) . 0 D i u 2 G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems How minimizers look like? One can show (Spolaor-Trey-Velichkov): - There are no triple points ∂ D i ∩ ∂ D j ∩ ∂ D k = ∅ . - If u i , u j are the first (positive) eigenfunctions of D i , D j then v = u i − u j is a (local) minimizer of � |∇ v | 2 + m i |{ v > 0 }| + m j |{ v < 0 }| + H.O.T. G. De Philippis (CIMS): Two phase Bernoulli problem

Back to the Bernoulli free boundary problem We are interested in the regularity of u and of the free boundary: Γ = Γ + ∪ Γ − Γ + = ∂ { u > 0 } Γ − = ∂ { u < 0 } . u = 0 Γ + u > 0 u < 0 Γ − u = 0 G. De Philippis (CIMS): Two phase Bernoulli problem

Known results - u is Lipschitz, Alt-Caffarelli (one phase), Alt-Caffarelli-Friedmann (two-phase). - If u is a solution of the one-phase problem, then Γ + is smooth outside a (relatively) closed set Σ + with dim H ≤ d − 5 (Alt-Caffarelli, Weiss, Jerison-Savin, a recent new proof from De Silva). - There is a minimizer in dimension d = 7 with a point singularity (De Silva-Jerison). - If u is a solution of the two phase problem and λ 0 ≥ min { λ + , λ − } , then Γ + = Γ − = Γ is smooth. (Alt-Caffarelli-Friedmann, Caffarelli, De Silva-Ferrari-Salsa). G. De Philippis (CIMS): Two phase Bernoulli problem

The case λ 0 ≥ min { λ + , λ − } If λ − ≤ λ 0 , let v be the harmonic function which is equal to u − on ∂ ( D \ { u > 0 } ). Then w = u + − v satisfies J ( w , D ) ≤ J ( u , D ). since λ − |{ w < 0 }| ≤ λ − |{ u < 0 }| + λ 0 |{ u = 0 }| and � � |∇ v | 2 ≤ |∇ u − | 2 u > 0 u > 0 w < 0 u < 0 G. De Philippis (CIMS): Two phase Bernoulli problem

The case λ 0 < min { λ + , λ − } When λ 0 < min { λ + , λ − } the three phases may co-exist and branch points might appear. u = 0 P u > 0 Branch points u < 0 Q u = 0 G. De Philippis (CIMS): Two phase Bernoulli problem

Main result Theorem D.-Spolaor-Velichkov ’19 (Spolaor-Velichkov’16 for d = 2) Let u be a local minimizer of J. Let us define Γ ± = ∂ {± u > 0 } Γ DP = Γ + ∩ Γ − OP = Γ ± \ Γ DP , Γ ± Then - Γ ± are C 1, α manifolds outside relatively closed set Σ ± with dim H (Σ ± ) ≤ d − 5 . - Γ DP ∩ Σ ± = ∅ . In particular Γ DP is a closed subset of a C 1, α graph. Γ + OP Γ DP u > 0 u < 0 Γ − OP u = 0 G. De Philippis (CIMS): Two phase Bernoulli problem

Steps in the proof As it is customary in Geometric Measure Theory, the above result is based on two steps: - Blow up analysis. - ε -regularity theorem. G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality conditions Before detailing the proof, let us start by deriving the optimality conditions for minimizers. The first (trivial) one, one is that u is harmonic where � = 0 (which is open) on { u � = 0 } ∆ u = 0 What are the optimality conditions on the free boundary? They can be formally obtained by performing inner variations � d � X ∈ C c ( D ; R d ) ε =0 J ( u ε ) = 0 u ε ( x ) = u ( x + ε X ( x )) � d ε G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition Let us assume that u is one dimensional: u = α x + − β x − α − 1 1 − β G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition Let us assume that u is one dimensional: α β u = α x + − β x − u ε = 1 − ε ( x − ε ) + − 1 + ε ( x − ε ) − α α − 1 − 1 ε 1 1 − β − β 0 ≤ J ( u ε ) − J ( u ) = ( α 2 − β 2 ) ε − ( λ + − λ − ) ε + o ( ε ) G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition Moreover u = α x + − β x − α − 1 1 − β G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition Moreover α u ε = 1 − ε ( x − ε ) + − β x − u = α x + − β x − α α − 1 − 1 ε 1 1 − β − β G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition Moreover α u ε = 1 − ε ( x − ε ) + − β x − u = α x + − β x − α α − 1 − 1 ε 1 1 − β − β α 2 (1 − ε ) − α 2 − ( λ + − λ 0 ) ε 0 ≤ J ( u ε ) − J ( u ) = = α 2 ε − ( λ + − λ 0 ) ε + o ( ε ) G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality conditions We get the following problem   ∆ u = 0 on { u � = 0 }     |∇ u ± | 2 = λ ± − λ 0 on Γ ± OP |∇ u + | 2 − |∇ u − | 2 = λ + − λ −  on Γ DP     |∇ u ± | 2 ≥ λ ± − λ 0 on Γ ± G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary. G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary. Let x 0 ∈ Γ and r > 0. Let u x 0 , r ( x ) = u ( x 0 + rx ) ( u ( x 0 ) = 0). r Then { u x 0 , r } r > 0 is pre-compact in C 0 and every limit point is one-homogeneous (Weiss Monotonicity Formula). G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary. Let x 0 ∈ Γ and r > 0. Let u x 0 , r ( x ) = u ( x 0 + rx ) ( u ( x 0 ) = 0). r Then { u x 0 , r } r > 0 is pre-compact in C 0 and every limit point is one-homogeneous (Weiss Monotonicity Formula). If x 0 ∈ Γ is regular it is easy to see that there is a unique limit v x 0 and  � if x 0 ∈ Γ ±  ± λ ± − λ 0 ( x · e x 0 ) ±  OP v x 0 = α + ( x · e x 0 ) + − α − ( x · e x 0 ) − if x 0 ∈ Γ DP  �  α 2 + − α 2 α ± ≥ λ ± − λ 0 , − = λ + − λ − where e x 0 is the normal to Γ at x 0 . G. De Philippis (CIMS): Two phase Bernoulli problem