Monotonicity formulas and the singular set in the thin obstacle - PowerPoint PPT Presentation

Normalization: class S Assume � is flat: � = � n −  × {  } , φ =  If u solves Signorini problem, afer translation, rotation, and scaling, we may normalize u as follows: Definition We say u is a normalized solution of Signorini problem iff in B + ∆ u =   − ∂ x n u ≥ , on B ′ u ≥ , u ∂ x n u =    ∈ Γ ( u ) = ∂ Λ ( u ) = ∂ { u =  } . We denote the class of normalized solutions by S . + = � n −  ×( , +∞) ,  ∶ = B  ∩( � n −  ×{  })  ∶ = B  ∩ � n Notation : � n B + B ′ + , Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 8 / 1

Normalization: class S Every u ∈ S can be extended from B +  to B  by even symmetry u ( x ′ , − x n ) ∶ = u ( x ′ , x n ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 9 / 1

Normalization: class S Every u ∈ S can be extended from B +  to B  by even symmetry u ( x ′ , − x n ) ∶ = u ( x ′ , x n ) . Te resulting function will satisfy ∆ u ≤  in B  in B  ∖ Λ ( u ) ∆ u =  u ∆ u =  in B  . Here Λ ( u ) = { u =  } ⊂ B ′  . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 9 / 1

Normalization: class S Every u ∈ S can be extended from B +  to B  by even symmetry u ( x ′ , − x n ) ∶ = u ( x ′ , x n ) . Te resulting function will satisfy ∆ u ≤  in B  in B  ∖ Λ ( u ) ∆ u =  u ∆ u =  in B  . Here Λ ( u ) = { u =  } ⊂ B ′  . More specifically: ∆ u =  ( ∂ x n u ) � n −  ∣ Λ ( u ) in � ′ ( B  ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 9 / 1

Rescalings and blowups For u ∈ S and r >  consider rescalings u ( rx ) u r ( x ) ∶ = . ( r n −  ∫ ∂ B r u  )    Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 10 / 1

Rescalings and blowups For u ∈ S and r >  consider rescalings u ( rx ) u r ( x ) ∶ = . ( r n −  ∫ ∂ B r u  )    Te rescaling is normalized so that ∥ u r ∥ L  ( ∂ B  ) = . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 10 / 1

Rescalings and blowups For u ∈ S and r >  consider rescalings u ( rx ) u r ( x ) ∶ = . ( r n −  ∫ ∂ B r u  )    Te rescaling is normalized so that ∥ u r ∥ L  ( ∂ B  ) = . Limits of subsequences { u r j } for some r j →  + are known as blowups . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 10 / 1

Rescalings and blowups For u ∈ S and r >  consider rescalings u ( rx ) u r ( x ) ∶ = . ( r n −  ∫ ∂ B r u  )    Te rescaling is normalized so that ∥ u r ∥ L  ( ∂ B  ) = . Limits of subsequences { u r j } for some r j →  + are known as blowups . Generally the blowups may be different over different subsequences r = r j →  + . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 10 / 1

Almgren’s frequency function Teorem (Monotonicity of the frequency) Let u ∈ S . T en the frequency function r ∫ B r ∣ ∇ u ∣  r ↦ N ( r , u ) ∶ =  < r < . ↗ for ∫ ∂ B r u  Moreover, N ( r , u ) ≡ κ ⇐ ⇒ x ⋅ ∇ u − κ u =  in B  , i.e. u is homogeneous of degree κ in B  . [ Almgr en 1979] for harmonic u Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 11 / 1

Almgren’s frequency function Teorem (Monotonicity of the frequency) Let u ∈ S . T en the frequency function r ∫ B r ∣ ∇ u ∣  r ↦ N ( r , u ) ∶ =  < r < . ↗ for ∫ ∂ B r u  Moreover, N ( r , u ) ≡ κ ⇐ ⇒ x ⋅ ∇ u − κ u =  in B  , i.e. u is homogeneous of degree κ in B  . [ Almgr en 1979] for harmonic u [ G arofalo -L in 1986-87] for divergence form elliptic operators Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 11 / 1

Almgren’s frequency function Teorem (Monotonicity of the frequency) Let u ∈ S . T en the frequency function r ∫ B r ∣ ∇ u ∣  r ↦ N ( r , u ) ∶ =  < r < . ↗ for ∫ ∂ B r u  Moreover, N ( r , u ) ≡ κ ⇐ ⇒ x ⋅ ∇ u − κ u =  in B  , i.e. u is homogeneous of degree κ in B  . [ Almgr en 1979] for harmonic u [ G arofalo -L in 1986-87] for divergence form elliptic operators [ A thanasopoulos -C affarelli -S alsa 2007] for thin obstacle problem Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 11 / 1

Figure: Solution of the thin obstacle problem Re ( x  + i ∣ x  ∣)  /  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 12 / 1

Figure: Multi-valued harmonic function Re ( x  + ix  )  /  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 13 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Hence, ∃ blowup u  over a sequence r j →  + u r j → u  in W , ( B  ) Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Hence, ∃ blowup u  over a sequence r j →  + u r j → u  in L  ( ∂ B  ) Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Hence, ∃ blowup u  over a sequence r j →  + u r j → u   ∪ B ± in C  loc ( B ′  ) Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Hence, ∃ blowup u  over a sequence r j →  + u r j → u   ∪ B ± in C  loc ( B ′  ) Proposition (Homogeneity of blowups) Let u ∈ S and the blowup u  be as above. T en, u  is homogeneous of degree κ = N (  + , u ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Homogeneity of blowups Uniform estimates on rescalings { u r } : ∣ ∇ u r ∣  = N ( , u r ) = N ( r , u ) ≤ N ( , u ) . ∫ B  Hence, ∃ blowup u  over a sequence r j →  + u r j → u   ∪ B ± in C  loc ( B ′  ) Proposition (Homogeneity of blowups) Let u ∈ S and the blowup u  be as above. T en, u  is homogeneous of degree κ = N (  + , u ) . Proof. N ( r , u  ) = lim r j →  + N ( r , u r j ) = lim r j →  + N ( rr j , u ) = N (  + , u ) Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 14 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) ≥  −   . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) =  −  N (  + , u ) ≥  . or  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) =  −  N (  + , u ) ≥  . or  Proved by [ Sil vestre 2006], [ A thanasopoulos -C affarelli -S alsa 2007] Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) =  −  N (  + , u ) ≥  . or  Proved by [ Sil vestre 2006], [ A thanasopoulos -C affarelli -S alsa 2007] Teorem (Optimal regularity) loc ( B ′  ∪ B ±  ) ,  Let u ∈ S . T en u ∈ C  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) =  −  N (  + , u ) ≥  . or  Proved by [ Sil vestre 2006], [ A thanasopoulos -C affarelli -S alsa 2007] Teorem (Optimal regularity) loc ( B ′  ∪ B ±  ) ,  Let u ∈ S . T en u ∈ C  Originally proved by [ A thanasopoulos -C affarelli 2004] Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Optimal regularity Lemma (Minimal homogeneity) Let u ∈ S . T en N (  + , u ) =  −  N (  + , u ) ≥  . or  Proved by [ Sil vestre 2006], [ A thanasopoulos -C affarelli -S alsa 2007] Teorem (Optimal regularity) loc ( B ′  ∪ B ±  ) ,  Let u ∈ S . T en u ∈ C  Originally proved by [ A thanasopoulos -C affarelli 2004] u  /  ( x ) = Re ( x  + i ∣ x n ∣)  Achieved on ˆ  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 15 / 1

Classification of free boundary points Definition Given u ∈ S , for κ ≥  −   we define Γ κ ( u ) ∶ = { x  ∈ Γ ( u ) ∣ N x  (  + , u ) = κ } . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 16 / 1

Classification of free boundary points Definition Given u ∈ S , for κ ≥  −   we define Γ κ ( u ) ∶ = { x  ∈ Γ ( u ) ∣ N x  (  + , u ) = κ } . r ∫ B r ( x  ) ∣ ∇ u ∣  Here N x  ( r , u ) = . ∫ ∂ B r ( x  ) u  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 16 / 1

Classification of free boundary points Definition Given u ∈ S , for κ ≥  −   we define Γ κ ( u ) ∶ = { x  ∈ Γ ( u ) ∣ N x  (  + , u ) = κ } . r ∫ B r ( x  ) ∣ ∇ u ∣  Here N x  ( r , u ) = . ∫ ∂ B r ( x  ) u  Γ κ = ∅ whenever  −   < κ < . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 16 / 1

Classification of free boundary points Definition Given u ∈ S , for κ ≥  −   we define Γ κ ( u ) ∶ = { x  ∈ Γ ( u ) ∣ N x  (  + , u ) = κ } . r ∫ B r ( x  ) ∣ ∇ u ∣  Here N x  ( r , u ) = . ∫ ∂ B r ( x  ) u  Γ κ = ∅ whenever  −   < κ < . On the other hand,  ∈ Γ κ ( ˆ u κ ) for u κ ( x ) ∶ = Re ( x  + i ∣ x n ∣) κ , κ =  −   , , . . . ,  m −  ˆ  ,  m , . . . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 16 / 1

Classification of free boundary points Definition Given u ∈ S , for κ ≥  −   we define Γ κ ( u ) ∶ = { x  ∈ Γ ( u ) ∣ N x  (  + , u ) = κ } . r ∫ B r ( x  ) ∣ ∇ u ∣  Here N x  ( r , u ) = . ∫ ∂ B r ( x  ) u  Γ κ = ∅ whenever  −   < κ < . On the other hand,  ∈ Γ κ ( ˆ u κ ) for u κ ( x ) ∶ = Re ( x  + i ∣ x n ∣) κ , κ =  −   , , . . . ,  m −  ˆ  ,  m , . . . In dimension , these are the only possible values of κ . Not known in higher dimensions. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 16 / 1

Figure: Graphs of Re ( x  + i ∣ x  ∣)  and Re ( x  + i ∣ x  ∣)   Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 17 / 1

Regular free boundary points Of special interest is the case of the smallest possible value κ =  −   . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 18 / 1

Regular free boundary points Of special interest is the case of the smallest possible value κ =  −   . Definition (Regular points) For u ∈ S we say that x  ∈ Γ ( u ) is regular if N x  (  + , u ) =  −   , i.e., if  ( u ) . x  ∈ Γ  −  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 18 / 1

Regular free boundary points Of special interest is the case of the smallest possible value κ =  −   . Definition (Regular points) For u ∈ S we say that x  ∈ Γ ( u ) is regular if N x  (  + , u ) =  −   , i.e., if  ( u ) . x  ∈ Γ  −  Te following result was proved by [ Athanasopoulos-Caffar elli -S alsa 2007]. Teorem (Regularity of the regular set)  ( u ) is locally a C , α regular Let u ∈ S , then the free boundary Γ  −  ( n −  ) -dimensional surface. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 18 / 1

Singular free boundary points Definition (Singular points) Let u ∈ S . We say that x  is a singular point of the free boundary Γ ( u ) , if the coincidence set Λ ( u ) has vanishing ( n −  ) -dimensional density at x  , i.e. � n −  ( Λ ( u ) ∩ B ′ r ( x  )) = . lim � n −  ( B ′ r ( x  )) r →  + We denote by Σ ( u ) the subset of singular points of Γ ( u ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 19 / 1

Singular free boundary points Definition (Singular points) Let u ∈ S . We say that x  is a singular point of the free boundary Γ ( u ) , if the coincidence set Λ ( u ) has vanishing ( n −  ) -dimensional density at x  , i.e. � n −  ( Λ ( u ) ∩ B ′ r ( x  )) = . lim � n −  ( B ′ r ( x  )) r →  + We denote by Σ ( u ) the subset of singular points of Γ ( u ) . In terms of rescalings  ∈ Σ ( u ) ⇐ ⇒ lim r →  + � n −  ( Λ ( u r ) ∩ B ′  ) = . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 19 / 1

Singular free boundary points Definition (Singular points) Let u ∈ S . We say that x  is a singular point of the free boundary Γ ( u ) , if the coincidence set Λ ( u ) has vanishing ( n −  ) -dimensional density at x  , i.e. � n −  ( Λ ( u ) ∩ B ′ r ( x  )) = . lim � n −  ( B ′ r ( x  )) r →  + We denote by Σ ( u ) the subset of singular points of Γ ( u ) . In terms of rescalings  ∈ Σ ( u ) ⇐ ⇒ lim r →  + � n −  ( Λ ( u r ) ∩ B ′  ) = . Also define Σ κ ( u ) ∶ = Σ ( u ) ∩ Γ κ ( u ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 19 / 1

Singular free boundary points: example ✻ u ( x ′ ,  ) = x   x  x   Σ  ( ,  ) Σ  Σ  x  ✲ t � ✒ Σ  Σ   − ( x   + x   +   in �  with zero thin Figure: Free boundary for u ( x ) = x   x   ) x   x  obstacle on �  × {  } . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 20 / 1

Singular free boundary points: blowups Any blowup u  at a singular point x  ∈ Σ ( u ) belongs to the class P κ for κ = N x  (  + , u ) : P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } , i.e. u  is a homogeneous harmonic polynomial of degree κ , nonnegative on � n −  × {  } . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 21 / 1

Singular free boundary points: blowups Any blowup u  at a singular point x  ∈ Σ ( u ) belongs to the class P κ for κ = N x  (  + , u ) : P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } , i.e. u  is a homogeneous harmonic polynomial of degree κ , nonnegative on � n −  × {  } . Tis implies κ =  m , m ∈ � . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 21 / 1

Singular free boundary points: blowups Any blowup u  at a singular point x  ∈ Σ ( u ) belongs to the class P κ for κ = N x  (  + , u ) : P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } , i.e. u  is a homogeneous harmonic polynomial of degree κ , nonnegative on � n −  × {  } . Tis implies κ =  m , m ∈ � . Central question : Are blowups unique at x  ∈ Σ ( u ) ? Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 21 / 1

Singular free boundary points: blowups Any blowup u  at a singular point x  ∈ Σ ( u ) belongs to the class P κ for κ = N x  (  + , u ) : P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } , i.e. u  is a homogeneous harmonic polynomial of degree κ , nonnegative on � n −  × {  } . Tis implies κ =  m , m ∈ � . Central question : Are blowups unique at x  ∈ Σ ( u ) ? Equivalent to Taylor’s expansion: u ( x ′ , x n ) = p x  κ ( x − x  ) + o (∣ x − x  ∣ κ ) , with nonzero p x  κ ∈ P κ . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 21 / 1

Historical development: classical obstacle problem Normalized solution of classical obstacle problem:  ∈ Γ ( u ) = ∂ { u =  } ∆ u = χ { u >  } in B  , Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 22 / 1

Historical development: classical obstacle problem Normalized solution of classical obstacle problem:  ∈ Γ ( u ) = ∂ { u =  } ∆ u = χ { u >  } in B  , Singular free boundary points: � n -density of Λ ( u ) = { u =  } is zero. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 22 / 1

Historical development: classical obstacle problem Normalized solution of classical obstacle problem:  ∈ Γ ( u ) = ∂ { u =  } ∆ u = χ { u >  } in B  , Singular free boundary points: � n -density of Λ ( u ) = { u =  } is zero. Teorem (Taylor expansion at singular points) At singular points one has the Taylor expansion u ( x ) = p x  ( x − x  ) + o (∣ x − x  ∣  ) where p x  is a nonnegative homogeneous quadratic polynomial with ∆ p x  =  . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 22 / 1

Alt-Caffarelli-Friedman monotonicity formula First proved by [ Caffar elli -R iviere 1977] in dimension 2 Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 23 / 1

Alt-Caffarelli-Friedman monotonicity formula First proved by [ Caffar elli -R iviere 1977] in dimension 2 Proved in any dimension by [ C affarelli 1998] by using the following deep result of [ A lt -C affarelli -F riedman 1984] Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 23 / 1

Alt-Caffarelli-Friedman monotonicity formula First proved by [ Caffar elli -R iviere 1977] in dimension 2 Proved in any dimension by [ C affarelli 1998] by using the following deep result of [ A lt -C affarelli -F riedman 1984] Teorem (ACF monotonicity formula) If v ± ≥  are continuous subharmonic functions such that v + ⋅ v − =  , then ∣∇ v + ∣  ∣∇ v − ∣  r ↦ Φ ( r , v ± ) ∶ =  r  ∫ B r ∣ x ∣ n −  ∫ B r ∣ x ∣ n −  ↗ Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 23 / 1

Alt-Caffarelli-Friedman monotonicity formula First proved by [ Caffar elli -R iviere 1977] in dimension 2 Proved in any dimension by [ C affarelli 1998] by using the following deep result of [ A lt -C affarelli -F riedman 1984] Teorem (ACF monotonicity formula) If v ± ≥  are continuous subharmonic functions such that v + ⋅ v − =  , then ∣∇ v + ∣  ∣∇ v − ∣  r ↦ Φ ( r , v ± ) ∶ =  r  ∫ B r ∣ x ∣ n −  ∫ B r ∣ x ∣ n −  ↗ Applied to v ± = ( ∂ e u ) ± = max {± ∂ e u ,  } Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 23 / 1

Weiss’ monotonicity formula Later, [ W eiss 1999] discovered a simpler monotonicity formula, that can be used to prove the Taylor expansion at singular points. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 24 / 1

Weiss’ monotonicity formula Later, [ W eiss 1999] discovered a simpler monotonicity formula, that can be used to prove the Taylor expansion at singular points. Teorem (Weiss’ monotonicity formula) If u is a solution of the classical obstacle problem, then ∣ ∇ u ∣  +  u − r ↦ W ( r ) ∶ =   r n +  ∫ B r r n +  ∫ ∂ B r u  ↗ . In fact, d  r n +  ∫ ∂ B r ( x ⋅ ∇ u −  u )  . dr W ( r ) = Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 24 / 1

Monneau’s monotonicity formula at singular points More recently, [ Mo nneau 2003] derived yet another monotonicity formula from that of Weiss. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 25 / 1

Monneau’s monotonicity formula at singular points More recently, [ Mo nneau 2003] derived yet another monotonicity formula from that of Weiss. Tailor made for the study of singular free boundary points (in the classical obstacle problem). Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 25 / 1

Monneau’s monotonicity formula at singular points More recently, [ Mo nneau 2003] derived yet another monotonicity formula from that of Weiss. Tailor made for the study of singular free boundary points (in the classical obstacle problem). Teorem (Monneau’s monotonicity formula) Let u be a solution of the classical obstacle problem and  is a singular free boundary point. T en the function r ↦ M ( r , u , p ) ∶ = ( u − p )  ↗  r n +  ∫ ∂ B r for arbitrary nonnegative quadratic polynomial p with ∆ p =  . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 25 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Te monotonicity formulas of A-C-F , Weiss and Monneau are only suitable for κ = . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Te monotonicity formulas of A-C-F , Weiss and Monneau are only suitable for κ = . In the thin obstacle problem , frequencies κ take at least values κ =  m −   ,  m , m ∈ � . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Te monotonicity formulas of A-C-F , Weiss and Monneau are only suitable for κ = . In the thin obstacle problem , frequencies κ take at least values κ =  m −   ,  m , m ∈ � . In the thin obstacle problem, Almgren ’s monotonicity formula works regardless of κ . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Te monotonicity formulas of A-C-F , Weiss and Monneau are only suitable for κ = . In the thin obstacle problem , frequencies κ take at least values κ =  m −   ,  m , m ∈ � . In the thin obstacle problem, Almgren ’s monotonicity formula works regardless of κ . Initial idea : is there a Monneau type formula based on Almgen’s? Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Back to the thin obstacle problem In the classical obstacle problem the only frequency that appears is κ = . Te monotonicity formulas of A-C-F , Weiss and Monneau are only suitable for κ = . In the thin obstacle problem , frequencies κ take at least values κ =  m −   ,  m , m ∈ � . In the thin obstacle problem, Almgren ’s monotonicity formula works regardless of κ . Initial idea : is there a Monneau type formula based on Almgen’s? Solution found : there is a one-parameter family of monotonicity formulas { W κ } κ ≥  of Weiss type, which further generate a family of { M κ } κ =  m of Monneau type formulas. Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 26 / 1

Weiss type monotonicity formulas Teorem (Weiss type monotonicity formulas) Let u ∈ S and κ ≥  . T en ∣ ∇ u ∣  − r ↦ W κ ( r , u ) ∶ = κ  r n −  +  κ ∫ B r r n −  +  κ ∫ ∂ B r u  ↗ . In fact, d  r n +  κ ∫ ∂ B r ( x ⋅ ∇ u − κ u )  . dr W κ ( r , u ) = Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 27 / 1

Weiss type monotonicity formulas Teorem (Weiss type monotonicity formulas) Let u ∈ S and κ ≥  . T en ∣ ∇ u ∣  − r ↦ W κ ( r , u ) ∶ = κ  r n −  +  κ ∫ B r r n −  +  κ ∫ ∂ B r u  ↗ . In fact, d  r n +  κ ∫ ∂ B r ( x ⋅ ∇ u − κ u )  . dr W κ ( r , u ) = W κ ≡ const ⇐ ⇒ u is homogeneous of degree κ . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 27 / 1

Connection with Almgren’s formula For u ∈ S , let D ( r ) ∶ = ∫ B r ∣ ∇ u ∣  , H ( r ) ∶ = ∫ ∂ B r u  Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 28 / 1

Connection with Almgren’s formula For u ∈ S , let D ( r ) ∶ = ∫ B r ∣ ∇ u ∣  , H ( r ) ∶ = ∫ ∂ B r u  N ( r ) = rD ( r ) Almgren’s : H ( r ) Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 28 / 1

Connection with Almgren’s formula For u ∈ S , let D ( r ) ∶ = ∫ B r ∣ ∇ u ∣  , H ( r ) ∶ = ∫ ∂ B r u  N ( r ) = rD ( r ) Almgren’s : H ( r ) r n −  +  κ [ rD ( r ) − κ H ( r )] = H ( r ) r n −  +  κ [ N ( r ) − κ ]  W κ ( r ) = Weiss type : Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 28 / 1

Connection with Almgren’s formula For u ∈ S , let D ( r ) ∶ = ∫ B r ∣ ∇ u ∣  , H ( r ) ∶ = ∫ ∂ B r u  N ( r ) = rD ( r ) Almgren’s : H ( r ) r n −  +  κ [ rD ( r ) − κ H ( r )] = H ( r ) r n −  +  κ [ N ( r ) − κ ]  W κ ( r ) = Weiss type : Both follow from the same identities for D ′ ( r ) and H ′ ( r ) : H ′ ( r ) = n −  H ( r ) +  D ( r ) r D ′ ( r ) = n −  D ( r ) +  ∫ ∂ B r u  ν r Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 28 / 1

Monneau type monotonicity formulas Teorem (Monneau type monotonicity formulas) Let u ∈ S with  ∈ Σ κ ( u ) , κ =  m, m ∈ � . T en for arbitrary p κ ∈ P κ r ↦ M κ ( r , u , p κ ) ∶ =  ( u − p κ )  ↗ . r n −  +  κ ∫ ∂ B r Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 29 / 1

Monneau type monotonicity formulas Teorem (Monneau type monotonicity formulas) Let u ∈ S with  ∈ Σ κ ( u ) , κ =  m, m ∈ � . T en for arbitrary p κ ∈ P κ r ↦ M κ ( r , u , p κ ) ∶ =  ( u − p κ )  ↗ . r n −  +  κ ∫ ∂ B r Recall that for κ =  m P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 29 / 1

Monneau type monotonicity formulas Teorem (Monneau type monotonicity formulas) Let u ∈ S with  ∈ Σ κ ( u ) , κ =  m, m ∈ � . T en for arbitrary p κ ∈ P κ r ↦ M κ ( r , u , p κ ) ∶ =  ( u − p κ )  ↗ . r n −  +  κ ∫ ∂ B r Recall that for κ =  m P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } . Important observation : Te polynomial p κ ∈ P κ in the monotonicity formula M κ is arbitrary . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 29 / 1

Monneau type monotonicity formulas Teorem (Monneau type monotonicity formulas) Let u ∈ S with  ∈ Σ κ ( u ) , κ =  m, m ∈ � . T en for arbitrary p κ ∈ P κ r ↦ M κ ( r , u , p κ ) ∶ =  ( u − p κ )  ↗ . r n −  +  κ ∫ ∂ B r Recall that for κ =  m P κ = { p κ ( x ) ∣ ∆ p κ = , x ⋅ ∇ p κ − κ p κ = , p κ ( x ′ ,  ) ≥  } . Important observation : Te polynomial p κ ∈ P κ in the monotonicity formula M κ is arbitrary . Every blowup at a singular point  ∈ Σ k ( u ) is an element of P κ . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 29 / 1

Taylor expansion at singular points Teorem (Taylor expansion at singular points) Let u ∈ S . T en for any x  ∈ Σ κ ( u ) there exists a nonzero p x  κ ∈ P κ such that u ( x ) = p x  κ ( x − x  ) + o (∣ x − x  ∣ κ ) . Moreover, the mapping x  ↦ p x  κ is continuous on Σ κ ( u ) . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 30 / 1

Taylor expansion at singular points Teorem (Taylor expansion at singular points) Let u ∈ S . T en for any x  ∈ Σ κ ( u ) there exists a nonzero p x  κ ∈ P κ such that u ( x ) = p x  κ ( x − x  ) + o (∣ x − x  ∣ κ ) . Moreover, the mapping x  ↦ p x  κ is continuous on Σ κ ( u ) . Idea of the proof. Assume x  = . Let p κ be a blowup of u over a sequence r j → . Ten M κ ( r j , u , p κ ) → . ⇒ M κ ( r , u , p κ ) →  as r → . Monotonicity of M κ Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 30 / 1

Structure of the singular set Definition (Dimension at the singular point) For x  ∈ Σ κ ( u ) denote κ ∶ = dim { ξ ∈ � n −  ∣ ξ ⋅ ∇ x ′ p x  d x  κ ≡  } , which we call the dimension of Σ κ ( u ) at x  . For d = , , . . . , n −  define Σ d κ ( u ) ∶ = { x  ∈ Σ κ ( u ) ∣ d x  κ = d } . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 31 / 1

Structure of the singular set Definition (Dimension at the singular point) For x  ∈ Σ κ ( u ) denote κ ∶ = dim { ξ ∈ � n −  ∣ ξ ⋅ ∇ x ′ p x  d x  κ ≡  } , which we call the dimension of Σ κ ( u ) at x  . For d = , , . . . , n −  define Σ d κ ( u ) ∶ = { x  ∈ Σ κ ( u ) ∣ d x  κ = d } . ≡  on � n −  × {  } one has Note that since p x  κ /  ≤ d x  κ ≤ n − . Garofalo, Petrosyan (Purdue) Monotonicity formulas and the singular set University of Chicago 31 / 1