Parabolic Signorini Problem

Arshak Petrosyan

Free Boundary Problems in Biology Math Biosciences Institute, OSU November Õ¦–՘, óþÕÕ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ/ìÕ Semipermeable Membranes and Osmosis

Semipermeable membrane is a membrane that is permeable only for a certain type of molecules (solvents) and blocks other molecules (solutes). Because of the chemical imbalance, the solvent žows through the membrane from the region of smaller concentration of solute to the region of higher Picture Source: Wikipedia concentation (osmotic pressure). e žow occurs in one direction. e žow continues until a su›cient pressure builds up on the other side of the membrane (to compensate for osmotic pressure), which then shuts the žow. is process is known as osmosis.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó/ìÕ ᏹ ⊂ ∂Ω semipermeable part of the ᏹT boundary φ no žow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ u T pressure ∶ ΩT ∶= Ω × (, ] → ޒ the of ᏹ žow( − ∂ )u = the chemical solution, that satises a ∆ t  ΩT dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn

Ω

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹT φ no žow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ u T pressure ∶ ΩT ∶= Ω × (, ] → ޒ the of žow( − ∂ )u = the chemical solution, that satises a ∆ t  ΩT dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn ᏹ ⊂ ∂Ω semipermeable part of the boundary Ω

ᏹ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no žow

u T pressure ∶ ΩT ∶= Ω × (, ] → ޒ the of žow( − ∂ )u = the chemical solution, that satises a ∆ t  ΩT dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ

ᏹ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no žow

žow

ΩT

nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ u T pressure ∶ ΩT ∶= Ω × (, ] → ޒ the of ᏹ ( − ∂ )u = the chemical solution, that satises a ∆ t  dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no žow

žow

ΩT

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ u T pressure ∶ ΩT ∶= Ω × (, ] → ޒ the of ᏹ ( − ∂ )u = the chemical solution, that satises a ∆ t  dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ φ Ω

ᏹ ∂ u (∆ − t ) = 

Mathematical Formulation: Unilateral Problem

Given open Ω ⊂ ޒn ᏹ ⊂ ∂Ω semipermeable part of the ᏹ boundary T no žow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → ޒ u Ω Ω , T ޒ the pressure of ∶ T ∶= × ( ] → žow the chemical solution, that satises a ΩT dišusion equation (slightly compressible žuid) u ∂ u ∆ − t =  in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν =  (no žow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (žow)

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ u > φ ∂ u ν =  u = φ ∂ u ν ≥ 

ese are known as the Signorini boundary conditions u φ φ Since should stay above on ᏹT , is known as the thin obstacle. e problem is known as Parabolic Signorini Problem or Parabolic in Obstacle Problem.

Parabolic Signorini Problem

Letting λ → ∞ we obtain the following innite permeability conditions on ᏹT ( ) ᏹ T no žow u ≥ φ ∂ u ν ≥  u φ ∂ u ( − ) ν =  žow ΩT

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¦/ìÕ no žow

žow

u φ φ Since should stay above on ᏹT , is known as the thin obstacle. e problem is known as Parabolic Signorini Problem or Parabolic in Obstacle Problem.

Parabolic Signorini Problem

Letting λ → ∞ we obtain the following innite permeability conditions on ᏹT ( ) ᏹT u > φ u φ ≥ ∂ u ν =  ∂ u ν ≥  u = φ u φ ∂ u ∂ u ( − ) ν =  ν ≥  ΩT ese are known as the Signorini boundary conditions

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¦/ìÕ no žow

žow

Parabolic Signorini Problem

Letting λ → ∞ we obtain the following innite permeability conditions on ᏹT ( ) ᏹT u > φ u φ ≥ ∂ u ν =  ∂ u ν ≥  u = φ u φ ∂ u ∂ u ( − ) ν =  ν ≥  ΩT ese are known as the Signorini boundary conditions u φ φ Since should stay above on ᏹT , is known as the thin obstacle. e problem is known as Parabolic Signorini Problem or Parabolic in Obstacle Problem.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¦/ìÕ en for any (reasonable) initial condition u φ =  on Ω = Ω × {}

the solution exist and unique. See [D¶êZ¶±-L†™•« Õɘä].

Parabolic Signorini Problem

e function u(x, t) the solves the following variational inequality: ᏹT u > φ u u v ∂ u u v ∂ u S ∇ ⋅ ∇( − ) + t ( − ) ≥  ν =  Ω u =  u ∂ u L u = φ ∈ K, t ∈ (Ω) ∂ u ν ≥  for all v K ∈ ΩT where u φ =  K v W, v φ v  = { ∈ (Ω) ∶ Tᏹ ≥ , T∂Ω∖ᏹ = }

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¢/ìÕ Parabolic Signorini Problem

e function u(x, t) the solves the following variational inequality: ᏹT u > φ u u v ∂ u u v ∂ u S ∇ ⋅ ∇( − ) + t ( − ) ≥  ν =  Ω u =  u ∂ u L u = φ ∈ K, t ∈ (Ω) ∂ u ν ≥  for all v K ∈ ΩT where u φ =  K v W, v φ v  = { ∈ (Ω) ∶ Tᏹ ≥ , T∂Ω∖ᏹ = } en for any (reasonable) initial condition u φ =  on Ω = Ω × {}

the solution exist and unique. See [D¶êZ¶±-L†™•« Õɘä].

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¢/ìÕ x t u φ Let Λ ∶= {( , ) ∈ ᏹT ∶ = } be the so-called coincidence set. en Γ

∂ Λ Γ ∶= ᏹΛ

is the free boundary. One then interested in the structure, geometric properties and the regularity of the free boundary.

In order to do so one has to know the optimal regularity of the solution u in ΩT up to ᏹT .

Free Boundary Problem

e parabolic Signorini problem is a free boundary problem.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ä/ìÕ One then interested in the structure, geometric properties and the regularity of the free boundary.

In order to do so one has to know the optimal regularity of the solution u in ΩT up to ᏹT .

Free Boundary Problem

e parabolic Signorini problem is a free boundary problem. x t u φ Let Λ ∶= {( , ) ∈ ᏹT ∶ = } be the so-called coincidence set. en Γ

∂ Λ Γ ∶= ᏹΛ

is the free boundary.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ä/ìÕ In order to do so one has to know the optimal regularity of the solution u in ΩT up to ᏹT .

Free Boundary Problem

e parabolic Signorini problem is a free boundary problem. x t u φ Let Λ ∶= {( , ) ∈ ᏹT ∶ = } be the so-called coincidence set. en Γ

∂ Λ Γ ∶= ᏹΛ

is the free boundary. One then interested in the structure, geometric properties and the regularity of the free boundary.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ä/ìÕ Free Boundary Problem

e parabolic Signorini problem is a free boundary problem. x t u φ Let Λ ∶= {( , ) ∈ ᏹT ∶ = } be the so-called coincidence set. en Γ

∂ Λ Γ ∶= ᏹΛ

is the free boundary. One then interested in the structure, geometric properties and the regularity of the free boundary.

In order to do so one has to know the optimal regularity of the solution u in ΩT up to ᏹT .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ä/ìÕ In the elliptic case a similar result has been proved by [CZ€€Z§u† ÕÉßÉ] Proof in [U§Z’±«uêZ Õɘ¢] in the elliptic case worked also for ∞ nonhomogeneous equation ∆u = f , f ∈ L (Ω), with Signorini boundary conditions. at fact then implies the regularity in the parabolic case. Except some specic cases, no general results have been known for the free boundary.

Parabolic Signorini Problem: Known Results

eorem (“C,α-regularity” [U§Z’±«uêZ Õɘ¢]) Let u be a solution of the Parabolic Signorini Problem with φ C, C, , ∈ x ∩ t (ᏹT ) φ , and  L . en u Cα,α~ K for any K  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) ⋐ ΩT ∪ ᏹT and

u ~ C φ , , φ   Y∇ Y α,α ( ) ≤ K(Y YC ∩C (ᏹ ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x t T

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ß/ìÕ Proof in [U§Z’±«uêZ Õɘ¢] in the elliptic case worked also for ∞ nonhomogeneous equation ∆u = f , f ∈ L (Ω), with Signorini boundary conditions. at fact then implies the regularity in the parabolic case. Except some specic cases, no general results have been known for the free boundary.

Parabolic Signorini Problem: Known Results

eorem (“C,α-regularity” [U§Z’±«uêZ Õɘ¢]) Let u be a solution of the Parabolic Signorini Problem with φ C, C, , ∈ x ∩ t (ᏹT ) φ , and  L . en u Cα,α~ K for any K  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) ⋐ ΩT ∪ ᏹT and

u ~ C φ , , φ   Y∇ Y α,α ( ) ≤ K(Y YC ∩C (ᏹ ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x t T

In the elliptic case a similar result has been proved by [CZ€€Z§u† ÕÉßÉ]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ß/ìÕ Except some specic cases, no general results have been known for the free boundary.

Parabolic Signorini Problem: Known Results

eorem (“C,α-regularity” [U§Z’±«uêZ Õɘ¢]) Let u be a solution of the Parabolic Signorini Problem with φ C, C, , ∈ x ∩ t (ᏹT ) φ , and  L . en u Cα,α~ K for any K  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) ⋐ ΩT ∪ ᏹT and

u ~ C φ , , φ   Y∇ Y α,α ( ) ≤ K(Y YC ∩C (ᏹ ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x t T

In the elliptic case a similar result has been proved by [CZ€€Z§u† ÕÉßÉ] Proof in [U§Z’±«uêZ Õɘ¢] in the elliptic case worked also for ∞ nonhomogeneous equation ∆u = f , f ∈ L (Ω), with Signorini boundary conditions. at fact then implies the regularity in the parabolic case.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ß/ìÕ Parabolic Signorini Problem: Known Results

eorem (“C,α-regularity” [U§Z’±«uêZ Õɘ¢]) Let u be a solution of the Parabolic Signorini Problem with φ C, C, , ∈ x ∩ t (ᏹT ) φ , and  L . en u Cα,α~ K for any K  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) ⋐ ΩT ∪ ᏹT and

u ~ C φ , , φ   Y∇ Y α,α ( ) ≤ K(Y YC ∩C (ᏹ ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x t T

In the elliptic case a similar result has been proved by [CZ€€Z§u† ÕÉßÉ] Proof in [U§Z’±«uêZ Õɘ¢] in the elliptic case worked also for ∞ nonhomogeneous equation ∆u = f , f ∈ L (Ω), with Signorini boundary conditions. at fact then implies the regularity in the parabolic case. Except some specic cases, no general results have been known for the free boundary.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ß/ìÕ ~ eorem (“C, -regularity” [DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let u be a solution of the Parabolic Signorini Problem with žat ᏹ and φ C, C, , φ , and  L . en u C~,~ K for ∈ x ∩ t (ᏹT )  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) any K and ⋐ ΩT ∪ ᏹT

u ~ ~ C φ , , φ   Y∇ Y  , ( ) ≤ K(Y YC ∩C (M ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x,t t T

is theorem is precise in the sense that it gives the best regularity possible, even in time-independet case:

u x t x ix ~ ( , ) = Re(  + n)

n n− solves the Signorini problem in ޒ+ × ޒ with ᏹ = ޒ .

Parabolic Signorini Problem: Optimal Regularity

In the case when ᏹ is žat, we have the following theorem.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ˜/ìÕ is theorem is precise in the sense that it gives the best regularity possible, even in time-independet case:

u x t x ix ~ ( , ) = Re(  + n)

n n− solves the Signorini problem in ޒ+ × ޒ with ᏹ = ޒ .

Parabolic Signorini Problem: Optimal Regularity

In the case when ᏹ is žat, we have the following theorem.

~ eorem (“C, -regularity” [DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let u be a solution of the Parabolic Signorini Problem with žat ᏹ and φ C, C, , φ , and  L . en u C~,~ K for ∈ x ∩ t (ᏹT )  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) any K and ⋐ ΩT ∪ ᏹT

u ~ ~ C φ , , φ   Y∇ Y  , ( ) ≤ K(Y YC ∩C (M ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x,t t T

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ˜/ìÕ Parabolic Signorini Problem: Optimal Regularity

In the case when ᏹ is žat, we have the following theorem.

~ eorem (“C, -regularity” [DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let u be a solution of the Parabolic Signorini Problem with žat ᏹ and φ C, C, , φ , and  L . en u C~,~ K for ∈ x ∩ t (ᏹT )  ∈ Lip(Ω) ∈ (᏿T ) ∇ ∈ x,t ( ) any K and ⋐ ΩT ∪ ᏹT

u ~ ~ C φ , , φ   Y∇ Y  , ( ) ≤ K(Y YC ∩C (M ) + Y YLip(Ω) + Y YL (᏿T )) Cx,t K x,t t T

is theorem is precise in the sense that it gives the best regularity possible, even in time-independet case:

u x t x ix ~ ( , ) = Re(  + n)

n n− solves the Signorini problem in ޒ+ × ޒ with ᏹ = ޒ .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ˜/ìÕ φ ∶ ᏹ → ޒ (thin obstacle)  ∶ ∂Ω ∖ ᏹ → ޒ,  > φ on ᏹ ∩ ∂Ω. φ Minimize the Dirichlet integral u φ D u u dx Ω( ) = S∇ S SΩ on the closed convex set ᏹ Ω

K u W, u φ u  = { ∈ (Ω)S Tᏹ ≥ , T∂Ω∖ᏹ = }. e minimizer u satises ∆u =  in Ω u φ ∂ u u φ ∂ u ≥ , ν ≥ , ( − ) ν =  on ᏹ

Elliptic Case: in Obstacle Problem

Given Ω ⊂ ޒn, ᏹ ⊂ ∂Ω

Ω

ᏹ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ É/ìÕ Minimize the Dirichlet integral u φ D u u dx Ω( ) = S∇ S SΩ on the closed convex set ᏹ Ω

K u W, u φ u  = { ∈ (Ω)S Tᏹ ≥ , T∂Ω∖ᏹ = }. e minimizer u satises ∆u =  in Ω u φ ∂ u u φ ∂ u ≥ , ν ≥ , ( − ) ν =  on ᏹ

Elliptic Case: in Obstacle Problem

Given Ω ⊂ ޒn, ᏹ ⊂ ∂Ω φ ∶ ᏹ → ޒ (thin obstacle)  ∂ ᏹ ޒ  φ ᏹ ∂ ∶ Ω ∖ → , > on ∩ Ω. φ Ω

ᏹ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ É/ìÕ φ Ω u

ᏹ

e minimizer u satises ∆u =  in Ω u φ ∂ u u φ ∂ u ≥ , ν ≥ , ( − ) ν =  on ᏹ

Elliptic Case: in Obstacle Problem

Given Ω ⊂ ޒn, ᏹ ⊂ ∂Ω φ ∶ ᏹ → ޒ (thin obstacle)  ∶ ∂Ω ∖ ᏹ → ޒ,  > φ on ᏹ ∩ ∂Ω. Minimize the Dirichlet integral φ D u u dx Ω( ) = S∇ S SΩ on the closed convex set ᏹ Ω

K u W, u φ u  = { ∈ (Ω)S Tᏹ ≥ , T∂Ω∖ᏹ = }.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ É/ìÕ φ Ω

ᏹ

Elliptic Case: in Obstacle Problem

Given Ω ⊂ ޒn, ᏹ ⊂ ∂Ω φ ∶ ᏹ → ޒ (thin obstacle)  ∶ ∂Ω ∖ ᏹ → ޒ,  > φ on ᏹ ∩ ∂Ω. Minimize the Dirichlet integral u φ D u u dx Ω( ) = S∇ S SΩ on the closed convex set ᏹ Ω

K u W, u φ u  = { ∈ (Ω)S Tᏹ ≥ , T∂Ω∖ᏹ = }. e minimizer u satises ∆u =  in Ω u φ ∂ u u φ ∂ u ≥ , ν ≥ , ( − ) ν =  on ᏹ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ É/ìÕ φ = : [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] φ ∈ C,:[A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] φ ∈ C,:[ᏼ-T™ óþÕþ]

Elliptic Case: Optimal Regularity

e progress in parabolic case was motivated by the breakthrough result ~ of [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] establishing the C,  regularity in the elliptic thin obstacle problem.

eorem Let u be a solution of the thin obstacle problem for žat ᏹ, with φ ∈ C,(ᏹ) and ~  ∈ L(∂Ω ∖ ᏹ). en u ∈ C, (K) for any K ⋐ Ω ∪ ᏹ and u C φ  Y YC,~(K) ≤ K ‰Y YC,(ᏹ) + Y YL Ž .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õþ/ìÕ φ ∈ C,:[A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] φ ∈ C,:[ᏼ-T™ óþÕþ]

Elliptic Case: Optimal Regularity

e progress in parabolic case was motivated by the breakthrough result ~ of [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] establishing the C,  regularity in the elliptic thin obstacle problem.

eorem Let u be a solution of the thin obstacle problem for žat ᏹ, with φ ∈ C,(ᏹ) and ~  ∈ L(∂Ω ∖ ᏹ). en u ∈ C, (K) for any K ⋐ Ω ∪ ᏹ and u C φ  Y YC,~(K) ≤ K ‰Y YC,(ᏹ) + Y YL Ž .

φ = : [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õþ/ìÕ φ ∈ C,:[ᏼ-T™ óþÕþ]

Elliptic Case: Optimal Regularity

e progress in parabolic case was motivated by the breakthrough result ~ of [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] establishing the C,  regularity in the elliptic thin obstacle problem.

eorem Let u be a solution of the thin obstacle problem for žat ᏹ, with φ ∈ C,(ᏹ) and ~  ∈ L(∂Ω ∖ ᏹ). en u ∈ C, (K) for any K ⋐ Ω ∪ ᏹ and u C φ  Y YC,~(K) ≤ K ‰Y YC,(ᏹ) + Y YL Ž .

φ = : [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] φ ∈ C,:[A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õþ/ìÕ Elliptic Case: Optimal Regularity

e progress in parabolic case was motivated by the breakthrough result ~ of [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] establishing the C,  regularity in the elliptic thin obstacle problem.

eorem Let u be a solution of the thin obstacle problem for žat ᏹ, with φ ∈ C,(ᏹ) and ~  ∈ L(∂Ω ∖ ᏹ). en u ∈ C, (K) for any K ⋐ Ω ∪ ᏹ and u C φ  Y YC,~(K) ≤ K ‰Y YC,(ᏹ) + Y YL Ž .

φ = : [A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ] φ ∈ C,:[A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] φ ∈ C,:[ᏼ-T™ óþÕþ]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õþ/ìÕ If u solves Signorini problem, aŸer translation, rotation, and scaling, we may normalize u as follows:

Denition (Class S) We say u is a normalized solution of Signorini problem iš u B+ ∆ =  in  u ∂ u u ∂ u B′ ≥ , − xn ≥ , xn =  on   ∈ Γ(u) = ∂Λ(u) = ∂{u = }.

We denote the class of normalized solutions by S.

Notation n n− B+ B n B′ B n− : ޒ+ = ޒ × (, +∞),  ∶=  ∩ ޒ+,  ∶=  ∩ (ޒ × {})

Zero Obstacle φ: Normalization

− Assume ᏹ is žat: ᏹ = ޒn  × {}, φ = 

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÕ/ìÕ Denition (Class S) We say u is a normalized solution of Signorini problem iš u B+ ∆ =  in  u ∂ u u ∂ u B′ ≥ , − xn ≥ , xn =  on   ∈ Γ(u) = ∂Λ(u) = ∂{u = }.

We denote the class of normalized solutions by S.

Notation n n− B+ B n B′ B n− : ޒ+ = ޒ × (, +∞),  ∶=  ∩ ޒ+,  ∶=  ∩ (ޒ × {})

Zero Obstacle φ: Normalization

− Assume ᏹ is žat: ᏹ = ޒn  × {}, φ =  If u solves Signorini problem, aŸer translation, rotation, and scaling, we may normalize u as follows:

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÕ/ìÕ Notation n n− B+ B n B′ B n− : ޒ+ = ޒ × (, +∞),  ∶=  ∩ ޒ+,  ∶=  ∩ (ޒ × {})

Zero Obstacle φ: Normalization

− Assume ᏹ is žat: ᏹ = ޒn  × {}, φ =  If u solves Signorini problem, aŸer translation, rotation, and scaling, we may normalize u as follows:

Denition (Class S) We say u is a normalized solution of Signorini problem iš u B+ ∆ =  in  u ∂ u u ∂ u B′ ≥ , − xn ≥ , xn =  on   ∈ Γ(u) = ∂Λ(u) = ∂{u = }.

We denote the class of normalized solutions by S.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÕ/ìÕ Zero Obstacle φ: Normalization

− Assume ᏹ is žat: ᏹ = ޒn  × {}, φ =  If u solves Signorini problem, aŸer translation, rotation, and scaling, we may normalize u as follows:

Denition (Class S) We say u is a normalized solution of Signorini problem iš u B+ ∆ =  in  u ∂ u u ∂ u B′ ≥ , − xn ≥ , xn =  on   ∈ Γ(u) = ∂Λ(u) = ∂{u = }.

We denote the class of normalized solutions by S.

Notation n n− B+ B n B′ B n− : ޒ+ = ޒ × (, +∞),  ∶=  ∩ ޒ+,  ∶=  ∩ (ޒ × {})

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÕ/ìÕ e resulting function will satisfy u B ∆ ≤  in  u B u ∆ =  in  ∖ Λ( ) u u B ∆ =  in . u u B′ Here Λ( ) = { = } ⊂ . More specically:

u ∂ u Ᏼn− Ᏸ′ B ∆ = ( xn ) TΛ(u) in ( ).

Zero Obstacle φ: Normalization

u B+ B Every ∈ S can be extended from  to  by even symmetry u x′ x u x′ x ( , − n) ∶= ( , n).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õó/ìÕ More specically:

u ∂ u Ᏼn− Ᏸ′ B ∆ = ( xn ) TΛ(u) in ( ).

Zero Obstacle φ: Normalization

u B+ B Every ∈ S can be extended from  to  by even symmetry u x′ x u x′ x ( , − n) ∶= ( , n).

e resulting function will satisfy u B ∆ ≤  in  u B u ∆ =  in  ∖ Λ( ) u u B ∆ =  in . u u B′ Here Λ( ) = { = } ⊂ .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õó/ìÕ Zero Obstacle φ: Normalization

u B+ B Every ∈ S can be extended from  to  by even symmetry u x′ x u x′ x ( , − n) ∶= ( , n).

e resulting function will satisfy u B ∆ ≤  in  u B u ∆ =  in  ∖ Λ( ) u u B ∆ =  in . u u B′ Here Λ( ) = { = } ⊂ . More specically:

u ∂ u Ᏼn− Ᏸ′ B ∆ = ( xn ) TΛ(u) in ( ).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õó/ìÕ [GZ§™€Z™-L†• Õɘä-˜ß] for divergence form elliptic operators [A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] for thin obstacle problem

Almgren’s Frequency Function

eorem (Monotonicity of the frequency) Let u ∈ S. en the frequency function

r S∇uS r N r u ∫Br ↗ for r ↦ ( , ) ∶= u  < < . ∫∂Br Moreover, N r u κ x u κu in B , i.e. u is homogeneous of ( , ) ≡ ⇐⇒ ⋅ ∇ − =   degree κ in B . 

[A“§u• ÕÉßÉ] for harmonic u

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õì/ìÕ [A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] for thin obstacle problem

Almgren’s Frequency Function

eorem (Monotonicity of the frequency) Let u ∈ S. en the frequency function

r S∇uS r N r u ∫Br ↗ for r ↦ ( , ) ∶= u  < < . ∫∂Br Moreover, N r u κ x u κu in B , i.e. u is homogeneous of ( , ) ≡ ⇐⇒ ⋅ ∇ − =   degree κ in B . 

[A“§u• ÕÉßÉ] for harmonic u [GZ§™€Z™-L†• Õɘä-˜ß] for divergence form elliptic operators

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õì/ìÕ Almgren’s Frequency Function

eorem (Monotonicity of the frequency) Let u ∈ S. en the frequency function

r S∇uS r N r u ∫Br ↗ for r ↦ ( , ) ∶= u  < < . ∫∂Br Moreover, N r u κ x u κu in B , i.e. u is homogeneous of ( , ) ≡ ⇐⇒ ⋅ ∇ − =   degree κ in B . 

[A“§u• ÕÉßÉ] for harmonic u [GZ§™€Z™-L†• Õɘä-˜ß] for divergence form elliptic operators [A±„Z•Z«™£™¶™«-CZ€€Z§u†-SZ«Z óþþß] for thin obstacle problem

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õì/ìÕ x i x ~ Figure: Solution of the thin obstacle problem Re(  + S S)

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ¦/ìÕ x ix ~ Figure: Multi-valued harmonic function Re(  + )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ¢/ìÕ e rescaling is normalized so that

u  Y rYL (∂B) = . u r Limits of subsequences { r j } for some j → + are known as blowups. Generally the blowups may be dišerent over dišerent subsequences r r = j → +.

Rescalings and Blowups

For a solution u ∈ S and r >  consider rescalings u rx u x ( ) r( ) ∶=  .    − u ‰ rn  ∫∂Br Ž

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õä/ìÕ u r Limits of subsequences { r j } for some j → + are known as blowups. Generally the blowups may be dišerent over dišerent subsequences r r = j → +.

Rescalings and Blowups

For a solution u ∈ S and r >  consider rescalings u rx u x ( ) r( ) ∶=  .    − u ‰ rn  ∫∂Br Ž e rescaling is normalized so that

u  Y rYL (∂B) = .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õä/ìÕ Generally the blowups may be dišerent over dišerent subsequences r r = j → +.

Rescalings and Blowups

For a solution u ∈ S and r >  consider rescalings u rx u x ( ) r( ) ∶=  .    − u ‰ rn  ∫∂Br Ž e rescaling is normalized so that

u  Y rYL (∂B) = . u r Limits of subsequences { r j } for some j → + are known as blowups.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õä/ìÕ Rescalings and Blowups

For a solution u ∈ S and r >  consider rescalings u rx u x ( ) r( ) ∶=  .    − u ‰ rn  ∫∂Br Ž e rescaling is normalized so that

u  Y rYL (∂B) = . u r Limits of subsequences { r j } for some j → + are known as blowups. Generally the blowups may be dišerent over dišerent subsequences r r = j → +.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õä/ìÕ Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

u r Hence, ∃ blowup  over a sequence j → + u u W, B r j →  in ( )

Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B u r Hence, ∃ blowup  over a sequence j → + u u W, B r j →  in ( )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B u r Hence, ∃ blowup  over a sequence j → + u u L ∂B r j →  in ( )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B u r Hence, ∃ blowup  over a sequence j → + u u C B′ B± r j →  in loc(  ∪  )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B u r Hence, ∃ blowup  over a sequence j → + u u C B′ B± r j →  in loc(  ∪  ) Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ Homogeneity of Blowups

u Uniform estimates on rescalings { r}:

u  N u N r u N u S S∇ rS = (, r) = ( , ) ≤ (, ). B u r Hence, ∃ blowup  over a sequence j → + u u C B′ B± r j →  in loc(  ∪  ) Proposition (Homogeneity of blowups) Let u and the blowup u be as above. en, u is homogeneous of degree ∈ S   κ = N(+, u).

Proof. N r u N r u N rr u N u ( , ) = limr j→+ ( , r j ) = limr j→+ ( j, ) = (+, )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õß/ìÕ From Lemma we obtain that N(+, u) = κ ≥ ~ for any u ∈ S. From here one can show that

u Crn+ r S ≤ ,  < <  ∂Br and consequently that

u C,~ B± B′ ∈ ( ~ ∪ ~).

Proof of C,~ regularity

Lemma ([A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ]) Let u be a homogeneous global solution of the thin obstacle problem with  homogeneity κ. en κ ≥ ~.

κ x i x ~ Explicit solution for which = ~ is achieved is Re(  + S nS)

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ˜/ìÕ From here one can show that

u Crn+ r S ≤ ,  < <  ∂Br and consequently that

u C,~ B± B′ ∈ ( ~ ∪ ~).

Proof of C,~ regularity

Lemma ([A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ]) Let u be a homogeneous global solution of the thin obstacle problem with  homogeneity κ. en κ ≥ ~.

κ x i x ~ Explicit solution for which = ~ is achieved is Re(  + S nS) From Lemma we obtain that N(+, u) = κ ≥ ~ for any u ∈ S.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ˜/ìÕ Proof of C,~ regularity

Lemma ([A±„Z•Z«™£™¶™«-CZ€€Z§u† óþþþ]) Let u be a homogeneous global solution of the thin obstacle problem with  homogeneity κ. en κ ≥ ~.

κ x i x ~ Explicit solution for which = ~ is achieved is Re(  + S nS) From Lemma we obtain that N(+, u) = κ ≥ ~ for any u ∈ S. From here one can show that

u Crn+ r S ≤ ,  < <  ∂Br and consequently that

u C,~ B± B′ ∈ ( ~ ∪ ~).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ Õ˜/ìÕ Consider the dišerence

′ v(x) = u(x) − φ(x ).

f f ′φ L∞ B+ en it will be the class S with = ∆ ∈ (  ). Denition (Class S f ) v f f L∞ B+ We say that ∈ S for some ∈ (  ) if v f B+ ∆ = in  v ∂ v v ∂ v B′ ≥ , − xn ≥ , xn =  on .

Nonzero Obstacle φ: Normalization

Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C,: u B+ ∆ =  in  u φ ∂ u u φ ∂ u B′ ≥ , − xn ≥ , ( − ) xn =  on .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÉ/ìÕ f f ′φ L∞ B+ en it will be the class S with = ∆ ∈ (  ). Denition (Class S f ) v f f L∞ B+ We say that ∈ S for some ∈ (  ) if v f B+ ∆ = in  v ∂ v v ∂ v B′ ≥ , − xn ≥ , xn =  on .

Nonzero Obstacle φ: Normalization

Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C,: u B+ ∆ =  in  u φ ∂ u u φ ∂ u B′ ≥ , − xn ≥ , ( − ) xn =  on .

Consider the dišerence

′ v(x) = u(x) − φ(x ).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÉ/ìÕ Nonzero Obstacle φ: Normalization

Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C,: u B+ ∆ =  in  u φ ∂ u u φ ∂ u B′ ≥ , − xn ≥ , ( − ) xn =  on .

Consider the dišerence

′ v(x) = u(x) − φ(x ).

f f ′φ L∞ B+ en it will be the class S with = ∆ ∈ (  ). Denition (Class S f ) v f f L∞ B+ We say that ∈ S for some ∈ (  ) if v f B+ ∆ = in  v ∂ v v ∂ v B′ ≥ , − xn ≥ , xn =  on .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ÕÉ/ìÕ Originally due to [CZ€€Z§u†-SZ«Z-S†êu«±§u óþþ˜] in the thin obstacle ′ problem, under the additional assumption S f (x)S ≤ CSx S. In this form, essentially in [ᏼ.-T™ óþÕþ]. Proof consists in estimating the error terms. e truncation of the growth is needed to absorb those terms.

Nonzero Obstacle φ: Truncated Frequency Function

eorem (Monotonicity of truncated frequency)

Let v f . en for any δ there exists C C f δ such that ∈ S >  = (Y YL∞ , ) >  d r r v reCrδ v rn+−δ eCrδ ↗ ↦ Φ( , ) = dr log max ›S , + ( − ) ∂Br

for  < r < .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óþ/ìÕ In this form, essentially in [ᏼ.-T™ óþÕþ]. Proof consists in estimating the error terms. e truncation of the growth is needed to absorb those terms.

Nonzero Obstacle φ: Truncated Frequency Function

eorem (Monotonicity of truncated frequency)

Let v f . en for any δ there exists C C f δ such that ∈ S >  = (Y YL∞ , ) >  d r r v reCrδ v rn+−δ eCrδ ↗ ↦ Φ( , ) = dr log max ›S , + ( − ) ∂Br

for  < r < .

Originally due to [CZ€€Z§u†-SZ«Z-S†êu«±§u óþþ˜] in the thin obstacle ′ problem, under the additional assumption S f (x)S ≤ CSx S.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óþ/ìÕ Proof consists in estimating the error terms. e truncation of the growth is needed to absorb those terms.

Nonzero Obstacle φ: Truncated Frequency Function

eorem (Monotonicity of truncated frequency)

Let v f . en for any δ there exists C C f δ such that ∈ S >  = (Y YL∞ , ) >  d r r v reCrδ v rn+−δ eCrδ ↗ ↦ Φ( , ) = dr log max ›S , + ( − ) ∂Br

for  < r < .

Originally due to [CZ€€Z§u†-SZ«Z-S†êu«±§u óþþ˜] in the thin obstacle ′ problem, under the additional assumption S f (x)S ≤ CSx S. In this form, essentially in [ᏼ.-T™ óþÕþ].

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óþ/ìÕ Nonzero Obstacle φ: Truncated Frequency Function

eorem (Monotonicity of truncated frequency)

Let v f . en for any δ there exists C C f δ such that ∈ S >  = (Y YL∞ , ) >  d r r v reCrδ v rn+−δ eCrδ ↗ ↦ Φ( , ) = dr log max ›S , + ( − ) ∂Br

for  < r < .

Originally due to [CZ€€Z§u†-SZ«Z-S†êu«±§u óþþ˜] in the thin obstacle ′ problem, under the additional assumption S f (x)S ≤ CSx S. In this form, essentially in [ᏼ.-T™ óþÕþ]. Proof consists in estimating the error terms. e truncation of the growth is needed to absorb those terms.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óþ/ìÕ eorem ([P™™• ÕÉÉä]) Let u be a caloric function (solution of the heat equation) in the strip S n R . en R = ޒ × (− , ]

r u G x r dx N r u ∫t=−r S∇ S ( , ) ↗ for r R ( , ) = uG x r dx  < < . ∫t=−r ( , ) Moreover, N(r, u) ≡ κ ⇐⇒ u is parabolically homogeneous of degree κ, i.e. u(λx, λt) = λκu(x, t). − ~ −S S~ Here G(x, t) = (πt) n e x t, t >  is the heat (Gaussian) kernel.

Parabolic Case: Poon’s Monotonicity Formula

e optimal regularity in the elliptic case was obtained with the help of Almgren’s Frequency Function. So we need a parabolic analogue of the frequency.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óÕ/ìÕ − ~ −S S~ Here G(x, t) = (πt) n e x t, t >  is the heat (Gaussian) kernel.

Parabolic Case: Poon’s Monotonicity Formula

e optimal regularity in the elliptic case was obtained with the help of Almgren’s Frequency Function. So we need a parabolic analogue of the frequency.

eorem ([P™™• ÕÉÉä]) Let u be a caloric function (solution of the heat equation) in the strip S n R . en R = ޒ × (− , ]

r u G x r dx N r u ∫t=−r S∇ S ( , ) ↗ for r R ( , ) = uG x r dx  < < . ∫t=−r ( , ) Moreover, N(r, u) ≡ κ ⇐⇒ u is parabolically homogeneous of degree κ, i.e. u(λx, λt) = λκu(x, t).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óÕ/ìÕ Parabolic Case: Poon’s Monotonicity Formula

e optimal regularity in the elliptic case was obtained with the help of Almgren’s Frequency Function. So we need a parabolic analogue of the frequency.

eorem ([P™™• ÕÉÉä]) Let u be a caloric function (solution of the heat equation) in the strip S n R . en R = ޒ × (− , ]

r u G x r dx N r u ∫t=−r S∇ S ( , ) ↗ for r R ( , ) = uG x r dx  < < . ∫t=−r ( , ) Moreover, N(r, u) ≡ κ ⇐⇒ u is parabolically homogeneous of degree κ, i.e. u(λx, λt) = λκu(x, t). − ~ −S S~ Here G(x, t) = (πt) n e x t, t >  is the heat (Gaussian) kernel.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óÕ/ìÕ v S+ n We want to “extend” to the half-strip  = ޒ+ × (−, ] in the following way. η C∞ B Let ∈  ( ) be a cutoš function such that η η x η η η B = (S S),  ≤ ≤ , T = , supp ⊂ ~ B~ and consider ′ v(x, t) = [u(x, t) − φ(x , , t)]η(x). v S+ n en solves the Signorini problem in the half-strip  = ޒ+ × (−, ] with a nonzero right-hand side v ∂ v f η x ′φ ∂ φ u φ x′ t η u η ∆ − t = ∶= ( )[−∆ + t ] + [ − ( , )]∆ + ∇ ∇

Important note: the right-hand side f is nonzero even if φ ≡ .

Parabolic Case: Normalization

Suppose now u solves the Parabolic Signorini Problem in Q+ B+ B′ φ C, C,  =  × (−, ] with ᏹ =  and ∈ x ∩ t .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óó/ìÕ η C∞ B Let ∈  ( ) be a cutoš function such that η η x η η η B = (S S),  ≤ ≤ , T = , supp ⊂ ~ B~ and consider ′ v(x, t) = [u(x, t) − φ(x , , t)]η(x). v S+ n en solves the Signorini problem in the half-strip  = ޒ+ × (−, ] with a nonzero right-hand side v ∂ v f η x ′φ ∂ φ u φ x′ t η u η ∆ − t = ∶= ( )[−∆ + t ] + [ − ( , )]∆ + ∇ ∇

Important note: the right-hand side f is nonzero even if φ ≡ .

Parabolic Case: Normalization

Suppose now u solves the Parabolic Signorini Problem in Q+ B+ B′ φ C, C,  =  × (−, ] with ᏹ =  and ∈ x ∩ t . v S+ n We want to “extend” to the half-strip  = ޒ+ × (−, ] in the following way.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óó/ìÕ v S+ n en solves the Signorini problem in the half-strip  = ޒ+ × (−, ] with a nonzero right-hand side v ∂ v f η x ′φ ∂ φ u φ x′ t η u η ∆ − t = ∶= ( )[−∆ + t ] + [ − ( , )]∆ + ∇ ∇

Important note: the right-hand side f is nonzero even if φ ≡ .

Parabolic Case: Normalization

Suppose now u solves the Parabolic Signorini Problem in Q+ B+ B′ φ C, C,  =  × (−, ] with ᏹ =  and ∈ x ∩ t . v S+ n We want to “extend” to the half-strip  = ޒ+ × (−, ] in the following way. η C∞ B Let ∈  ( ) be a cutoš function such that η η x η η η B = (S S),  ≤ ≤ , T = , supp ⊂ ~ B~ and consider ′ v(x, t) = [u(x, t) − φ(x , , t)]η(x).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óó/ìÕ Important note: the right-hand side f is nonzero even if φ ≡ .

Parabolic Case: Normalization

Suppose now u solves the Parabolic Signorini Problem in Q+ B+ B′ φ C, C,  =  × (−, ] with ᏹ =  and ∈ x ∩ t . v S+ n We want to “extend” to the half-strip  = ޒ+ × (−, ] in the following way. η C∞ B Let ∈  ( ) be a cutoš function such that η η x η η η B = (S S),  ≤ ≤ , T = , supp ⊂ ~ B~ and consider ′ v(x, t) = [u(x, t) − φ(x , , t)]η(x). v S+ n en solves the Signorini problem in the half-strip  = ޒ+ × (−, ] with a nonzero right-hand side v ∂ v f η x ′φ ∂ φ u φ x′ t η u η ∆ − t = ∶= ( )[−∆ + t ] + [ − ( , )]∆ + ∇ ∇

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óó/ìÕ Parabolic Case: Normalization

Suppose now u solves the Parabolic Signorini Problem in Q+ B+ B′ φ C, C,  =  × (−, ] with ᏹ =  and ∈ x ∩ t . v S+ n We want to “extend” to the half-strip  = ޒ+ × (−, ] in the following way. η C∞ B Let ∈  ( ) be a cutoš function such that η η x η η η B = (S S),  ≤ ≤ , T = , supp ⊂ ~ B~ and consider ′ v(x, t) = [u(x, t) − φ(x , , t)]η(x). v S+ n en solves the Signorini problem in the half-strip  = ޒ+ × (−, ] with a nonzero right-hand side v ∂ v f η x ′φ ∂ φ u φ x′ t η u η ∆ − t = ∶= ( )[−∆ + t ] + [ − ( , )]∆ + ∇ ∇

Important note: the right-hand side f is nonzero even if φ ≡ .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óó/ìÕ Poon’s frequency is now given by i r N r u u(− ) ( , ) = h r . u(− ) i h For our generalization, however, u and u are too irregular and we have to average them to regain missing regularity:  H r  h t dt  u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + ( ) ( − ) −r Sr  I r  i t dt  t u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + S SS∇ ( )S ( − ) −r Sr

Averaged and Truncated Poon’s Formula

For the extended u dene h t u x, t G x, t dx u( ) = S n ( ) ( − ) ޒ+ i t t u x, t G x, t dx, u( ) = − S n S∇ ( )S ( − ) ޒ+

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óì/ìÕ i h For our generalization, however, u and u are too irregular and we have to average them to regain missing regularity:  H r  h t dt  u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + ( ) ( − ) −r Sr  I r  i t dt  t u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + S SS∇ ( )S ( − ) −r Sr

Averaged and Truncated Poon’s Formula

For the extended u dene h t u x, t G x, t dx u( ) = S n ( ) ( − ) ޒ+ i t t u x, t G x, t dx, u( ) = − S n S∇ ( )S ( − ) ޒ+ Poon’s frequency is now given by i r N r u u(− ) ( , ) = h r . u(− )

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óì/ìÕ Averaged and Truncated Poon’s Formula

For the extended u dene h t u x, t G x, t dx u( ) = S n ( ) ( − ) ޒ+ i t t u x, t G x, t dx, u( ) = − S n S∇ ( )S ( − ) ޒ+ Poon’s frequency is now given by i r N r u u(− ) ( , ) = h r . u(− ) i h For our generalization, however, u and u are too irregular and we have to average them to regain missing regularity:  H r  h t dt  u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + ( ) ( − ) −r Sr  I r  i t dt  t u x t G x t dxdt u u , , ( ) = r S  ( ) = r S + S SS∇ ( )S ( − ) −r Sr

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óì/ìÕ Using this generalized frequency formula, as well as an estimation on parabolic homogeneity of blowups we obtain the optimal regularity.

Averaged and Truncated Poon’s Formula

eorem ([DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let v f S+ . en for any δ there exist C such that ∈ S (  ) >  d r v  reCrδ H r r−δ  eCrδ Φ( , ) = log max{ v( ), } + ( − ) ↗  dr  for  < r < .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó¦/ìÕ Averaged and Truncated Poon’s Formula

eorem ([DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let v f S+ . en for any δ there exist C such that ∈ S (  ) >  d r v  reCrδ H r r−δ  eCrδ Φ( , ) = log max{ v( ), } + ( − ) ↗  dr  for  < r < .

Using this generalized frequency formula, as well as an estimation on parabolic homogeneity of blowups we obtain the optimal regularity.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó¦/ìÕ δ u If Φu(+) <  −  then one can show that the family { r} is convergent n in suitable sense on ޒ+ × (−∞, ] to a parabolically homogeneous u solution  of the Parabolic Signorini Problem u ∂ u n ∆  − t  =  in ޒ+ × (−∞, ] u ∂ u u ∂ u n−  ≥ , − xn  ≥ ,  xn  =  on ޒ × (−∞, ] u κ  δ Parabolic homogeneity  is =  Φ(+) <  − < . Besides, because of C,α-regularity, also κ ≥  + α > . us:  < κ < .

Parabolic Rescalings and Blowups

As in the elliptic case, we consider the rescalings u(rx, rt) r f (rx, rt) u x, t , f x, t , r( ) = H r ~ r( ) = H r ~ u( ) u( ) x t S+ ޒn r for ( , ) ∈ ~r = + × (−~ , ]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó¢/ìÕ u κ  δ Parabolic homogeneity  is =  Φ(+) <  − < . Besides, because of C,α-regularity, also κ ≥  + α > . us:  < κ < .

Parabolic Rescalings and Blowups

As in the elliptic case, we consider the rescalings u(rx, rt) r f (rx, rt) u x, t , f x, t , r( ) = H r ~ r( ) = H r ~ u( ) u( ) x t S+ ޒn r for ( , ) ∈ ~r = + × (−~ , ] δ u If Φu(+) <  −  then one can show that the family { r} is convergent n in suitable sense on ޒ+ × (−∞, ] to a parabolically homogeneous u solution  of the Parabolic Signorini Problem u ∂ u n ∆  − t  =  in ޒ+ × (−∞, ] u ∂ u u ∂ u n−  ≥ , − xn  ≥ ,  xn  =  on ޒ × (−∞, ]

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó¢/ìÕ Parabolic Rescalings and Blowups

As in the elliptic case, we consider the rescalings u(rx, rt) r f (rx, rt) u x, t , f x, t , r( ) = H r ~ r( ) = H r ~ u( ) u( ) x t S+ ޒn r for ( , ) ∈ ~r = + × (−~ , ] δ u If Φu(+) <  −  then one can show that the family { r} is convergent n in suitable sense on ޒ+ × (−∞, ] to a parabolically homogeneous u solution  of the Parabolic Signorini Problem u ∂ u n ∆  − t  =  in ޒ+ × (−∞, ] u ∂ u u ∂ u n−  ≥ , − xn  ≥ ,  xn  =  on ޒ × (−∞, ] u κ  δ Parabolic homogeneity  is =  Φ(+) <  − < . Besides, because of C,α-regularity, also κ ≥  + α > . us:  < κ < .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó¢/ìÕ e proof is based on a rather deep monotonicity formula of Cašarelli to reduce it to dimension n =  and then analysing of the principal  x ޒ eigenvalues of the Ornstein-Uhlenbeck operator −∆ +  ⋅ ∇ in for the slit planes  a Ωa ∶= ޒ ∖ ((−∞, ] × {}).

Parabolically Homogeneous Global Solutions

Lemma ([DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let u be a parabolically homogeneous solution of the Parabolic Signorini  Problem in n with homogeneity κ . en necessarily κ ޒ+ × (−∞, ]  < <  = ~ and u x t C x ix ~ ( , ) = Re(  + n) , aŸer a possible rotation in ޒn−.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óä/ìÕ Parabolically Homogeneous Global Solutions

Lemma ([DZ•†u†-GZ§™€Z™-ᏼ.-T™ óþÕÕ]) Let u be a parabolically homogeneous solution of the Parabolic Signorini  Problem in n with homogeneity κ . en necessarily κ ޒ+ × (−∞, ]  < <  = ~ and u x t C x ix ~ ( , ) = Re(  + n) , aŸer a possible rotation in ޒn−.

e proof is based on a rather deep monotonicity formula of Cašarelli to reduce it to dimension n =  and then analysing of the principal  x ޒ eigenvalues of the Ornstein-Uhlenbeck operator −∆ +  ⋅ ∇ in for the slit planes  a Ωa ∶= ޒ ∖ ((−∞, ] × {}).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óä/ìÕ   H r n u G x t dxdt Cr is implies u( ) = ∫ޒ+ ( , − ) ≤ is further implies that

~ sup SuS ≤ Cr  + ( ) Qr~ x,t x t Q′ u x t for any ( , ) ∈ ~ such that ( , ) = . Using interior parabolic estimates one then obtains

u C~,~ Q+ ∇ ∈ x,t ( ~).

Proof of Optimal Regularity

δ From Lemma we obtain that Φu(+) ≥ , if Φu(+) <  −  . us, always Φu(+) ≥ .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óß/ìÕ is further implies that

~ sup SuS ≤ Cr  + ( ) Qr~ x,t x t Q′ u x t for any ( , ) ∈ ~ such that ( , ) = . Using interior parabolic estimates one then obtains

u C~,~ Q+ ∇ ∈ x,t ( ~).

Proof of Optimal Regularity

δ From Lemma we obtain that Φu(+) ≥ , if Φu(+) <  −  . us, always Φu(+) ≥ .   H r n u G x t dxdt Cr is implies u( ) = ∫ޒ+ ( , − ) ≤

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óß/ìÕ Using interior parabolic estimates one then obtains

u C~,~ Q+ ∇ ∈ x,t ( ~).

Proof of Optimal Regularity

δ From Lemma we obtain that Φu(+) ≥ , if Φu(+) <  −  . us, always Φu(+) ≥ .   H r n u G x t dxdt Cr is implies u( ) = ∫ޒ+ ( , − ) ≤ is further implies that

~ sup SuS ≤ Cr  + ( ) Qr~ x,t x t Q′ u x t for any ( , ) ∈ ~ such that ( , ) = .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óß/ìÕ Proof of Optimal Regularity

δ From Lemma we obtain that Φu(+) ≥ , if Φu(+) <  −  . us, always Φu(+) ≥ .   H r n u G x t dxdt Cr is implies u( ) = ∫ޒ+ ( , − ) ≤ is further implies that

~ sup SuS ≤ Cr  + ( ) Qr~ x,t x t Q′ u x t for any ( , ) ∈ ~ such that ( , ) = . Using interior parabolic estimates one then obtains

u C~,~ Q+ ∇ ∈ x,t ( ~).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óß/ìÕ q x′ t k en the parabolic Taylor polynomial k( , ) of degree satises φ x′ t q x′ t C x′  t ℓ~ S ( , ) − k( , )S ≤ (S S + ) .

q n− Q x t n Extend k from ޒ × ޒ to a caloric polynomial k( , ) on ޒ × ޒ: Q ∂ Q Q x′ t q x′ t ∆ k − t k = , k( , , ) = k( , ).

Consider then v x t u x t Q x t φ x′ t q x′ t η x k( , ) = [ ( , ) − k( , ) − ( , ) − k( , )] ( )

with a cutoš function η(x).

Finer Study of the Free Boundary

φ Hℓ,ℓ~ B′ ℓ k γ Assume now is in parabolic Holder¨ class x′,t ( ), with = + ≥ , k ∈ ގ,  < γ ≤ .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó˜/ìÕ q n− Q x t n Extend k from ޒ × ޒ to a caloric polynomial k( , ) on ޒ × ޒ: Q ∂ Q Q x′ t q x′ t ∆ k − t k = , k( , , ) = k( , ).

Consider then v x t u x t Q x t φ x′ t q x′ t η x k( , ) = [ ( , ) − k( , ) − ( , ) − k( , )] ( )

with a cutoš function η(x).

Finer Study of the Free Boundary

φ Hℓ,ℓ~ B′ ℓ k γ Assume now is in parabolic Holder¨ class x′,t ( ), with = + ≥ , k ∈ ގ,  < γ ≤ . q x′ t k en the parabolic Taylor polynomial k( , ) of degree satises φ x′ t q x′ t C x′  t ℓ~ S ( , ) − k( , )S ≤ (S S + ) .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó˜/ìÕ Consider then v x t u x t Q x t φ x′ t q x′ t η x k( , ) = [ ( , ) − k( , ) − ( , ) − k( , )] ( )

with a cutoš function η(x).

Finer Study of the Free Boundary

φ Hℓ,ℓ~ B′ ℓ k γ Assume now is in parabolic Holder¨ class x′,t ( ), with = + ≥ , k ∈ ގ,  < γ ≤ . q x′ t k en the parabolic Taylor polynomial k( , ) of degree satises φ x′ t q x′ t C x′  t ℓ~ S ( , ) − k( , )S ≤ (S S + ) .

q n− Q x t n Extend k from ޒ × ޒ to a caloric polynomial k( , ) on ޒ × ޒ: Q ∂ Q Q x′ t q x′ t ∆ k − t k = , k( , , ) = k( , ).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó˜/ìÕ Finer Study of the Free Boundary

φ Hℓ,ℓ~ B′ ℓ k γ Assume now is in parabolic Holder¨ class x′,t ( ), with = + ≥ , k ∈ ގ,  < γ ≤ . q x′ t k en the parabolic Taylor polynomial k( , ) of degree satises φ x′ t q x′ t C x′  t ℓ~ S ( , ) − k( , )S ≤ (S S + ) .

q n− Q x t n Extend k from ޒ × ޒ to a caloric polynomial k( , ) on ޒ × ޒ: Q ∂ Q Q x′ t q x′ t ∆ k − t k = , k( , , ) = k( , ).

Consider then v x t u x t Q x t φ x′ t q x′ t η x k( , ) = [ ( , ) − k( , ) − ( , ) − k( , )] ( )

with a cutoš function η(x).

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ó˜/ìÕ eorem (Better Truncated Monotonicity Formula,[D-G-ᏼ-T óþÕÕ]) For v as above and δ γ there exist C C such that k < = δ d (ℓ) r v  reCrδ H r rℓ−δ  eCrδ Φ ( , k) = log max{ v ( ), } + ( − ) ↗  dr k  for  < r < .

Finer Study of the Free Boundary

v + en k solves the Signorini problem in S with nonzero right-hand-side v ∂ v f S+ ∆ k − t k = k in 

with f x t C x  t (ℓ−)~ S k( , )S ≤ (S S + S S) .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óÉ/ìÕ Finer Study of the Free Boundary

v + en k solves the Signorini problem in S with nonzero right-hand-side v ∂ v f S+ ∆ k − t k = k in 

with f x t C x  t (ℓ−)~ S k( , )S ≤ (S S + S S) .

eorem (Better Truncated Monotonicity Formula,[D-G-ᏼ-T óþÕÕ]) For v as above and δ γ there exist C C such that k < = δ d (ℓ) r v  reCrδ H r rℓ−δ  eCrδ Φ ( , k) = log max{ v ( ), } + ( − ) ↗  dr k  for  < r < .

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ óÉ/ìÕ e more regular is the thin obstacle φ, the larger is ℓ, the ner we can classify free boundary points. It is conjectured that κ can take only discrete values κ = ~, ~, . . . , m − ~, . . .. As of now it is known only that there is no κ in (~, ), so κ = ~ is isolated.

Classication of Free Boundary Points

Denition (ℓ) (ℓ) v κ We say (, ) ∈ Γκ iš Φ (+, k) = .

One can show that ~ ≤ κ ≤ ℓ and therefore we have a foliation

(ℓ) Γ = Γκ ~≤κ≤ℓ

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìþ/ìÕ It is conjectured that κ can take only discrete values κ = ~, ~, . . . , m − ~, . . .. As of now it is known only that there is no κ in (~, ), so κ = ~ is isolated.

Classication of Free Boundary Points

Denition (ℓ) (ℓ) v κ We say (, ) ∈ Γκ iš Φ (+, k) = .

One can show that ~ ≤ κ ≤ ℓ and therefore we have a foliation

(ℓ) Γ = Γκ ~≤κ≤ℓ

e more regular is the thin obstacle φ, the larger is ℓ, the ner we can classify free boundary points.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìþ/ìÕ As of now it is known only that there is no κ in (~, ), so κ = ~ is isolated.

Classication of Free Boundary Points

Denition (ℓ) (ℓ) v κ We say (, ) ∈ Γκ iš Φ (+, k) = .

One can show that ~ ≤ κ ≤ ℓ and therefore we have a foliation

(ℓ) Γ = Γκ ~≤κ≤ℓ

e more regular is the thin obstacle φ, the larger is ℓ, the ner we can classify free boundary points. It is conjectured that κ can take only discrete values κ = ~, ~, . . . , m − ~, . . ..

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìþ/ìÕ Classication of Free Boundary Points

Denition (ℓ) (ℓ) v κ We say (, ) ∈ Γκ iš Φ (+, k) = .

One can show that ~ ≤ κ ≤ ℓ and therefore we have a foliation

(ℓ) Γ = Γκ ~≤κ≤ℓ

e more regular is the thin obstacle φ, the larger is ℓ, the ner we can classify free boundary points. It is conjectured that κ can take only discrete values κ = ~, ~, . . . , m − ~, . . .. As of now it is known only that there is no κ in (~, ), so κ = ~ is isolated.

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìþ/ìÕ One rst shows that  ∈ Lip(, ~) Lip ⇒ C,α follows from a special version of the Parabolic Boundary Harnack Principle by [S„† óþÕÕ].

Regularity of the Free Boundary

Regular Set e set Γ~ is known as the .

u = φ Γ ~ eorem ([DZ•†u†-GZ§™€Z™-ᏼ-T™ óþÕÕ])   Let φ H,~ Q′ . If then there ∈ ( ) (, ) ∈ Γ~ Λ exists δ such that x@I  x′′ t >  n− = ( , ) Q Q x  x′′ t Q Γ ∩ δ = Γ~ ∩ δ = { n− = ( , )} ∩ δ, where  is such that  Cα,α~. ∇ ∈ x′′,t

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìÕ/ìÕ Lip ⇒ C,α follows from a special version of the Parabolic Boundary Harnack Principle by [S„† óþÕÕ].

Regularity of the Free Boundary

Regular Set e set Γ~ is known as the .

u = φ Γ ~ eorem ([DZ•†u†-GZ§™€Z™-ᏼ-T™ óþÕÕ])   Let φ H,~ Q′ . If then there ∈ ( ) (, ) ∈ Γ~ Λ exists δ such that x@I  x′′ t >  n− = ( , ) Q Q x  x′′ t Q Γ ∩ δ = Γ~ ∩ δ = { n− = ( , )} ∩ δ, where  is such that  Cα,α~. ∇ ∈ x′′,t

One rst shows that  ∈ Lip(, ~)

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìÕ/ìÕ Regularity of the Free Boundary

Regular Set e set Γ~ is known as the .

u = φ Γ ~ eorem ([DZ•†u†-GZ§™€Z™-ᏼ-T™ óþÕÕ])   Let φ H,~ Q′ . If then there ∈ ( ) (, ) ∈ Γ~ Λ exists δ such that x@I  x′′ t >  n− = ( , ) Q Q x  x′′ t Q Γ ∩ δ = Γ~ ∩ δ = { n− = ( , )} ∩ δ, where  is such that  Cα,α~. ∇ ∈ x′′,t

One rst shows that  ∈ Lip(, ~) Lip ⇒ C,α follows from a special version of the Parabolic Boundary Harnack Principle by [S„† óþÕÕ].

Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ìÕ/ìÕ