Parabolic Signorini Problem
Arshak Petrosyan
Free Boundary Problems in Biology
Math Biosciences Institute, OSU
November 14–18, 2011
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
1 / 31
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).
Picture Source: Wikipedia
Because of the chemical
imbalance, the solvent flows
through the membrane from the
region of smaller concentration of
solute to the region of higher
concentation (osmotic pressure).
The flow occurs in one direction. The flow continues until a sufficient
pressure builds up on the other side of the membrane (to compensate for
osmotic pressure), which then shuts the flow. This process is known as
osmosis.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
Ω
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
M ⊂ ∂Ω semipermeable part of the
boundary
Ω
M
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
M ⊂ ∂Ω semipermeable part of the
boundary
φ
φ ∶ MT ∶= M × (, T] → R osmotic pressure
Ω
M
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
M ⊂ ∂Ω semipermeable part of the
boundary
φ
φ ∶ MT ∶= M × (, T] → R osmotic pressure
u ∶ Ω T ∶= Ω × (, T] → R the pressure of
the chemical solution, that satisfies a
diffusion equation (slightly compressible
fluid)
∆u − ∂ t u = 
Arshak Petrosyan (Purdue)
M
Ω
(∆ − ∂ t )u = 
in Ω T
Parabolic Signorini Problem
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
M ⊂ ∂Ω semipermeable part of the
boundary
φ
φ ∶ MT ∶= M × (, T] → R osmotic pressure
u ∶ Ω T ∶= Ω × (, T] → R the pressure of
the chemical solution, that satisfies a
diffusion equation (slightly compressible
fluid)
∆u − ∂ t u = 
M
Ω
(∆ − ∂ t )u = 
in Ω T
On MT we have the following boundary conditions (finite permeability)
Arshak Petrosyan (Purdue)
u>φ
⇒
∂ν u = 
u≤φ
⇒
∂ν u = λ(u − φ)
Parabolic Signorini Problem
(no flow)
(flow)
FBP in Biology, MBI, Nov 2011
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Mathematical Formulation: Unilateral Problem
Given open Ω ⊂ Rn
M ⊂ ∂Ω semipermeable part of the
boundary
MT
no flow
φ ∶ MT ∶= M × (, T] → R osmotic pressure
u ∶ Ω T ∶= Ω × (, T] → R the pressure of
the chemical solution, that satisfies a
diffusion equation (slightly compressible
fluid)
∆u − ∂ t u = 
flow
ΩT
in Ω T
On MT we have the following boundary conditions (finite permeability)
Arshak Petrosyan (Purdue)
u>φ
⇒
∂ν u = 
u≤φ
⇒
∂ν u = λ(u − φ)
Parabolic Signorini Problem
(no flow)
(flow)
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Parabolic Signorini Problem
Letting λ → ∞ we obtain the following
conditions on MT (infinite permeability)
MT
u≥φ
no flow
∂ν u ≥ 
(u − φ)∂ν u = 
flow
ΩT
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Parabolic Signorini Problem
Letting λ → ∞ we obtain the following
conditions on MT (infinite permeability)
MT
u≥φ
u>φ
∂ν u = 
∂ν u ≥ 
u=φ
(u − φ)∂ν u = 
∂ν u ≥ 
ΩT
These are known as the Signorini
boundary conditions
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Parabolic Signorini Problem
Letting λ → ∞ we obtain the following
conditions on MT (infinite permeability)
MT
u≥φ
u>φ
∂ν u = 
∂ν u ≥ 
u=φ
(u − φ)∂ν u = 
∂ν u ≥ 
ΩT
These are known as the Signorini
boundary conditions
Since u should stay above φ on MT , φ is
known as the thin obstacle. The problem
is known as Parabolic Signorini Problem
or Parabolic Thin Obstacle Problem.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Parabolic Signorini Problem
The function u(x, t) the solves the
following variational inequality:
MT
∫ ∇u ⋅ ∇(u − v) + ∂ t u(u − v) ≥ 
∂ν u = 
Ω
u ∈ K,
∂ t u ∈ L (Ω)

for all v ∈ K
where
u>φ
u=
u=φ
∂ν u ≥ 
ΩT
u = φ
K = {v ∈ W , (Ω) ∶ v∣M ≥ φ, v∣∂Ω∖M = }
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Parabolic Signorini Problem
The function u(x, t) the solves the
following variational inequality:
MT
∫ ∇u ⋅ ∇(u − v) + ∂ t u(u − v) ≥ 
∂ν u = 
Ω
u ∈ K,
∂ t u ∈ L (Ω)

for all v ∈ K
u>φ
u=
u=φ
∂ν u ≥ 
where
ΩT
u = φ
K = {v ∈ W , (Ω) ∶ v∣M ≥ φ, v∣∂Ω∖M = }
Then for any (reasonable) initial condition
u = φ
on Ω = Ω × {}
the solution exist and unique. See [Duvaut-Lions 1986].
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Free Boundary Problem
The parabolic Signorini problem is a free
boundary problem.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Free Boundary Problem
The parabolic Signorini problem is a free
boundary problem.
Let Λ ∶= {(x, t) ∈ MT ∶ u = φ} be the
so-called coincidence set. Then
Γ ∶= ∂M Λ
Γ
Λ
is the free boundary.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
6 / 31
Free Boundary Problem
The parabolic Signorini problem is a free
boundary problem.
Let Λ ∶= {(x, t) ∈ MT ∶ u = φ} be the
so-called coincidence set. Then
Γ ∶= ∂M Λ
Γ
Λ
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
FBP in Biology, MBI, Nov 2011
6 / 31
Free Boundary Problem
The parabolic Signorini problem is a free
boundary problem.
Let Λ ∶= {(x, t) ∈ MT ∶ u = φ} be the
so-called coincidence set. Then
Γ ∶= ∂M Λ
Γ
Λ
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 Ω T up to MT .
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Parabolic Signorini Problem: Known Results
Theorem (“C ,α -regularity” [Ural’tseva 1985])
Let u be a solution of the Parabolic Signorini Problem with φ ∈ C x, ∩ C t, (MT ),
α,α/
φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for any K ⋐ Ω T ∪ MT
and
∥∇u∥C α ,α/ (K) ≤ C K (∥φ∥C x, ∩C , (MT ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
Arshak Petrosyan (Purdue)
t
Parabolic Signorini Problem
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Parabolic Signorini Problem: Known Results
Theorem (“C ,α -regularity” [Ural’tseva 1985])
Let u be a solution of the Parabolic Signorini Problem with φ ∈ C x, ∩ C t, (MT ),
α,α/
φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for any K ⋐ Ω T ∪ MT
and
∥∇u∥C α ,α/ (K) ≤ C K (∥φ∥C x, ∩C , (MT ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
t
In the elliptic case a similar result has been proved by [Caffarelli 1979]
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Parabolic Signorini Problem: Known Results
Theorem (“C ,α -regularity” [Ural’tseva 1985])
Let u be a solution of the Parabolic Signorini Problem with φ ∈ C x, ∩ C t, (MT ),
α,α/
φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for any K ⋐ Ω T ∪ MT
and
∥∇u∥C α ,α/ (K) ≤ C K (∥φ∥C x, ∩C , (MT ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
t
In the elliptic case a similar result has been proved by [Caffarelli 1979]
Proof in [Ural’tseva 1985] in the elliptic case worked also for
nonhomogeneous equation ∆u = f , f ∈ L∞ (Ω), with Signorini boundary
conditions. That fact then implies the regularity in the parabolic case.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
7 / 31
Parabolic Signorini Problem: Known Results
Theorem (“C ,α -regularity” [Ural’tseva 1985])
Let u be a solution of the Parabolic Signorini Problem with φ ∈ C x, ∩ C t, (MT ),
α,α/
φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for any K ⋐ Ω T ∪ MT
and
∥∇u∥C α ,α/ (K) ≤ C K (∥φ∥C x, ∩C , (MT ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
t
In the elliptic case a similar result has been proved by [Caffarelli 1979]
Proof in [Ural’tseva 1985] in the elliptic case worked also for
nonhomogeneous equation ∆u = f , f ∈ L∞ (Ω), with Signorini boundary
conditions. That fact then implies the regularity in the parabolic case.
Except some specific cases, no general results have been known for the
free boundary.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Parabolic Signorini Problem: Optimal Regularity
In the case when M is flat, we have the following theorem.
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Parabolic Signorini Problem
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Parabolic Signorini Problem: Optimal Regularity
In the case when M is flat, we have the following theorem.
Theorem (“C ,/ -regularity” [Danielli-Garofalo-P.-To 2011])
Let u be a solution of the Parabolic Signorini Problem with flat M and
/,/
φ ∈ C x, ∩ C t, (MT ), φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for
any K ⋐ Ω T ∪ MT and
∥∇u∥C /,/ (K) ≤ C K (∥φ∥C , ∩C , (M T ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
Arshak Petrosyan (Purdue)
x ,t
t
Parabolic Signorini Problem
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Parabolic Signorini Problem: Optimal Regularity
In the case when M is flat, we have the following theorem.
Theorem (“C ,/ -regularity” [Danielli-Garofalo-P.-To 2011])
Let u be a solution of the Parabolic Signorini Problem with flat M and
/,/
φ ∈ C x, ∩ C t, (MT ), φ ∈ Lip(Ω ), and  ∈ L (ST ). Then ∇u ∈ C x,t (K) for
any K ⋐ Ω T ∪ MT and
∥∇u∥C /,/ (K) ≤ C K (∥φ∥C , ∩C , (M T ) + ∥φ ∥Lip(Ω ) + ∥∥L (ST ) )
x ,t
x ,t
t
This theorem is precise in the sense that it gives the best regularity
possible, even in time-independet case:
u(x, t) = Re(x + ix n )/
solves the Signorini problem in R+n × R with M = Rn− .
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Parabolic Signorini Problem
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Elliptic Case: Thin Obstacle Problem
Given Ω ⊂ Rn , M ⊂ ∂Ω
Ω
M
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Parabolic Signorini Problem
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Elliptic Case: Thin Obstacle Problem
Given Ω ⊂ Rn , M ⊂ ∂Ω
φ ∶ M → R (thin obstacle)
 ∶ ∂Ω ∖ M → R,  > φ on M ∩ ∂Ω.
φ
Ω
M
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Parabolic Signorini Problem
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Elliptic Case: Thin Obstacle Problem
Given Ω ⊂ Rn , M ⊂ ∂Ω
φ ∶ M → R (thin obstacle)
 ∶ ∂Ω ∖ M → R,  > φ on M ∩ ∂Ω.
Minimize the Dirichlet integral
D Ω (u) = ∫ ∣∇u∣ dx
φ
Ω
M
on the closed convex set
Ω
K = {u ∈ W , (Ω) ∣ u∣M ≥ φ, u∣∂Ω∖M = }.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Elliptic Case: Thin Obstacle Problem
Given Ω ⊂ Rn , M ⊂ ∂Ω
φ ∶ M → R (thin obstacle)
 ∶ ∂Ω ∖ M → R,  > φ on M ∩ ∂Ω.
Minimize the Dirichlet integral
u
φ
D Ω (u) = ∫ ∣∇u∣ dx
Ω
M
on the closed convex set
Ω
K = {u ∈ W , (Ω) ∣ u∣M ≥ φ, u∣∂Ω∖M = }.
The minimizer u satisfies
∆u = 
u ≥ φ,
Arshak Petrosyan (Purdue)
∂ν u ≥ ,
in Ω
(u − φ)∂ν u = 
Parabolic Signorini Problem
on M
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Elliptic Case: Optimal Regularity
The progress in parabolic case was motivated by the breakthrough result
of [Athanasopoulos-Caffarelli 2000] establishing the C ,/ regularity
in the elliptic thin obstacle problem.
Theorem
Let u be a solution of the thin obstacle problem for flat M, with φ ∈ C , (M) and
 ∈ L (∂Ω ∖ M). Then u ∈ C ,/ (K) for any K ⋐ Ω ∪ M and
∥u∥C ,/ (K) ≤ C K (∥φ∥C , (M) + ∥∥L ) .
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Elliptic Case: Optimal Regularity
The progress in parabolic case was motivated by the breakthrough result
of [Athanasopoulos-Caffarelli 2000] establishing the C ,/ regularity
in the elliptic thin obstacle problem.
Theorem
Let u be a solution of the thin obstacle problem for flat M, with φ ∈ C , (M) and
 ∈ L (∂Ω ∖ M). Then u ∈ C ,/ (K) for any K ⋐ Ω ∪ M and
∥u∥C ,/ (K) ≤ C K (∥φ∥C , (M) + ∥∥L ) .
φ = : [Athanasopoulos-Caffarelli 2000]
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Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Elliptic Case: Optimal Regularity
The progress in parabolic case was motivated by the breakthrough result
of [Athanasopoulos-Caffarelli 2000] establishing the C ,/ regularity
in the elliptic thin obstacle problem.
Theorem
Let u be a solution of the thin obstacle problem for flat M, with φ ∈ C , (M) and
 ∈ L (∂Ω ∖ M). Then u ∈ C ,/ (K) for any K ⋐ Ω ∪ M and
∥u∥C ,/ (K) ≤ C K (∥φ∥C , (M) + ∥∥L ) .
φ = : [Athanasopoulos-Caffarelli 2000]
φ ∈ C , : [Athanasopoulos-Caffarelli-Salsa 2007]
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Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
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Elliptic Case: Optimal Regularity
The progress in parabolic case was motivated by the breakthrough result
of [Athanasopoulos-Caffarelli 2000] establishing the C ,/ regularity
in the elliptic thin obstacle problem.
Theorem
Let u be a solution of the thin obstacle problem for flat M, with φ ∈ C , (M) and
 ∈ L (∂Ω ∖ M). Then u ∈ C ,/ (K) for any K ⋐ Ω ∪ M and
∥u∥C ,/ (K) ≤ C K (∥φ∥C , (M) + ∥∥L ) .
φ = : [Athanasopoulos-Caffarelli 2000]
φ ∈ C , : [Athanasopoulos-Caffarelli-Salsa 2007]
φ ∈ C , : [P-To 2010]
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Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Assume M is flat: M = Rn− × {}, φ = 
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Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Assume M is flat: M = Rn− × {}, φ = 
If u solves Signorini problem, after translation, rotation, and scaling, we
may normalize u as follows:
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Assume M is flat: M = Rn− × {}, φ = 
If u solves Signorini problem, after translation, rotation, and scaling, we
may normalize u as follows:
Definition (Class S)
We say u is a normalized solution of Signorini problem iff
∆u = 
u ≥ ,
−∂ x n u ≥ ,
in B+
u ∂xn u = 
on B′
 ∈ Γ(u) = ∂Λ(u) = ∂{u = }.
We denote the class of normalized solutions by S.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Assume M is flat: M = Rn− × {}, φ = 
If u solves Signorini problem, after translation, rotation, and scaling, we
may normalize u as follows:
Definition (Class S)
We say u is a normalized solution of Signorini problem iff
∆u = 
u ≥ ,
−∂ x n u ≥ ,
in B+
u ∂xn u = 
on B′
 ∈ Γ(u) = ∂Λ(u) = ∂{u = }.
We denote the class of normalized solutions by S.
Notation: R+n = Rn− × (, +∞),
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B+ ∶= B ∩ R+n ,
Parabolic Signorini Problem
B′ ∶= B ∩ (Rn− × {})
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Zero Obstacle φ: Normalization
Every u ∈ S can be extended from B+ to B by even symmetry
u(x ′ , −x n ) ∶= u(x ′ , x n ).
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Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Every u ∈ S can be extended from B+ to B by even symmetry
u(x ′ , −x n ) ∶= u(x ′ , x n ).
The resulting function will satisfy
∆u ≤ 
in B
∆u = 
in B ∖ Λ(u)
u ∆u = 
in B .
Here Λ(u) = {u = } ⊂ B′ .
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Zero Obstacle φ: Normalization
Every u ∈ S can be extended from B+ to B by even symmetry
u(x ′ , −x n ) ∶= u(x ′ , x n ).
The resulting function will satisfy
∆u ≤ 
in B
∆u = 
in B ∖ Λ(u)
u ∆u = 
in B .
Here Λ(u) = {u = } ⊂ B′ .
More specifically:
∆u = (∂ x n u) Hn− ∣Λ(u)
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Parabolic Signorini Problem
in D′ (B ).
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Almgren’s Frequency Function
Theorem (Monotonicity of the frequency)
Let u ∈ S. Then the frequency function
r ↦ N(r, u) ∶=
r ∫B r ∣∇u∣
∫∂B r u 
↗
for
 < r < .
Moreover, N(r, u) ≡ κ ⇐⇒ x ⋅ ∇u − κu =  in B , i.e. u is homogeneous of
degree κ in B .
[Almgren 1979] for harmonic u
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Parabolic Signorini Problem
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Almgren’s Frequency Function
Theorem (Monotonicity of the frequency)
Let u ∈ S. Then the frequency function
r ↦ N(r, u) ∶=
r ∫B r ∣∇u∣
∫∂B r u 
↗
for
 < r < .
Moreover, N(r, u) ≡ κ ⇐⇒ x ⋅ ∇u − κu =  in B , i.e. u is homogeneous of
degree κ in B .
[Almgren 1979] for harmonic u
[Garofalo-Lin 1986-87] for divergence form elliptic operators
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Parabolic Signorini Problem
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Almgren’s Frequency Function
Theorem (Monotonicity of the frequency)
Let u ∈ S. Then the frequency function
r ↦ N(r, u) ∶=
r ∫B r ∣∇u∣
∫∂B r u 
↗
for
 < r < .
Moreover, N(r, u) ≡ κ ⇐⇒ x ⋅ ∇u − κu =  in B , i.e. u is homogeneous of
degree κ in B .
[Almgren 1979] for harmonic u
[Garofalo-Lin 1986-87] for divergence form elliptic operators
[Athanasopoulos-Caffarelli-Salsa 2007] for thin obstacle problem
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Figure: Solution of the thin obstacle problem Re(x  + i∣x  ∣)/
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Figure: Multi-valued harmonic function Re(x  + ix  )/
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Rescalings and Blowups
For a solution u ∈ S and r >  consider rescalings
ur (x) ∶=
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u(rx)
( r n− ∫∂B r

Parabolic Signorini Problem

u ) 
.
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Rescalings and Blowups
For a solution u ∈ S and r >  consider rescalings
ur (x) ∶=
u(rx)
( r n− ∫∂B r


u ) 
.
The rescaling is normalized so that
∥ur ∥L (∂B ) = .
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Rescalings and Blowups
For a solution u ∈ S and r >  consider rescalings
ur (x) ∶=
u(rx)
( r n− ∫∂B r


u ) 
.
The rescaling is normalized so that
∥ur ∥L (∂B ) = .
Limits of subsequences {ur j } for some r j → + are known as blowups.
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Parabolic Signorini Problem
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Rescalings and Blowups
For a solution u ∈ S and r >  consider rescalings
ur (x) ∶=
u(rx)
( r n− ∫∂B r


u ) 
.
The rescaling is normalized so that
∥ur ∥L (∂B ) = .
Limits of subsequences {ur j } for some r j → + are known as blowups.
Generally the blowups may be different over different subsequences
r = r j → +.
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
Hence, ∃ blowup u over a sequence r j → +
u r j → u
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in W , (B )
Parabolic Signorini Problem
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
Hence, ∃ blowup u over a sequence r j → +
u r j → u
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in L (∂B )
Parabolic Signorini Problem
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
Hence, ∃ blowup u over a sequence r j → +
u r j → u
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
in Cloc
(B′ ∪ B± )
Parabolic Signorini Problem
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
Hence, ∃ blowup u over a sequence r j → +
u r j → u

in Cloc
(B′ ∪ B± )
Proposition (Homogeneity of blowups)
Let u ∈ S and the blowup u be as above. Then, u is homogeneous of degree
κ = N(+, u).
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Homogeneity of Blowups
Uniform estimates on rescalings {ur }:

∫ ∣∇ur ∣ = N(, ur ) = N(r, u) ≤ N(, u).
B
Hence, ∃ blowup u over a sequence r j → +
u r j → u

in Cloc
(B′ ∪ B± )
Proposition (Homogeneity of blowups)
Let u ∈ S and the blowup u be as above. Then, u is homogeneous of degree
κ = N(+, u).
Proof.
N(r, u ) = limr j →+ N(r, ur j ) = limr j →+ N(rr j , u) = N(+, u)
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Proof of C ,/ regularity
Lemma ([Athanasopoulos-Caffarelli 2000])
Let u be a homogeneous global solution of the thin obstacle problem with
homogeneity κ. Then κ ≥ /.
Explicit solution for which κ = / is achieved is Re(x + i∣x n ∣)/
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Proof of C ,/ regularity
Lemma ([Athanasopoulos-Caffarelli 2000])
Let u be a homogeneous global solution of the thin obstacle problem with
homogeneity κ. Then κ ≥ /.
Explicit solution for which κ = / is achieved is Re(x + i∣x n ∣)/
From Lemma we obtain that N(+, u) = κ ≥ / for any u ∈ S.
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Proof of C ,/ regularity
Lemma ([Athanasopoulos-Caffarelli 2000])
Let u be a homogeneous global solution of the thin obstacle problem with
homogeneity κ. Then κ ≥ /.
Explicit solution for which κ = / is achieved is Re(x + i∣x n ∣)/
From Lemma we obtain that N(+, u) = κ ≥ / for any u ∈ S.
From here one can show that
∫
∂B r
u  ≤ Cr n+ ,
<r<
and consequently that
u ∈ C ,/ (B±/ ∪ B′/ ).
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Nonzero Obstacle φ: Normalization
Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C , :
∆u = 
u ≥ φ,
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−∂ x n u ≥ ,
in B+
(u − φ) ∂ x n u = 
Parabolic Signorini Problem
on B′ .
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Nonzero Obstacle φ: Normalization
Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C , :
∆u = 
u ≥ φ,
−∂ x n u ≥ ,
in B+
(u − φ) ∂ x n u = 
on B′ .
Consider the difference
v(x) = u(x) − φ(x ′ ).
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Nonzero Obstacle φ: Normalization
Let now u solve the thin obstacle problem with nonzero obstacle φ ∈ C , :
∆u = 
u ≥ φ,
−∂ x n u ≥ ,
in B+
(u − φ) ∂ x n u = 
on B′ .
Consider the difference
v(x) = u(x) − φ(x ′ ).
Then it will be the class S f with f = ∆′ φ ∈ L∞ (B+ ).
Definition (Class S f )
We say that v ∈ S f for some f ∈ L∞ (B+ ) if
∆v = f
v ≥ ,
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−∂ x n v ≥ ,
in B+
v ∂xn v = 
Parabolic Signorini Problem
on B′ .
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Nonzero Obstacle φ: Truncated Frequency Function
Theorem (Monotonicity of truncated frequency)
Let v ∈ S f . Then for any δ >  there exists C = C(∥ f ∥L∞ , δ) >  such that
r ↦ Φ(r, v) = re Cr
δ
δ
d
log max {∫ v  , r n+−δ } + (e Cr − )↗
dr
∂B r
for  < r < .
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Nonzero Obstacle φ: Truncated Frequency Function
Theorem (Monotonicity of truncated frequency)
Let v ∈ S f . Then for any δ >  there exists C = C(∥ f ∥L∞ , δ) >  such that
r ↦ Φ(r, v) = re Cr
δ
δ
d
log max {∫ v  , r n+−δ } + (e Cr − )↗
dr
∂B r
for  < r < .
Originally due to [Caffarelli-Salsa-Silvestre 2008] in the thin obstacle
problem, under the additional assumption ∣ f (x)∣ ≤ C∣x ′ ∣.
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Parabolic Signorini Problem
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Nonzero Obstacle φ: Truncated Frequency Function
Theorem (Monotonicity of truncated frequency)
Let v ∈ S f . Then for any δ >  there exists C = C(∥ f ∥L∞ , δ) >  such that
r ↦ Φ(r, v) = re Cr
δ
δ
d
log max {∫ v  , r n+−δ } + (e Cr − )↗
dr
∂B r
for  < r < .
Originally due to [Caffarelli-Salsa-Silvestre 2008] in the thin obstacle
problem, under the additional assumption ∣ f (x)∣ ≤ C∣x ′ ∣.
In this form, essentially in [P.-To 2010].
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Parabolic Signorini Problem
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Nonzero Obstacle φ: Truncated Frequency Function
Theorem (Monotonicity of truncated frequency)
Let v ∈ S f . Then for any δ >  there exists C = C(∥ f ∥L∞ , δ) >  such that
r ↦ Φ(r, v) = re Cr
δ
δ
d
log max {∫ v  , r n+−δ } + (e Cr − )↗
dr
∂B r
for  < r < .
Originally due to [Caffarelli-Salsa-Silvestre 2008] in the thin obstacle
problem, under the additional assumption ∣ f (x)∣ ≤ C∣x ′ ∣.
In this form, essentially in [P.-To 2010].
Proof consists in estimating the error terms. The truncation of the growth
is needed to absorb those terms.
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Parabolic Signorini Problem
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Parabolic Case: Poon’s Monotonicity Formula
The 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.
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Parabolic Signorini Problem
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Parabolic Case: Poon’s Monotonicity Formula
The 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.
Theorem ([Poon 1996])
Let u be a caloric function (solution of the heat equation) in the strip
S R = Rn × (−R  , ]. Then
N(r, u) =
r  ∫t=−r  ∣∇u∣ G(x, r  )dx
∫t=−r  u  G(x, r  )dx
↗
for  < r < R.
Moreover, N(r, u) ≡ κ ⇐⇒ u is parabolically homogeneous of degree κ, i.e.
u(λx, λ t) = λκ u(x, t).
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Parabolic Case: Poon’s Monotonicity Formula
The 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.
Theorem ([Poon 1996])
Let u be a caloric function (solution of the heat equation) in the strip
S R = Rn × (−R  , ]. Then
N(r, u) =
r  ∫t=−r  ∣∇u∣ G(x, r  )dx
∫t=−r  u  G(x, r  )dx
↗
for  < r < R.
Moreover, N(r, u) ≡ κ ⇐⇒ u is parabolically homogeneous of degree κ, i.e.
u(λx, λ t) = λκ u(x, t).
Here G(x, t) = (πt)−n/ e −∣x∣
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 /t
, t >  is the heat (Gaussian) kernel.
Parabolic Signorini Problem
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Parabolic Case: Normalization
Suppose now u solves the Parabolic Signorini Problem in
Q+ = B+ × (−, ] with M = B′ and φ ∈ C x, ∩ C t, .
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Parabolic Signorini Problem
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Parabolic Case: Normalization
Suppose now u solves the Parabolic Signorini Problem in
Q+ = B+ × (−, ] with M = B′ and φ ∈ C x, ∩ C t, .
We want to “extend” v to the half-strip S+ = R+n × (−, ] in the following
way.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Parabolic Case: Normalization
Suppose now u solves the Parabolic Signorini Problem in
Q+ = B+ × (−, ] with M = B′ and φ ∈ C x, ∩ C t, .
We want to “extend” v to the half-strip S+ = R+n × (−, ] in the following
way.
Let η ∈ C∞ (B ) be a cutoff function such that
η = η(∣x∣),
and consider
Arshak Petrosyan (Purdue)
 ≤ η ≤ ,
η∣B
/
= ,
supp η ⊂ B/
v(x, t) = [u(x, t) − φ(x ′ , , t)]η(x).
Parabolic Signorini Problem
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Parabolic Case: Normalization
Suppose now u solves the Parabolic Signorini Problem in
Q+ = B+ × (−, ] with M = B′ and φ ∈ C x, ∩ C t, .
We want to “extend” v to the half-strip S+ = R+n × (−, ] in the following
way.
Let η ∈ C∞ (B ) be a cutoff function such that
η = η(∣x∣),
and consider
 ≤ η ≤ ,
η∣B
/
= ,
supp η ⊂ B/
v(x, t) = [u(x, t) − φ(x ′ , , t)]η(x).
Then v solves the Signorini problem in the half-strip S+ = R+n × (−, ]
with a nonzero right-hand side
∆v − ∂ t v = f ∶= η(x)[−∆′ φ + ∂ t φ] + [u − φ(x ′ , t)]∆η + ∇u∇η
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Parabolic Case: Normalization
Suppose now u solves the Parabolic Signorini Problem in
Q+ = B+ × (−, ] with M = B′ and φ ∈ C x, ∩ C t, .
We want to “extend” v to the half-strip S+ = R+n × (−, ] in the following
way.
Let η ∈ C∞ (B ) be a cutoff function such that
η = η(∣x∣),
and consider
 ≤ η ≤ ,
η∣B
/
= ,
supp η ⊂ B/
v(x, t) = [u(x, t) − φ(x ′ , , t)]η(x).
Then v solves the Signorini problem in the half-strip S+ = R+n × (−, ]
with a nonzero right-hand side
∆v − ∂ t v = f ∶= η(x)[−∆′ φ + ∂ t φ] + [u − φ(x ′ , t)]∆η + ∇u∇η
Important note: the right-hand side f is nonzero even if φ ≡ .
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Averaged and Truncated Poon’s Formula
For the extended u define
hu (t) = ∫
n
R+
u(x, t) G(x, −t)dx
iu (t) = −t ∫
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n
R+
∣∇u(x, t)∣ G(x, −t)dx,
Parabolic Signorini Problem
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Averaged and Truncated Poon’s Formula
For the extended u define
hu (t) = ∫
n
R+
u(x, t) G(x, −t)dx
iu (t) = −t ∫
n
R+
∣∇u(x, t)∣ G(x, −t)dx,
Poon’s frequency is now given by
N(r, u) =
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iu (−r  )
.
hu (−r  )
Parabolic Signorini Problem
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Averaged and Truncated Poon’s Formula
For the extended u define
hu (t) = ∫
n
R+
u(x, t) G(x, −t)dx
iu (t) = −t ∫
n
R+
∣∇u(x, t)∣ G(x, −t)dx,
Poon’s frequency is now given by
N(r, u) =
iu (−r  )
.
hu (−r  )
For our generalization, however, iu and hu are too irregular and we have
to average them to regain missing regularity:



hu (t)dt =  ∫ + u(x, t) G(x, −t)dxdt
∫


r −r
r Sr



Iu (r) =  ∫ iu (t)dt =  ∫ + ∣t∣∣∇u(x, t)∣ G(x, −t)dxdt
r −r 
r Sr
Hu (r) =
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Averaged and Truncated Poon’s Formula
Theorem ([Danielli-Garofalo-P.-To 2011])
Let v ∈ S f (S+ ). Then for any δ >  there exist C such that
δ d
δ


Φ(r, v) = re Cr
log max{Hv (r), r −δ } + (e Cr − )

dr

↗
for  < r < .
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Averaged and Truncated Poon’s Formula
Theorem ([Danielli-Garofalo-P.-To 2011])
Let v ∈ S f (S+ ). Then for any δ >  there exist C such that
δ d
δ


Φ(r, v) = re Cr
log max{Hv (r), r −δ } + (e Cr − )

dr

↗
for  < r < .
Using this generalized frequency formula, as well as an estimation on
parabolic homogeneity of blowups we obtain the optimal regularity.
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Parabolic Signorini Problem
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Parabolic Rescalings and Blowups
As in the elliptic case, we consider the rescalings
ur (x, t) =
u(rx, r  t)
,
Hu (r)/
fr (x, t) =
r  f (rx, r  t)
,
Hu (r)/
+
for (x, t) ∈ S/r
= R+n × (−/r  , ]
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Parabolic Signorini Problem
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Parabolic Rescalings and Blowups
As in the elliptic case, we consider the rescalings
ur (x, t) =
u(rx, r  t)
,
Hu (r)/
fr (x, t) =
r  f (rx, r  t)
,
Hu (r)/
+
for (x, t) ∈ S/r
= R+n × (−/r  , ]
If Φu (+) <  − δ then one can show that the family {ur } is convergent
in suitable sense on R+n × (−∞, ] to a parabolically homogeneous
solution u of the Parabolic Signorini Problem
∆u − ∂ t u = 
u ≥ ,
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−∂ x n u ≥ ,
u ∂ x n u = 
Parabolic Signorini Problem
in R+n × (−∞, ]
on Rn− × (−∞, ]
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Parabolic Rescalings and Blowups
As in the elliptic case, we consider the rescalings
ur (x, t) =
u(rx, r  t)
,
Hu (r)/
fr (x, t) =
r  f (rx, r  t)
,
Hu (r)/
+
for (x, t) ∈ S/r
= R+n × (−/r  , ]
If Φu (+) <  − δ then one can show that the family {ur } is convergent
in suitable sense on R+n × (−∞, ] to a parabolically homogeneous
solution u of the Parabolic Signorini Problem
∆u − ∂ t u = 
u ≥ ,
−∂ x n u ≥ ,
u ∂ x n u = 
in R+n × (−∞, ]
on Rn− × (−∞, ]
Parabolic homogeneity u is κ =  Φ (+) <  − δ < . Besides, because of
C ,α -regularity, also κ ≥  + α > . Thus:
 < κ < .
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Parabolic Signorini Problem
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Parabolically Homogeneous Global Solutions
Lemma ([Danielli-Garofalo-P.-To 2011])
Let u be a parabolically homogeneous solution of the Parabolic Signorini
Problem in R+n × (−∞, ] with homogeneity  < κ < . Then necessarily κ = /
and
u (x, t) = C Re(x + ix n )/ ,
after a possible rotation in Rn− .
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Parabolic Signorini Problem
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Parabolically Homogeneous Global Solutions
Lemma ([Danielli-Garofalo-P.-To 2011])
Let u be a parabolically homogeneous solution of the Parabolic Signorini
Problem in R+n × (−∞, ] with homogeneity  < κ < . Then necessarily κ = /
and
u (x, t) = C Re(x + ix n )/ ,
after a possible rotation in Rn− .
The proof is based on a rather deep monotonicity formula of Caffarelli to
reduce it to dimension n =  and then analysing of the principal
eigenvalues of the Ornstein-Uhlenbeck operator −∆ +  x ⋅ ∇ in R for the
slit planes
Ω a ∶= R ∖ ((−∞, a] × {}).
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Parabolic Signorini Problem
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Proof of Optimal Regularity
From Lemma we obtain that Φu (+) ≥ , if Φu (+) <  − δ. Thus,
always Φu (+) ≥ .
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Parabolic Signorini Problem
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Proof of Optimal Regularity
From Lemma we obtain that Φu (+) ≥ , if Φu (+) <  − δ. Thus,
always Φu (+) ≥ .
This implies Hu (r) = ∫R+n u  G(x, −t)dxdt ≤ Cr 
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
27 / 31
Proof of Optimal Regularity
From Lemma we obtain that Φu (+) ≥ , if Φu (+) <  − δ. Thus,
always Φu (+) ≥ .
This implies Hu (r) = ∫R+n u  G(x, −t)dxdt ≤ Cr 
This further implies that
sup
∣u∣ ≤ Cr /
+ (x ,t )
Q r/
 
′
for any (x , t ) ∈ Q/
such that u(x , t ) = .
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
27 / 31
Proof of Optimal Regularity
From Lemma we obtain that Φu (+) ≥ , if Φu (+) <  − δ. Thus,
always Φu (+) ≥ .
This implies Hu (r) = ∫R+n u  G(x, −t)dxdt ≤ Cr 
This further implies that
∣u∣ ≤ Cr /
sup
+ (x ,t )
Q r/
 
′
for any (x , t ) ∈ Q/
such that u(x , t ) = .
Using interior parabolic estimates one then obtains
∇u ∈ C x,t
/,/
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+
(Q/
).
Parabolic Signorini Problem
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Finer Study of the Free Boundary
ℓ,ℓ/
Assume now φ is in parabolic Hölder class H x ′ ,t (B′ ), with ℓ = k + γ ≥ ,
k ∈ N,  < γ ≤ .
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Parabolic Signorini Problem
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Finer Study of the Free Boundary
ℓ,ℓ/
Assume now φ is in parabolic Hölder class H x ′ ,t (B′ ), with ℓ = k + γ ≥ ,
k ∈ N,  < γ ≤ .
Then the parabolic Taylor polynomial q k (x ′ , t) of degree k satisfies
∣φ(x ′ , t) − q k (x ′ , t)∣ ≤ C(∣x ′ ∣ + t)ℓ/ .
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
28 / 31
Finer Study of the Free Boundary
ℓ,ℓ/
Assume now φ is in parabolic Hölder class H x ′ ,t (B′ ), with ℓ = k + γ ≥ ,
k ∈ N,  < γ ≤ .
Then the parabolic Taylor polynomial q k (x ′ , t) of degree k satisfies
∣φ(x ′ , t) − q k (x ′ , t)∣ ≤ C(∣x ′ ∣ + t)ℓ/ .
Extend q k from Rn− × R to a caloric polynomial Q k (x, t) on Rn × R:
∆Q k − ∂ t Q k = ,
Arshak Petrosyan (Purdue)
Q k (x ′ , , t) = q k (x ′ , t).
Parabolic Signorini Problem
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Finer Study of the Free Boundary
ℓ,ℓ/
Assume now φ is in parabolic Hölder class H x ′ ,t (B′ ), with ℓ = k + γ ≥ ,
k ∈ N,  < γ ≤ .
Then the parabolic Taylor polynomial q k (x ′ , t) of degree k satisfies
∣φ(x ′ , t) − q k (x ′ , t)∣ ≤ C(∣x ′ ∣ + t)ℓ/ .
Extend q k from Rn− × R to a caloric polynomial Q k (x, t) on Rn × R:
∆Q k − ∂ t Q k = ,
Q k (x ′ , , t) = q k (x ′ , t).
Consider then
v k (x, t) = [u(x, t) − Q k (x, t) − φ(x ′ , t) − q k (x ′ , t)]η(x)
with a cutoff function η(x).
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Parabolic Signorini Problem
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Finer Study of the Free Boundary
Then v k solves the Signorini problem in S+ with nonzero right-hand-side
∆v k − ∂ t v k = f k
with
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in S+
∣ f k (x, t)∣ ≤ C(∣x∣ + ∣t∣)(ℓ−)/ .
Parabolic Signorini Problem
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Finer Study of the Free Boundary
Then v k solves the Signorini problem in S+ with nonzero right-hand-side
∆v k − ∂ t v k = f k
with
in S+
∣ f k (x, t)∣ ≤ C(∣x∣ + ∣t∣)(ℓ−)/ .
Theorem (Better Truncated Monotonicity Formula,[D-G-P-T 2011])
For v k as above and δ < γ there exist C = C δ such that
δ d
δ


Φ(ℓ) (r, v k ) = re Cr
log max{Hv k (r), r ℓ−δ } + (e Cr − )

dr

↗
for  < r < .
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Parabolic Signorini Problem
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Classification of Free Boundary Points
Definition
We say (, ) ∈ Γκ
(ℓ)
iff Φ(ℓ) (+, v k ) = κ.
One can show that / ≤ κ ≤ ℓ and therefore we have a foliation
Γ=
⋃ Γκ
(ℓ)
/≤κ≤ℓ
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Parabolic Signorini Problem
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Classification of Free Boundary Points
Definition
We say (, ) ∈ Γκ
(ℓ)
iff Φ(ℓ) (+, v k ) = κ.
One can show that / ≤ κ ≤ ℓ and therefore we have a foliation
Γ=
⋃ Γκ
(ℓ)
/≤κ≤ℓ
The more regular is the thin obstacle φ, the larger is ℓ, the finer we can
classify free boundary points.
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
30 / 31
Classification of Free Boundary Points
Definition
We say (, ) ∈ Γκ
(ℓ)
iff Φ(ℓ) (+, v k ) = κ.
One can show that / ≤ κ ≤ ℓ and therefore we have a foliation
Γ=
⋃ Γκ
(ℓ)
/≤κ≤ℓ
The more regular is the thin obstacle φ, the larger is ℓ, the finer we can
classify free boundary points.
It is conjectured that κ can take only discrete values
κ = /, /, . . . , m − /, . . ..
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
FBP in Biology, MBI, Nov 2011
30 / 31
Classification of Free Boundary Points
Definition
We say (, ) ∈ Γκ
(ℓ)
iff Φ(ℓ) (+, v k ) = κ.
One can show that / ≤ κ ≤ ℓ and therefore we have a foliation
Γ=
⋃ Γκ
(ℓ)
/≤κ≤ℓ
The more regular is the thin obstacle φ, the larger is ℓ, the finer 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.
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Parabolic Signorini Problem
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Regularity of the Free Boundary
The set Γ/ is known as the Regular Set.
Theorem ([Danielli-Garofalo-P-To 2011])
Let φ ∈ H ,/ (Q′ ). If (, ) ∈ Γ/ then there
exists δ >  such that
u=φ
Γ/
Λ
I
@
x n− = (x ′′ , t)
Γ ∩ Q δ = Γ/ ∩ Q δ = {x n− = (x ′′ , t)} ∩ Q δ ,
α,α/
where  is such that ∇ ∈ C x ′′ ,t .
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Parabolic Signorini Problem
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Regularity of the Free Boundary
The set Γ/ is known as the Regular Set.
Theorem ([Danielli-Garofalo-P-To 2011])
Let φ ∈ H ,/ (Q′ ). If (, ) ∈ Γ/ then there
exists δ >  such that
u=φ
Γ/
Λ
I
@
x n− = (x ′′ , t)
Γ ∩ Q δ = Γ/ ∩ Q δ = {x n− = (x ′′ , t)} ∩ Q δ ,
α,α/
where  is such that ∇ ∈ C x ′′ ,t .
One first shows that  ∈ Lip(, /)
Arshak Petrosyan (Purdue)
Parabolic Signorini Problem
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Regularity of the Free Boundary
The set Γ/ is known as the Regular Set.
Theorem ([Danielli-Garofalo-P-To 2011])
Let φ ∈ H ,/ (Q′ ). If (, ) ∈ Γ/ then there
exists δ >  such that
u=φ
Γ/
Λ
I
@
x n− = (x ′′ , t)
Γ ∩ Q δ = Γ/ ∩ Q δ = {x n− = (x ′′ , t)} ∩ Q δ ,
α,α/
where  is such that ∇ ∈ C x ′′ ,t .
One first shows that  ∈ Lip(, /)
Lip ⇒ C ,α follows from a special version of the Parabolic Boundary
Harnack Principle by [Shi 2011].
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