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Parabolic Signorini Problem Parabolic Signorini Problem Arshak Petrosyan Free Boundary Problems in Biology Math Biosciences Institute, OSU November Õ¦–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 ows through the membrane from the region of smaller concentration of solute to the region of higher Picture Source: Wikipedia concentation (osmotic pressure). e ow occurs in one direction. e ow continues until a sucient pressure builds up on the other side of the membrane (to compensate for osmotic pressure), which then shuts the ow. 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 ow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R u T pressure ∶ ΩT ∶= Ω × (, ] → R the of ᏹ ow( − ∂ )u = the chemical solution, that satises a ∆ t ΩT diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn Ω Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹT φ no ow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R u T pressure ∶ ΩT ∶= Ω × (, ] → R the of ow( − ∂ )u = the chemical solution, that satises a ∆ t ΩT diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn ᏹ ⊂ ∂Ω semipermeable part of the boundary Ω ᏹ Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no ow u T pressure ∶ ΩT ∶= Ω × (, ] → R the of ow( − ∂ )u = the chemical solution, that satises a ∆ t ΩT diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R ᏹ Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no ow ow ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R u T pressure ∶ ΩT ∶= Ω × (, ] → R the of ᏹ ( − ∂ )u = the chemical solution, that satises a ∆ t diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ ᏹ T no ow ow ΩT Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn ᏹ ⊂ ∂Ω semipermeable part of the boundary φ Ω φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R u T pressure ∶ ΩT ∶= Ω × (, ] → R the of ᏹ ( − ∂ )u = the chemical solution, that satises a ∆ t diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ì/ìÕ φ Ω ᏹ ∂ u (∆ − t ) = Mathematical Formulation: Unilateral Problem Given open Ω ⊂ Rn ᏹ ⊂ ∂Ω semipermeable part of the ᏹ boundary T no ow φ T osmotic pressure ∶ ᏹT ∶= ᏹ × (, ] → R u Ω Ω , T R the pressure of ∶ T ∶= × ( ] → ow the chemical solution, that satises a ΩT diusion equation (slightly compressible uid) u ∂ u ∆ − t = in ΩT nite permeability On ᏹT we have the following boundary conditions ( ) u φ ∂ u > ⇒ ν = (no ow) u φ ∂ u λ u φ ≤ ⇒ ν = ( − ) (ow) 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 innite permeability conditions on ᏹT ( ) ᏹ T no ow u ≥ φ ∂ u ν ≥ u φ ∂ u ( − ) ν = ow ΩT Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBPinBiology,MBI,NovóþÕÕ ¦/ìÕ no ow ow 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 innite 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 ow ow Parabolic Signorini Problem Letting λ → ∞ we obtain the following innite 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« ÕÉä]. 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« ÕÉä]. 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 [CZZ§u ÕÉßÉ] Proof in [U§Z’±«uêZ ÕÉ¢] 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 specic cases, no general results have been known for the free boundary. Parabolic Signorini Problem: Known Results eorem (“C,α-regularity” [U§Z’±«uêZ ÕÉ¢]) 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 ÕÉ¢] in the elliptic case worked also for ∞ nonhomogeneous equation ∆u = f , f ∈ L (Ω), with Signorini boundary conditions.
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