� 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 ). Because of the chemical imbalance, the solvent flows through the membrane from the region of smaller concentration of solute to the region of higher Picture Source: Wikipedia concentation ( osmotic pressure ). Te flow occurs in one direction. Te 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. Tis process is known as osmosis . Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 2 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n Ω Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n � ⊂ ∂ Ω semipermeable part of the boundary Ω � Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n � ⊂ ∂ Ω semipermeable part of the boundary φ Ω φ ∶ � T ∶ = � × ( , T ] → � osmotic pressure � Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n � ⊂ ∂ Ω semipermeable part of the boundary φ Ω φ ∶ � T ∶ = � × ( , T ] → � osmotic pressure u ∶ Ω T ∶ = Ω × ( , T ] → � the pressure of � ( ∆ − ∂ t ) u = the chemical solution, that satisfies a diffusion equation ( slightly compressible fluid ) ∆ u − ∂ t u = in Ω T Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n � ⊂ ∂ Ω semipermeable part of the boundary φ Ω φ ∶ � T ∶ = � × ( , T ] → � osmotic pressure u ∶ Ω T ∶ = Ω × ( , T ] → � the pressure of � ( ∆ − ∂ t ) u = the chemical solution, that satisfies a diffusion equation ( slightly compressible fluid ) ∆ u − ∂ t u = in Ω T On � T we have the following boundary conditions ( finite permeability ) ⇒ u > φ ∂ ν u = ( no flow ) ⇒ ∂ ν u = λ ( u − φ ) u ≤ φ ( flow ) Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Mathematical Formulation: Unilateral Problem Given open Ω ⊂ � n � ⊂ ∂ Ω semipermeable part of the � T boundary no flow φ ∶ � T ∶ = � × ( , T ] → � osmotic pressure u ∶ Ω T ∶ = Ω × ( , T ] → � the pressure of flow the chemical solution, that satisfies a Ω T diffusion equation ( slightly compressible fluid ) ∆ u − ∂ t u = in Ω T On � T we have the following boundary conditions ( finite permeability ) ⇒ u > φ ∂ ν u = ( no flow ) ⇒ ∂ ν u = λ ( u − φ ) u ≤ φ ( flow ) Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 3 / 31
Parabolic Signorini Problem Letting λ → ∞ we obtain the following conditions on � T ( infinite permeability ) � T u ≥ φ no flow ∂ ν u ≥ ( u − φ ) ∂ ν u = flow Ω T Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 4 / 31
Parabolic Signorini Problem Letting λ → ∞ we obtain the following conditions on � T ( infinite permeability ) � T u ≥ φ u > φ ∂ ν u = ∂ ν u ≥ u = φ ( u − φ ) ∂ ν u = ∂ ν u ≥ Ω T Tese are known as the Signorini boundary conditions Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 4 / 31
Parabolic Signorini Problem Letting λ → ∞ we obtain the following conditions on � T ( infinite permeability ) � T u ≥ φ u > φ ∂ ν u = ∂ ν u ≥ u = φ ( u − φ ) ∂ ν u = ∂ ν u ≥ Ω T Tese are known as the Signorini boundary conditions Since u should stay above φ on � T , φ is known as the thin obstacle . Te problem is known as Parabolic Signorini Problem or Parabolic Tin Obstacle Problem . Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 4 / 31
Parabolic Signorini Problem Te function u ( x , t ) the solves the following variational inequality: � T u > φ ∫ Ω ∇ u ⋅ ∇ ( u − v ) + ∂ t u ( u − v ) ≥ ∂ ν u = u = u ∈ K , ∂ t u ∈ L ( Ω ) u = φ for all v ∈ K ∂ ν u ≥ Ω T where u = φ K = { v ∈ W , ( Ω ) ∶ v ∣ � ≥ φ , v ∣ ∂ Ω ∖ � = } Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 5 / 31
Parabolic Signorini Problem Te function u ( x , t ) the solves the following variational inequality: � T u > φ ∫ Ω ∇ u ⋅ ∇ ( u − v ) + ∂ t u ( u − v ) ≥ ∂ ν u = u = u ∈ K , ∂ t u ∈ L ( Ω ) u = φ for all v ∈ K ∂ ν u ≥ Ω T where u = φ K = { v ∈ W , ( Ω ) ∶ v ∣ � ≥ φ , v ∣ ∂ Ω ∖ � = } Ten for any (reasonable) initial condition on Ω = Ω × { } u = φ the solution exist and unique. See [ Duvau t -L ions 1986]. Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 5 / 31
Free Boundary Problem Te parabolic Signorini problem is a free boundary problem. Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 6 / 31
Free Boundary Problem Te parabolic Signorini problem is a free boundary problem. Let Λ ∶ = {( x , t ) ∈ � T ∶ u = φ } be the so-called coincidence set . Ten Γ Γ ∶ = ∂ � Λ Λ is the free boundary . Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 6 / 31
Free Boundary Problem Te parabolic Signorini problem is a free boundary problem. Let Λ ∶ = {( x , t ) ∈ � T ∶ u = φ } be the so-called coincidence set . Ten Γ Γ ∶ = ∂ � Λ 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 Te parabolic Signorini problem is a free boundary problem. Let Λ ∶ = {( x , t ) ∈ � T ∶ u = φ } be the so-called coincidence set . Ten Γ Γ ∶ = ∂ � Λ 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 � T . Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 6 / 31
Parabolic Signorini Problem: Known Results Teorem (“ C , α -regularity” [ Ural’ tseva 1985]) Let u be a solution of the Parabolic Signorini Problem with φ ∈ C , x ∩ C , t ( � T ) , φ ∈ Lip ( Ω ) , and ∈ L ( � T ) . T en ∇ u ∈ C α , α / ( K ) for any K ⋐ Ω T ∪ � T x , t and ∥∇ u ∥ C α , α / ( K ) ≤ C K (∥ φ ∥ C , t ( � T ) + ∥ φ ∥ Lip ( Ω ) + ∥ ∥ L ( � T ) ) x ∩ C , x , t Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 7 / 31
Parabolic Signorini Problem: Known Results Teorem (“ C , α -regularity” [ Ural’ tseva 1985]) Let u be a solution of the Parabolic Signorini Problem with φ ∈ C , x ∩ C , t ( � T ) , φ ∈ Lip ( Ω ) , and ∈ L ( � T ) . Ten ∇ u ∈ C α , α / ( K ) for any K ⋐ Ω T ∪ � T x , t and ∥∇ u ∥ C α , α / ( K ) ≤ C K (∥ φ ∥ C , t ( � T ) + ∥ φ ∥ Lip ( Ω ) + ∥ ∥ L ( � T ) ) x ∩ C , x , t In the elliptic case a similar result has been proved by [ C affarelli 1979] Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 7 / 31
Parabolic Signorini Problem: Known Results Teorem (“ C , α -regularity” [ Ural’ tseva 1985]) Let u be a solution of the Parabolic Signorini Problem with φ ∈ C , x ∩ C , t ( � T ) , φ ∈ Lip ( Ω ) , and ∈ L ( � T ) . Ten ∇ u ∈ C α , α / ( K ) for any K ⋐ Ω T ∪ � T x , t and ∥∇ u ∥ C α , α / ( K ) ≤ C K (∥ φ ∥ C , t ( � T ) + ∥ φ ∥ Lip ( Ω ) + ∥ ∥ L ( � T ) ) x ∩ C , x , t In the elliptic case a similar result has been proved by [ C affarelli 1979] Proof in [ U ral ’ tseva 1985] in the elliptic case worked also for nonhomogeneous equation ∆ u = f , f ∈ L ∞ ( Ω ) , with Signorini boundary conditions. Tat fact then implies the regularity in the parabolic case. Arshak Petrosyan (Purdue) Parabolic Signorini Problem FBP in Biology, MBI, Nov 2011 7 / 31
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