Regularity of weakly well posed characteristic boundary value - - PowerPoint PPT Presentation

Introduction From IBVPs to BVPs The main result

Regularity of weakly well posed characteristic boundary value problems Alessandro Morando

Department of Mathematics University of Brescia Joint work with P. Secchi

Intensive Research Month

• n Hyperbolic Conservation Laws and Fluid Dynamics

Department of Mathematics, University of Parma February 1-28, 2010

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Characteristic IBVP for hyperbolic systems

Consider the problem      Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω , where Ω := Rn

+ := {x = (x1, x2, . . . , xn) ∈ Rn : x1 > 0},

QT := Ω × (0, T), ΣT := ∂Ω × (0, T) L := ∂t + n

j=1 Aj(x, t)∂xj + B(x, t), Aj, B ∈ MN×N

M = M(x, t) ∈ Md×N, rank(M) = d (maximal rank) u(x, t) ∈ RN, F(x, t) ∈ RN, f(x) ∈ RN, G(x, t) ∈ Rd

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Characteristic IBVP for hyperbolic systems

Consider the problem      Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω , where Ω := Rn

+ := {x = (x1, x2, . . . , xn) ∈ Rn : x1 > 0},

QT := Ω × (0, T), ΣT := ∂Ω × (0, T) L := ∂t + n

j=1 Aj(x, t)∂xj + B(x, t), Aj, B ∈ MN×N

M = M(x, t) ∈ Md×N, rank(M) = d (maximal rank) u(x, t) ∈ RN, F(x, t) ∈ RN, f(x) ∈ RN, G(x, t) ∈ Rd

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Characteristic boundary

The boundary ∂Ω is characteristic with constant multiplicity if the boundary matrix Aν :=

n

• j=1

Ajνj is singular with constant rank 1 ≤ r = rank Aν < N at ∂Ω (ν = ν(x)

• utward normal vector to ∂Ω).

When Ω = {x1 > 0} then Aν = −A1 |∂Ω (since ν = (−1, 0, . . . , 0)) d = #{positive eigenvalues of A1 at ∂Ω} Full regularity (existence in usual Sobolev spaces Hm(Ω)) can’t be expected, in general, because of the possible loss of normal regularity at ∂Ω. [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998].

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Characteristic boundary

The boundary ∂Ω is characteristic with constant multiplicity if the boundary matrix Aν :=

n

• j=1

Ajνj is singular with constant rank 1 ≤ r = rank Aν < N at ∂Ω (ν = ν(x)

• utward normal vector to ∂Ω).

When Ω = {x1 > 0} then Aν = −A1 |∂Ω (since ν = (−1, 0, . . . , 0)) d = #{positive eigenvalues of A1 at ∂Ω} Full regularity (existence in usual Sobolev spaces Hm(Ω)) can’t be expected, in general, because of the possible loss of normal regularity at ∂Ω. [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998].

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Characteristic boundary

The boundary ∂Ω is characteristic with constant multiplicity if the boundary matrix Aν :=

n

• j=1

Ajνj is singular with constant rank 1 ≤ r = rank Aν < N at ∂Ω (ν = ν(x)

• utward normal vector to ∂Ω).

When Ω = {x1 > 0} then Aν = −A1 |∂Ω (since ν = (−1, 0, . . . , 0)) d = #{positive eigenvalues of A1 at ∂Ω} Full regularity (existence in usual Sobolev spaces Hm(Ω)) can’t be expected, in general, because of the possible loss of normal regularity at ∂Ω. [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998].

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Generally speaking, one normal derivative (w.r.t. ∂Ω) is controlled by two tangential derivatives. Natural function space is the weighted anisotropic Sobolev space Hm

∗ (Ω) := {u ∈ L2(Ω) : Zα∂k x1u ∈ L2(Ω), |α| + 2k ≤ m},

where Zα := Zα1

1 . . . Zαn n ,

α = (α1, . . . , αn) , Z1 = x1∂x1 and Zj = ∂xj for j = 2, . . . , n, if Ω = {x1 > 0}. [Chen Shuxing, Chinese Ann. Math. 1982], [Yanagisawa & Matsumura, CMP 1991].

back to Hm

tan

back to m = 1

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition

Consider the BVP

• Lu = F ,

in {x1 > 0} , Mu = G ,

• n {x1 = 0} .

L := ∂t + n

j=1 Aj∂xj, hyperbolic operator (with

eigenvalues of constant multiplicity); Aj ∈ MN×N, j = 1, . . . , n, and det A1 = 0 (i.e. non characteristic boundary); M ∈ Md×N, rank(M) = d = #{positive eigenvalues of A1}.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition

Consider the BVP

• Lu = F ,

in {x1 > 0} , Mu = G ,

• n {x1 = 0} .

L := ∂t + n

j=1 Aj∂xj, hyperbolic operator (with

eigenvalues of constant multiplicity); Aj ∈ MN×N, j = 1, . . . , n, and det A1 = 0 (i.e. non characteristic boundary); M ∈ Md×N, rank(M) = d = #{positive eigenvalues of A1}.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

• Let u = u(x1, x′, t) (x′ = (x2, . . . , xn)) be a solution to (??) for

F = 0 and G = 0.

• Let

u = u(x1, η, τ) be Fourier-Laplace transform of u w.r.t. x′ and t respectively (η and τ dual variables of x′ and t respectively).

u solves the ODE problem

• db

u dx1 = A(η, τ)

u , x1 > 0 , M u(0) = 0 , where A(η, τ) := −(A1)−1

• τIn + i

n

• j=2

Ajηj

• .
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

• Let u = u(x1, x′, t) (x′ = (x2, . . . , xn)) be a solution to (??) for

F = 0 and G = 0.

• Let

u = u(x1, η, τ) be Fourier-Laplace transform of u w.r.t. x′ and t respectively (η and τ dual variables of x′ and t respectively).

u solves the ODE problem

• db

u dx1 = A(η, τ)

u , x1 > 0 , M u(0) = 0 , where A(η, τ) := −(A1)−1

• τIn + i

n

• j=2

Ajηj

• .
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

• Let u = u(x1, x′, t) (x′ = (x2, . . . , xn)) be a solution to (??) for

F = 0 and G = 0.

• Let

u = u(x1, η, τ) be Fourier-Laplace transform of u w.r.t. x′ and t respectively (η and τ dual variables of x′ and t respectively).

u solves the ODE problem

• db

u dx1 = A(η, τ)

u , x1 > 0 , M u(0) = 0 , where A(η, τ) := −(A1)−1

• τIn + i

n

• j=2

Ajηj

• .
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Let E−(η, τ) be the stable subspace of A(η, τ).

• Kreiss-Lopatinskii condition (KL):

kerM ∩ E−(η, τ) = {0}, ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0.

• ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0, ∃C = C(η, τ) > 0 :

|A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• Uniform Kreiss-Lopatinskii condition (UKL):

∃C > 0 : ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0 : |A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Let E−(η, τ) be the stable subspace of A(η, τ).

• Kreiss-Lopatinskii condition (KL):

kerM ∩ E−(η, τ) = {0}, ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0.

• ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0, ∃C = C(η, τ) > 0 :

|A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• Uniform Kreiss-Lopatinskii condition (UKL):

∃C > 0 : ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0 : |A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Let E−(η, τ) be the stable subspace of A(η, τ).

• Kreiss-Lopatinskii condition (KL):

kerM ∩ E−(η, τ) = {0}, ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0.

• ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0, ∃C = C(η, τ) > 0 :

|A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• Uniform Kreiss-Lopatinskii condition (UKL):

∃C > 0 : ∀(η, τ) ∈ Rn−1 × C, ℜτ > 0 : |A1V | ≤ C|MV | ∀V ∈ E−(η, τ).

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition and well posedness

1. det Aν = 0 (i.e. non characteristic boundary)

• L2−strong well posedness

(Existence + Uniqueness + (L2, L2)−estimate) ⇔ (UKL);

2. det Aν = 0 (i.e. characteristic boundary)

• L2−strong well posedness ⇒ (UKL);
• (UKL) + structural assumptions on L ⇒ L2−strong well

posedness;

• (KL) but NOT (UKL) ⇒ Weak well posedness (Energy estimates

with finite loss of regularity)?

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition and well posedness

1. det Aν = 0 (i.e. non characteristic boundary)

• L2−strong well posedness

(Existence + Uniqueness + (L2, L2)−estimate) ⇔ (UKL);

2. det Aν = 0 (i.e. characteristic boundary)

• L2−strong well posedness ⇒ (UKL);
• (UKL) + structural assumptions on L ⇒ L2−strong well

posedness;

• (KL) but NOT (UKL) ⇒ Weak well posedness (Energy estimates

with finite loss of regularity)?

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition and well posedness

1. det Aν = 0 (i.e. non characteristic boundary)

• L2−strong well posedness

(Existence + Uniqueness + (L2, L2)−estimate) ⇔ (UKL);

2. det Aν = 0 (i.e. characteristic boundary)

• L2−strong well posedness ⇒ (UKL);
• (UKL) + structural assumptions on L ⇒ L2−strong well

posedness;

• (KL) but NOT (UKL) ⇒ Weak well posedness (Energy estimates

with finite loss of regularity)?

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition and well posedness

1. det Aν = 0 (i.e. non characteristic boundary)

• L2−strong well posedness

(Existence + Uniqueness + (L2, L2)−estimate) ⇔ (UKL);

2. det Aν = 0 (i.e. characteristic boundary)

• L2−strong well posedness ⇒ (UKL);
• (UKL) + structural assumptions on L ⇒ L2−strong well

posedness;

• (KL) but NOT (UKL) ⇒ Weak well posedness (Energy estimates

with finite loss of regularity)?

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Kreiss-Lopatinskii condition and well posedness

1. det Aν = 0 (i.e. non characteristic boundary)

• L2−strong well posedness

(Existence + Uniqueness + (L2, L2)−estimate) ⇔ (UKL);

2. det Aν = 0 (i.e. characteristic boundary)

• L2−strong well posedness ⇒ (UKL);
• (UKL) + structural assumptions on L ⇒ L2−strong well

posedness;

• (KL) but NOT (UKL) ⇒ Weak well posedness (Energy estimates

with finite loss of regularity)?

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Compressible vortex sheets

Characteristic free boundary value problems for piecewise smooth solutions: 2D vortex sheets for compressible Euler equations:

• ∂tρ + ∇x · (ρ u) = 0 ,

∂t(ρ u) + ∇x · (ρ u ⊗ u) + ∇x p(ρ) = 0 , where t ≥ 0, x ∈ R2. At the unknown discontinuity front Σ = {x1 = ϕ(x2, t)} ∂tϕ = v± · ν, [p] = 0, where [p] = p+ − p− denotes the jump across it. Here the mass flux j = j± := ρ±(v± · ν − ∂tϕ) = 0 at Σ. [Coulombel & Secchi, Indiana UMJ 2004, Ann. Sci. ENS 2008]. [M. & Trebeschi, J. Hyperbolic Differ. Equ. 2008: non isentropic case]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Compressible vortex sheets

Characteristic free boundary value problems for piecewise smooth solutions: 2D vortex sheets for compressible Euler equations:

• ∂tρ + ∇x · (ρ u) = 0 ,

∂t(ρ u) + ∇x · (ρ u ⊗ u) + ∇x p(ρ) = 0 , where t ≥ 0, x ∈ R2. At the unknown discontinuity front Σ = {x1 = ϕ(x2, t)} ∂tϕ = v± · ν, [p] = 0, where [p] = p+ − p− denotes the jump across it. Here the mass flux j = j± := ρ±(v± · ν − ∂tϕ) = 0 at Σ. [Coulombel & Secchi, Indiana UMJ 2004, Ann. Sci. ENS 2008]. [M. & Trebeschi, J. Hyperbolic Differ. Equ. 2008: non isentropic case]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Compressible vortex sheets

Characteristic free boundary value problems for piecewise smooth solutions: 2D vortex sheets for compressible Euler equations:

• ∂tρ + ∇x · (ρ u) = 0 ,

∂t(ρ u) + ∇x · (ρ u ⊗ u) + ∇x p(ρ) = 0 , where t ≥ 0, x ∈ R2. At the unknown discontinuity front Σ = {x1 = ϕ(x2, t)} ∂tϕ = v± · ν, [p] = 0, where [p] = p+ − p− denotes the jump across it. Here the mass flux j = j± := ρ±(v± · ν − ∂tϕ) = 0 at Σ. [Coulombel & Secchi, Indiana UMJ 2004, Ann. Sci. ENS 2008]. [M. & Trebeschi, J. Hyperbolic Differ. Equ. 2008: non isentropic case]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

The above problem is a (non standard) characteristic free boundary value problem for a symmetrizable hyperbolic system.

• The boundary conditions are not maximally non-negative.
• For this problem the Uniform Kreiss-Lopatinskii condition (UKL) is

never satisfied. The Kreiss-Lopatinskii condition is either violated (Hadamard ill-posedness) or satisfied in weak form.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Linearized problem around a given basic state Wr,l := (ρ, vr,l, 0)T , ϕ = 0, satifying: vr − vl > 2 √ 2c .      LW := A0∂tW + A1∂1W + A2∂2W = F , in R3

+ :=]0, +∞[x1×R2 x2,t ,

B(W nc, ψ) := MW nc

|x1=0 + b

• ∂tψ

∂x2ψ

• = G
• n {0} × R2

x2,t .

[Coulombel & Secchi, Indiana UMJ 2004]. γ||W||2

L2

γ(R3 +) + ||W nc

|x1=0||2 L2

γ(R2) + ||ψ||2

H1

γ(R2)

≤ C

• 1

γ3||LW||2 L2(H1

γ) + 1

γ2 ||B(W nc |x1=0, ψ)||2 H1

γ(R2)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Linearized problem around a given basic state Wr,l := (ρ, vr,l, 0)T , ϕ = 0, satifying: vr − vl > 2 √ 2c .      LW := A0∂tW + A1∂1W + A2∂2W = F , in R3

+ :=]0, +∞[x1×R2 x2,t ,

B(W nc, ψ) := MW nc

|x1=0 + b

• ∂tψ

∂x2ψ

• = G
• n {0} × R2

x2,t .

[Coulombel & Secchi, Indiana UMJ 2004]. γ||W||2

L2

γ(R3 +) + ||W nc

|x1=0||2 L2

γ(R2) + ||ψ||2

H1

γ(R2)

≤ C

• 1

γ3||LW||2 L2(H1

γ) + 1

γ2 ||B(W nc |x1=0, ψ)||2 H1

γ(R2)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Other characteristic problems

[Sabl´ e-Tougeron, ARMA 1988]: Neumann problem for nonlinear elastodynamic equations in 2D. (KL) but NOT (UKL) γ||u||2

L2

γ(R3 +) + γ||u| x1=0||2

H−1/2

γ

(R2)

≤ C

• 1

γ ||F||2 L2

γ(R3 +) + 1

γ ||G||2 H1/2

γ

(R2)

• The same estimate holds for the

free boundary motion of 3D-compressible fluids in physical vacuum [Lindblad, Comm. Math. Phys. 2003][Coutand, Lindblad & Shkoller, 2009], [Trakhinin, 2008]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Other characteristic problems

[Sabl´ e-Tougeron, ARMA 1988]: Neumann problem for nonlinear elastodynamic equations in 2D. (KL) but NOT (UKL) γ||u||2

L2

γ(R3 +) + γ||u| x1=0||2

H−1/2

γ

(R2)

≤ C

• 1

γ ||F||2 L2

γ(R3 +) + 1

γ ||G||2 H1/2

γ

(R2)

• The same estimate holds for the

free boundary motion of 3D-compressible fluids in physical vacuum [Lindblad, Comm. Math. Phys. 2003][Coutand, Lindblad & Shkoller, 2009], [Trakhinin, 2008]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Other characteristic problems

[Sabl´ e-Tougeron, ARMA 1988]: Neumann problem for nonlinear elastodynamic equations in 2D. (KL) but NOT (UKL) γ||u||2

L2

γ(R3 +) + γ||u| x1=0||2

H−1/2

γ

(R2)

≤ C

• 1

γ ||F||2 L2

γ(R3 +) + 1

γ ||G||2 H1/2

γ

(R2)

• The same estimate holds for the

free boundary motion of 3D-compressible fluids in physical vacuum [Lindblad, Comm. Math. Phys. 2003][Coutand, Lindblad & Shkoller, 2009], [Trakhinin, 2008]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

Other characteristic problems

[Sabl´ e-Tougeron, ARMA 1988]: Neumann problem for nonlinear elastodynamic equations in 2D. (KL) but NOT (UKL) γ||u||2

L2

γ(R3 +) + γ||u| x1=0||2

H−1/2

γ

(R2)

≤ C

• 1

γ ||F||2 L2

γ(R3 +) + 1

γ ||G||2 H1/2

γ

(R2)

• The same estimate holds for the

free boundary motion of 3D-compressible fluids in physical vacuum [Lindblad, Comm. Math. Phys. 2003][Coutand, Lindblad & Shkoller, 2009], [Trakhinin, 2008]

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Problem of regularity

For general boundary conditions, the current theory is mainly devoted to establish sufficient conditions for the L2 well-posedness. We consider the problem of regularity: Prove the regularity of any given L2 solution, satisfying an apriori energy estimate, for sufficiently smooth data. (Independently of the structural assumptions on L and M providing the L2 well-posedness).

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Characteristic IBVP

Consider the problem      Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω , where Ω ⊂ Rn, QT = Ω × (0, T), ΣT = ∂Ω × (0, T) L := ∂t + n

j=1 Aj(x, t)∂xj + B(x, t), Aj, B ∈ MN×N

M = M(x, t) ∈ Md×N, rank(M) = d (maximal rank) u(x, t) ∈ RN, F(x, t) ∈ RN, f(x) ∈ RN, G(x, t) ∈ Rd Aj, B ∈ C∞(QT ), j = 1, . . . , n.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Existence of the L2-weak solution: Assume Aj, for j = 1, . . . , n, are fixed and regular. For all B, there exists γ0 ≥ 1 such that for all F ∈ L2(QT ), G ∈ L2(ΣT ), f ∈ L2(Ω) problem (3) admits a unique solution u ∈ C([0, T]; L2(Ω)) with Pu| ΣT ∈ L2(ΣT ). u enjoys the energy estimate for all γ ≥ γ0, 0 < τ ≤ T, e−2γτ||u(τ)||2

L2 +

τ e−2γt γ||u(t)||2

L2 + ||Pu| ∂Ω(t)||2 L2(∂Ω)

• dt

≤ C0

• ||f||2

L2 +

τ e−2γt 1 γ ||F(t)||2

L2 + ||G(t)||2 L2(∂Ω)

• dt
• .
• Cfr. [Rauch, CPAM 1972].

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Existence of the L2-weak solution: Assume Aj, for j = 1, . . . , n, are fixed and regular. For all B, there exists γ0 ≥ 1 such that for all F ∈ L2(QT ), G ∈ L2(ΣT ), f ∈ L2(Ω) problem (3) admits a unique solution u ∈ C([0, T]; L2(Ω)) with Pu| ΣT ∈ L2(ΣT ). u enjoys the energy estimate for all γ ≥ γ0, 0 < τ ≤ T, e−2γτ||u(τ)||2

L2 +

τ e−2γt γ||u(t)||2

L2 + ||Pu| ∂Ω(t)||2 L2(∂Ω)

• dt

≤ C0

• ||f||2

L2 +

τ e−2γt 1 γ ||F(t)||2

L2 + ||G(t)||2 L2(∂Ω)

• dt
• .
• Cfr. [Rauch, CPAM 1972].

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Existence of the L2-weak solution: Assume Aj, for j = 1, . . . , n, are fixed and regular. For all B, there exists γ0 ≥ 1 such that for all F ∈ L2(QT ), G ∈ L2(ΣT ), f ∈ L2(Ω) problem (3) admits a unique solution u ∈ C([0, T]; L2(Ω)) with Pu| ΣT ∈ L2(ΣT ). u enjoys the energy estimate for all γ ≥ γ0, 0 < τ ≤ T, e−2γτ||u(τ)||2

L2 +

τ e−2γt γ||u(t)||2

L2 + ||Pu| ∂Ω(t)||2 L2(∂Ω)

• dt

≤ C0

• ||f||2

L2 +

τ e−2γt 1 γ ||F(t)||2

L2 + ||G(t)||2 L2(∂Ω)

• dt
• .
• Cfr. [Rauch, CPAM 1972].

back

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Regularity of strongly well posed IBVPs

Theorem (M., Secchi, Trebeschi, J. Hyperbolic Differ.

• Equ. 2009)

Let m ≥ 1 be an integer and s = max{m,

• (n + 1)/2
• + 5}.

Assume Aj ∈ CT (Hs

∗), for j = 0, . . . , n, and B ∈ CT (Hs−1 ∗

) (or B ∈ CT (Hs

∗) if m = s). For all F ∈ Hm ∗ (QT ), G ∈ Hm(ΣT ),

f ∈ Hm

∗ (Ω), with f (h) ∈ Hm−h ∗

(Ω) for h = 1, . . . , m, satisfying the compatibility conditions of order m − 1, the unique solution u to (3) belongs to CT(Hm

∗ ) and Pu| ΣT ∈ Hm(ΣT ).Moreover u enjoys the

a priori estimate ||u||CT (Hm

∗ ) + ||Pu| ΣT ||Hm(ΣT )

≤ Cm

• |||f|||m,∗ + ||F||Hm

∗ (QT ) + ||G||Hm(ΣT )

• .
• Cfr. [Tartakoff, Indiana UMJ 1972]
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Regularity of strongly well posed IBVPs

Theorem (M., Secchi, Trebeschi, J. Hyperbolic Differ.

• Equ. 2009)

Let m ≥ 1 be an integer and s = max{m,

• (n + 1)/2
• + 5}.

Assume Aj ∈ CT (Hs

∗), for j = 0, . . . , n, and B ∈ CT (Hs−1 ∗

) (or B ∈ CT (Hs

∗) if m = s). For all F ∈ Hm ∗ (QT ), G ∈ Hm(ΣT ),

f ∈ Hm

∗ (Ω), with f (h) ∈ Hm−h ∗

(Ω) for h = 1, . . . , m, satisfying the compatibility conditions of order m − 1, the unique solution u to (3) belongs to CT(Hm

∗ ) and Pu| ΣT ∈ Hm(ΣT ).Moreover u enjoys the

a priori estimate ||u||CT (Hm

∗ ) + ||Pu| ΣT ||Hm(ΣT )

≤ Cm

• |||f|||m,∗ + ||F||Hm

∗ (QT ) + ||G||Hm(ΣT )

• .
• Cfr. [Tartakoff, Indiana UMJ 1972]
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Regularity of strongly well posed IBVPs

Theorem (M., Secchi, Trebeschi, J. Hyperbolic Differ.

• Equ. 2009)

Let m ≥ 1 be an integer and s = max{m,

• (n + 1)/2
• + 5}.

Assume Aj ∈ CT (Hs

∗), for j = 0, . . . , n, and B ∈ CT (Hs−1 ∗

) (or B ∈ CT (Hs

∗) if m = s). For all F ∈ Hm ∗ (QT ), G ∈ Hm(ΣT ),

f ∈ Hm

∗ (Ω), with f (h) ∈ Hm−h ∗

(Ω) for h = 1, . . . , m, satisfying the compatibility conditions of order m − 1, the unique solution u to (3) belongs to CT(Hm

∗ ) and Pu| ΣT ∈ Hm(ΣT ).Moreover u enjoys the

a priori estimate ||u||CT (Hm

∗ ) + ||Pu| ΣT ||Hm(ΣT )

≤ Cm

• |||f|||m,∗ + ||F||Hm

∗ (QT ) + ||G||Hm(ΣT )

• .
• Cfr. [Tartakoff, Indiana UMJ 1972]
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Regularity of strongly well posed IBVPs

Theorem (M., Secchi, Trebeschi, J. Hyperbolic Differ.

• Equ. 2009)

Let m ≥ 1 be an integer and s = max{m,

• (n + 1)/2
• + 5}.

Assume Aj ∈ CT (Hs

∗), for j = 0, . . . , n, and B ∈ CT (Hs−1 ∗

) (or B ∈ CT (Hs

∗) if m = s). For all F ∈ Hm ∗ (QT ), G ∈ Hm(ΣT ),

f ∈ Hm

∗ (Ω), with f (h) ∈ Hm−h ∗

(Ω) for h = 1, . . . , m, satisfying the compatibility conditions of order m − 1, the unique solution u to (3) belongs to CT(Hm

∗ ) and Pu| ΣT ∈ Hm(ΣT ).Moreover u enjoys the

a priori estimate ||u||CT (Hm

∗ ) + ||Pu| ΣT ||Hm(ΣT )

≤ Cm

• |||f|||m,∗ + ||F||Hm

∗ (QT ) + ||G||Hm(ΣT )

• .
• Cfr. [Tartakoff, Indiana UMJ 1972]
• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

The general strategy

• 1. Study of the Homogeneous IBVP

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = 0 in Ω . Reduce the homogeneous IBVP to a stationary BVP. Study of the tangential regularity.

• 2. Study the general case

     Lu = F in QT , Mu = G

• n ΣT ,

u|t=0 = f in Ω . Full anisotropic regularity: case m = 1. Full anisotropic regularity: case m ≥ 2.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Reduction to a BVP

Reduce to the case (t = xn+1) Q := Rn+1

+

= {x1 > 0}, Σ := {x1 = 0} × Rn

x′,t,

Consider the BVP

• (γ + L)uγ = Fγ

in Q, Muγ = Gγ

• n Σ.

(1) The solution uγ to the BVP above enjoys the L2− energy estimate γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

for all γ > γ0 and fixed C0 > 0, γ0 ≥ 1.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Reduction to a BVP

Reduce to the case (t = xn+1) Q := Rn+1

+

= {x1 > 0}, Σ := {x1 = 0} × Rn

x′,t,

Consider the BVP

• (γ + L)uγ = Fγ

in Q, Muγ = Gγ

• n Σ.

(1) The solution uγ to the BVP above enjoys the L2− energy estimate γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

for all γ > γ0 and fixed C0 > 0, γ0 ≥ 1.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Problem of regularity The general strategy Reduction to a BVP

Reduction to a BVP

Reduce to the case (t = xn+1) Q := Rn+1

+

= {x1 > 0}, Σ := {x1 = 0} × Rn

x′,t,

Consider the BVP

• (γ + L)uγ = Fγ

in Q, Muγ = Gγ

• n Σ.

(1) The solution uγ to the BVP above enjoys the L2− energy estimate γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

for all γ > γ0 and fixed C0 > 0, γ0 ≥ 1.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

ր ր ||Fγ||2

L2(Hs

γ)

||Gγ||2

Hs

γ(Σ)

s ≥ 1 , where: ||Fγ||2

L2(Hs

γ) :=

• |α|≤s , α1=0

γ2(s−|α|)||∂αFγ||2

L2(Q) ,

||Gγ||2

Hs

γ(Σ) :=

• |α′|≤s

γ2(s−|α′|)||∂α′Gγ||2

L2(Σ) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

ր ր ||Fγ||2

L2(Hs

γ)

||Gγ||2

Hs

γ(Σ)

s ≥ 1 , where: ||Fγ||2

L2(Hs

γ) :=

• |α|≤s , α1=0

γ2(s−|α|)||∂αFγ||2

L2(Q) ,

||Gγ||2

Hs

γ(Σ) :=

• |α′|≤s

γ2(s−|α′|)||∂α′Gγ||2

L2(Σ) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

ր ր ||Fγ||2

L2(Hs

γ)

||Gγ||2

Hs

γ(Σ)

s ≥ 1 , where: ||Fγ||2

L2(Hs

γ) :=

• |α|≤s , α1=0

γ2(s−|α|)||∂αFγ||2

L2(Q) ,

||Gγ||2

Hs

γ(Σ) :=

• |α′|≤s

γ2(s−|α′|)||∂α′Gγ||2

L2(Σ) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

γ||uγ||2

L2(Q) + ||Puγ |Σ||2 L2(Σ) ≤ C0

1 γ ||Fγ||2

L2(Q) + ||Gγ||2 L2(Σ)

• ,

ր ր ||Fγ||2

L2(Hs

γ)

||Gγ||2

Hs

γ(Σ)

s ≥ 1 , where: ||Fγ||2

L2(Hs

γ) :=

• |α|≤s , α1=0

γ2(s−|α|)||∂αFγ||2

L2(Q) ,

||Gγ||2

Hs

γ(Σ) :=

• |α′|≤s

γ2(s−|α′|)||∂α′Gγ||2

L2(Σ) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential Regularity of the BVP

Define the ”conormal” Sobolev space Hm

tan(Q) = Hm(Q; Σ) := {u ∈ L2(Q) : Zαu ∈ L2(Q), |α| ≤ m} ,

with the norm ||u||2

Hm

tan(Q) :=

• |α|≤m

γ2(m−|α|)||Zαu||2

L2(Q).

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Regularity of weakly well posed BVPs

Assume that, for given Aj ∈ C∞

(0)(Rn+1 +

) (j = 1, ...n + 1) and suitable “lower order operators” B there exist C0 > 0 and γ0 ≥ 1, s. t. ∀ γ ≥ γ0, ∀ F ∈ L2(Hs

γ), G ∈ Hs γ(Rn) the BVP

• (γ + L + B)u = F ,

in Rn+1

+

, Mu = G ,

• n {0} × Rn

(L :=

n+1

• j=1

Aj∂j, ∂n+1 ≡ ∂t) admits a unique solution u ∈ L2(Rn+1

+

), with uI

|{x1=0} ∈ L2(Rn), s.t.

γ||u||2

L2(Rn+1

+

) + ||uI | {x1=0}||2 L2(Rn)

≤ C0

• 1

γ2s+1 ||F||2 L2(Hs

γ) +

1 γ2s ||G||2 Hs

γ(Rn)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Theorem (M., Secchi, 2009) Under all the previous assumptions, ∀ m ∈ N and ∀B ∈ C∞

(0)(Rn+1 +

), N × N−matrix-valued function, there exist Cm > 0 and γm ≥ 1 s.t. ∀ F ∈ Hs+m,m

tan,γ

(Rn+1

+

), G ∈ Hs+m

γ

(Rn), ∀ γ ≥ γm, the L2−solution u

• f (3) (with B = (multiplication by B) and data F, G) belongs to

Hm

tan,γ(Rn+1 +

), with uI

| {x1=0} ∈ Hm γ (Rn), and satisfies

γ||u||2

Hm

tan,γ(Rn+1 +

) + ||uI | {x1=0}||2 Hm

γ (Rn)

≤ Cm

• 1

γ2s+1||F||2 Hs+m,m

tan,γ

(Rn+1

+

) + 1 γ2s ||G||2 Hs+m

γ

(Rn)

• Notation:

||F||2

Hs+m,m

tan,γ

(Rn+1

+

) :=

• |α|≤s+m , 0≤α1≤m

γ2(s+m−|α|)||ZαF||2

L2(Rn+1

+

)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Introduce the (norm preserving) bijection ♯ : L2(Rn+1

+

) → L2(Rn+1) by w♯(x) := w(ex1, x′)ex1/2. For all m ∈ N, the map ♯ : Hm

tan(Rn+1 +

) → Hm(Rn+1) is a topological isomorphism.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Consider the family of norms [H¨

• rmander, 1963]

||w||2

Rn+1

+

,m−1,tan,γ,δ :=

• Rn+1 |(w♯)∧(ξ)|2λ2(m−1),γ

δ

(ξ)dξ, for 0 < δ ≤ 1, γ ≥ 1 where λm−1,γ

δ

(ξ) := (γ2 + |ξ|2)m/2(γ2 + |δξ|2)−1/2. Here (w♯)∧(ξ) denotes the Fourier transform of w♯(x) w.r.t. x. This norm is equivalent to wHm−1

tan , γ(Rn+1 +

) for each fixed 0 < δ ≤ 1.

Moreover, w ∈ Hm

tan , γ(Rn+1 +

) if and only if w ∈ Hm−1

tan , γ(Rn+1 +

) and wRn+1

+

,m−1,tan,γ,δ

remains bounded as δ ↓ 0.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Consider the family of norms [H¨

• rmander, 1963]

||w||2

Rn+1

+

,m−1,tan,γ,δ :=

• Rn+1 |(w♯)∧(ξ)|2λ2(m−1),γ

δ

(ξ)dξ, for 0 < δ ≤ 1, γ ≥ 1 where λm−1,γ

δ

(ξ) := (γ2 + |ξ|2)m/2(γ2 + |δξ|2)−1/2. Here (w♯)∧(ξ) denotes the Fourier transform of w♯(x) w.r.t. x. This norm is equivalent to wHm−1

tan , γ(Rn+1 +

) for each fixed 0 < δ ≤ 1.

Moreover, w ∈ Hm

tan , γ(Rn+1 +

) if and only if w ∈ Hm−1

tan , γ(Rn+1 +

) and wRn+1

+

,m−1,tan,γ,δ

remains bounded as δ ↓ 0.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

|w|Rn+1

+

,m−1,tan,γ,δ ≡ ||λm−1,γ δ

(Z)w||L2(Rn+1

+

)

where λm−1,γ

δ

(Z)w :=

• λm−1,γ

δ

(D)(w♯) ♯−1 λm−1,γ

δ

(D)v(x) = (2π)−(n+1)

• Rn+1 eix·ξλm−1,γ

δ

(ξ) v(ξ)dξ . “TANGENTIAL” (“CONORMAL”) OPERATOR ALGEBRA a = a(x, ξ, γ) ∈ Γm (m ∈ R) ⇔ |∂α

ξ ∂β xa(x, ξ, γ)| ≤ Cα,β(γ2 + |ξ|2)(m−|α|)/2 ∀ x, ξ ∈ Rn+1 , γ ≥ 1

For a ∈ Γm, define: a(x, Z, γ)w(x) :=

• a(x, D, γ)(w♯)

♯−1 , ∀ w ∈ C∞

(0)(Rn+1 +

)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

|w|Rn+1

+

,m−1,tan,γ,δ ≡ ||λm−1,γ δ

(Z)w||L2(Rn+1

+

)

where λm−1,γ

δ

(Z)w :=

• λm−1,γ

δ

(D)(w♯) ♯−1 λm−1,γ

δ

(D)v(x) = (2π)−(n+1)

• Rn+1 eix·ξλm−1,γ

δ

(ξ) v(ξ)dξ . “TANGENTIAL” (“CONORMAL”) OPERATOR ALGEBRA a = a(x, ξ, γ) ∈ Γm (m ∈ R) ⇔ |∂α

ξ ∂β xa(x, ξ, γ)| ≤ Cα,β(γ2 + |ξ|2)(m−|α|)/2 ∀ x, ξ ∈ Rn+1 , γ ≥ 1

For a ∈ Γm, define: a(x, Z, γ)w(x) :=

• a(x, D, γ)(w♯)

♯−1 , ∀ w ∈ C∞

(0)(Rn+1 +

)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

|w|Rn+1

+

,m−1,tan,γ,δ ≡ ||λm−1,γ δ

(Z)w||L2(Rn+1

+

)

where λm−1,γ

δ

(Z)w :=

• λm−1,γ

δ

(D)(w♯) ♯−1 λm−1,γ

δ

(D)v(x) = (2π)−(n+1)

• Rn+1 eix·ξλm−1,γ

δ

(ξ) v(ξ)dξ . “TANGENTIAL” (“CONORMAL”) OPERATOR ALGEBRA a = a(x, ξ, γ) ∈ Γm (m ∈ R) ⇔ |∂α

ξ ∂β xa(x, ξ, γ)| ≤ Cα,β(γ2 + |ξ|2)(m−|α|)/2 ∀ x, ξ ∈ Rn+1 , γ ≥ 1

For a ∈ Γm, define: a(x, Z, γ)w(x) :=

• a(x, D, γ)(w♯)

♯−1 , ∀ w ∈ C∞

(0)(Rn+1 +

)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

λm−1,γ

δ

(Z) is invertible, with inverse

• λ−m+1,γ

δ

(Z) : tangential operator with symbol

• λ−m+1,γ

δ

(ξ) := (λm−1,γ

δ

(ξ))−1 . Apply λm−1,γ

δ

(Z) to (γ + L + B)u = F, where L :=

n+1

• j=1

Aj∂j. Decompose γ + L + B = Lγ,tan

tangential derivatives

+ A1

1∂1 normal derivative

with A1

1 :=

• AI,I

1

• and det AI,I

1

= 0. Lγλm−1,γ

δ

(Z)u+[λm−1,γ

δ

(Z), Lγ,tan]u

• tangential commutator

+ [λm−1,γ

δ

(Z), A1

1∂1]

• normal commutator

u = λm−1,γ

δ

(Z)F

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

λm−1,γ

δ

(Z) is invertible, with inverse

• λ−m+1,γ

δ

(Z) : tangential operator with symbol

• λ−m+1,γ

δ

(ξ) := (λm−1,γ

δ

(ξ))−1 . Apply λm−1,γ

δ

(Z) to (γ + L + B)u = F, where L :=

n+1

• j=1

Aj∂j. Decompose γ + L + B = Lγ,tan

tangential derivatives

+ A1

1∂1 normal derivative

with A1

1 :=

• AI,I

1

• and det AI,I

1

= 0. Lγλm−1,γ

δ

(Z)u+[λm−1,γ

δ

(Z), Lγ,tan]u

• tangential commutator

+ [λm−1,γ

δ

(Z), A1

1∂1]

• normal commutator

u = λm−1,γ

δ

(Z)F

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

λm−1,γ

δ

(Z) is invertible, with inverse

• λ−m+1,γ

δ

(Z) : tangential operator with symbol

• λ−m+1,γ

δ

(ξ) := (λm−1,γ

δ

(ξ))−1 . Apply λm−1,γ

δ

(Z) to (γ + L + B)u = F, where L :=

n+1

• j=1

Aj∂j. Decompose γ + L + B = Lγ,tan

tangential derivatives

+ A1

1∂1 normal derivative

with A1

1 :=

• AI,I

1

• and det AI,I

1

= 0. Lγλm−1,γ

δ

(Z)u+[λm−1,γ

δ

(Z), Lγ,tan]u

• tangential commutator

+ [λm−1,γ

δ

(Z), A1

1∂1]

• normal commutator

u = λm−1,γ

δ

(Z)F

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

λm−1,γ

δ

(Z) is invertible, with inverse

• λ−m+1,γ

δ

(Z) : tangential operator with symbol

• λ−m+1,γ

δ

(ξ) := (λm−1,γ

δ

(ξ))−1 . Apply λm−1,γ

δ

(Z) to (γ + L + B)u = F, where L :=

n+1

• j=1

Aj∂j. Decompose γ + L + B = Lγ,tan

tangential derivatives

+ A1

1∂1 normal derivative

with A1

1 :=

• AI,I

1

• and det AI,I

1

= 0. Lγλm−1,γ

δ

(Z)u+[λm−1,γ

δ

(Z), Lγ,tan]u

• tangential commutator

+ [λm−1,γ

δ

(Z), A1

1∂1]

• normal commutator

u = λm−1,γ

δ

(Z)F

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Tangential commutator

[λm−1,γ

δ

(Z), Lγ,tan]u տ use λ−m+1,γ

δ

(Z)λm−1,γ

δ

(Z) = I [λm−1,γ

δ

(Z), Lγ,tan]u = [λm−1,γ

δ

(Z), Lγ,tan] λ−m+1,γ

δ

• =Tδ(x,Z,γ)

λm−1,γ

δ

(Z)u “Tangential” calculus + L2−continuity of zero-th order operators ⇒ Tδ(x, Z, γ) is L2−bounded operator with uniformly bounded operator norm with respect to δ and γ.

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Plan

1 Introduction

Characteristic IBVP for hyperbolic systems Characteristic free boundary problems

2 From IBVPs to BVPs

Problem of regularity The general strategy Reduction to a BVP

3 The main result

Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Normal commutator

[λm−1,γ

δ

(Z), A1

1∂1]u =

• [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

• Key step: [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI =

qδ(x, Z, γ)

• tangential operator of orderm−2

(∂1uI) Use the system (γ + L + B)u = F to write ∂1uI = (AI,I

1 )−1(F I) +

LI

tan(x, Z, γ)

• tangential derivatives

u ⇒ [λm−1,γ

δ

(Z), AI,I

1 ∂1]uI

= qδ(x, Z, γ)((AI,I

1 )−1F I) + qδ(x, Z, γ)

• rder m−2

LI

tan(x, Z, γ)

• rder 1
• λ−m+1,γ

δ

(Z)

• rder −m+1

λm−1,γ

δ

(Z ր source term ր L2 − bdd oprt, unif. w.r.t.δ, γ

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Applying λm−1,γ

δ

(Z) to brdy condition Mu = G yields Mλm−1,γ

δ

(Z)u = bm,δ(x′, Dx′, γ)G , (2) bm,δ(x′, Dx′, γ): pseudo-differential operator on the brdy Rn

x′

(x′ = (x2, . . . , xn+1)). Weak well posedness ⇒ γ||λm−1,γ

δ

(Z)u||2

L2(Rn+1

+

) + ||(λm−1,γ δ

(Z)u)I||2

L2(Rn)

≤ C0(

1 γ2s+1 ||λm−1,γ δ

(Z)F||2

L2(Hs

γ)

+

1 γ2s+1 ||qδ(x, Z, γ)((AI,I 1 )−1F I)||L2(Hs

γ)

+ 1

γ2s ||bm,δ(x′, Dx′, γ)G||2 Hs

γ(Rn)) .

(3)

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

F ∈ Hs+m,m

tan,γ

(Rn+1

+

), G ∈ Hs+m

γ

(Rn) ⇒ ||λm−1,γ

δ

(Z)F||2

L2(Hs

γ) ,

||qδ(x, Z, γ)((AI,I

1 )−1F I)||L2(Hs

γ) ,

||bm,δ(x′, Dx′, γ)G||2

Hs(Rn) uniformly bdd w.r.t. δ .

Then ||λm−1,γ

δ

(Z)u||2

L2(Rn+1

+

) ,

||(λm−1,γ

δ

(Z)u)I||2

L2(Rn) uniformly bdd w.r.t. δ

+ u ∈ Hm−1

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm−1 γ

(Rn) inductive hypothesis ⇒ u ∈ Hm

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm γ (Rn) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

F ∈ Hs+m,m

tan,γ

(Rn+1

+

), G ∈ Hs+m

γ

(Rn) ⇒ ||λm−1,γ

δ

(Z)F||2

L2(Hs

γ) ,

||qδ(x, Z, γ)((AI,I

1 )−1F I)||L2(Hs

γ) ,

||bm,δ(x′, Dx′, γ)G||2

Hs(Rn) uniformly bdd w.r.t. δ .

Then ||λm−1,γ

δ

(Z)u||2

L2(Rn+1

+

) ,

||(λm−1,γ

δ

(Z)u)I||2

L2(Rn) uniformly bdd w.r.t. δ

+ u ∈ Hm−1

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm−1 γ

(Rn) inductive hypothesis ⇒ u ∈ Hm

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm γ (Rn) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

F ∈ Hs+m,m

tan,γ

(Rn+1

+

), G ∈ Hs+m

γ

(Rn) ⇒ ||λm−1,γ

δ

(Z)F||2

L2(Hs

γ) ,

||qδ(x, Z, γ)((AI,I

1 )−1F I)||L2(Hs

γ) ,

||bm,δ(x′, Dx′, γ)G||2

Hs(Rn) uniformly bdd w.r.t. δ .

Then ||λm−1,γ

δ

(Z)u||2

L2(Rn+1

+

) ,

||(λm−1,γ

δ

(Z)u)I||2

L2(Rn) uniformly bdd w.r.t. δ

+ u ∈ Hm−1

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm−1 γ

(Rn) inductive hypothesis ⇒ u ∈ Hm

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm γ (Rn) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

F ∈ Hs+m,m

tan,γ

(Rn+1

+

), G ∈ Hs+m

γ

(Rn) ⇒ ||λm−1,γ

δ

(Z)F||2

L2(Hs

γ) ,

||qδ(x, Z, γ)((AI,I

1 )−1F I)||L2(Hs

γ) ,

||bm,δ(x′, Dx′, γ)G||2

Hs(Rn) uniformly bdd w.r.t. δ .

Then ||λm−1,γ

δ

(Z)u||2

L2(Rn+1

+

) ,

||(λm−1,γ

δ

(Z)u)I||2

L2(Rn) uniformly bdd w.r.t. δ

+ u ∈ Hm−1

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm−1 γ

(Rn) inductive hypothesis ⇒ u ∈ Hm

tan,γ(Rn+1 +

) , uI

| x1=0 ∈ Hm γ (Rn) .

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator

Thanks !!!

• A. Morando (Brescia University)

Characteristic Hyperbolic BVP’s