Extension symbols

N⁰:=Lq(Rⁿ⁻¹, X). For allλ∈Sθandh∈Lq(R+, Lq(Rⁿ⁻¹, X))we then derive

(T(λ)S(λ)h)(x_n) = Z ∞

0

k(λ,∇^N0, x_n, y_n)(S(λ)h)(y_n)dy_n[L

q(Rⁿ⁻¹,X)]

= Z ∞

0

Φ^−j_N(O)(λ,∇^N0)k(λ,∇^N0, x_n, y_n)h(y_n)dy_n[L

q(Rⁿ⁻¹,X)]

= (G^W[λ,Φ^−j_N(O)k]h)(xn).

Altogether, we obtainG^W[k]f =G^W[Φ^−j_N(O)k]g. The rest follows from Lemma 6.9.

Proposition 6.12. LetX be a Banach space of classHT and property(α). Letk∈KM(Sθ×Σⁿ⁻¹_δ ,O) andW:=Lp,%(R+, Lq(R+, Lq(Rⁿ⁻¹, X))). Then we get

(G^W[k])_|XM(O,0)∈L



XM(O,0),

M(O)

\

j=0

XM−j(O, j)



.

Proof. ForM = 0 the assertion follows from (6.4). LetΦ := Φ_N(O),m≤j ≤M ≥1, andf ∈X_M(O,0) be fixed. We defineN :=L_q(Rⁿ⁻¹, X),M:=L_q(Rⁿ⁻¹, L_q(R+, X))and derive

∂_n^m(G^W[k]f) =G^W[Φ^−m∂_n^mk](UΦ^m(∇^W₊⁰)U⁻¹f)

due to Lemma 6.11. It is easy to showUΦ^m(∇^W₊)U⁻¹∈L(XM(O,0), XM−m(O,0)) and Φ^−m∂_n^mk∈K0(S_θ×Σⁿ⁻¹_δ ,O).

We getG^W[Φ^−m(∂_n^mk)] ∈L(XM−m(O,0), XM−j(O,0)) due to (6.4) and X_M−m(O,0),→X_M−j(O,0).

Altogether, this yields

∂_n^mG^W[k]∈L(X_M(O,0), X_M−j(O,0)), for allm∈ {0, . . . , j}.

So we obtainG^W[k]∈L(XM(O,0), X_M−j(O, j))by Lemma 6.8.

6.2 Extension symbols

Definition 6.13(Extension symbols). LetO be a strictly positive order function andM ∈N0. Then we define

EM(Sθ×Σⁿ⁻¹_δ ,O), θ > π/2 as the set of all functionsh:Sθ×Σⁿ⁻¹_δ ×[0,∞)→Cwith

h(·,·, xn)∈H^∞(Sθ×Σⁿ⁻¹_δ ), xn≥0 andh(λ, z,·)∈C^M+1([0,∞),C)for all (λ, z)∈S_θ×Σⁿ⁻¹_δ such that

∂_n^jh(λ, z, x_n)

≤C·(Ξ_O(λ, z))^j−1 xn

for all(λ, z)∈Sθ×Σⁿ⁻¹_δ ,xn>0, andj≤M.

Example 6.14. Let ω(λ, z) := (ρλ+µ|z|²₋)^1/2,|z|−:=

−Pn−1 k=1z_k^21/2

,(ρ, µ >0), and let h(λ, z, xn) := exp(−ω(λ, z)xn), (λ, z)∈Sθ×Σⁿ⁻¹_δ , θ > π/2, xn>0.

Then we have h∈E2(S_θ×Σⁿ⁻¹_δ ,O) where O(γ) := max{1, γ/2},γ > 0. This typical symbol occurs in the solution formula of problems which are related to the heat equation.

The symbol classEM(Sθ×Σⁿ⁻¹_δ ,O)is highly related to the class of kernelsKM(Sθ×Σⁿ⁻¹_δ ,O). This is concretized by the next remark.

Remark 6.15. It is easy to see that

[(λ, z, xn, yn)7→h(λ, z, xn+yn)] ∈KM+1(Sθ×Σⁿ_δ,O), [(λ, z, x_n, y_n)7→Φ_N(O)(λ, z)h(λ, z, x_n+y_n)] ∈KM+1(S_θ×Σⁿ_δ,O), [(λ, z, xn, yn)7→(∂nh)(λ, z, xn+yn)] ∈KM(Sθ×Σⁿ_δ,O) for allh∈EM(Sθ×Σⁿ_δ,O).

Definition 6.16. LetX be a Banach space of classHT with property(α). Leth∈EM(S_θ×Σⁿ_δ,O)with M ≥1,

W :=L_p,%(R+, L_q(R+, L_q(Rⁿ⁻¹))), %≥0.

Then we define

E^W[h]f := −G^W[∂nh]f−G^W[h](∂nf)

forf ∈L_p,%(R+, H_q¹(R+, L_q(Rⁿ⁻¹))). Note that this definition is meaningful in the sense of Remark 6.15.

This definition is related to the so-called ’Volevich trick’.

Proposition 6.17. Let X be a Banach space of class HT with property (α). Leth∈EM(S_θ×Σⁿ⁻¹_δ ,O) withM ≥1and

W := Lp,%(R+, Lq(R+, Lq(Rⁿ⁻¹, X))), W⁰ := Lp,%(R+, Lq(Rⁿ⁻¹, Lq(R+, X))), W⁰⁰ := L_p,%(R+, L_q(Rⁿ⁻¹, X)).

(i) For all g∈Lp,%(R+, H_q¹(R+, Lq(Rⁿ⁻¹, X))) we have

E^W[h]g=−G^W[∂nh]g−G^W[Φ_N(O)h](UΦ⁻¹_N(O)(∇^W₊⁰)U⁻¹∂ng).

(ii) We have

[(E^W[h]g)(t)](xn) = (h(∇^W₊⁰⁰, xn)(γ⁺_0,ng))(t), a.a. t, xn>0

for all g ∈ Lp,%(R+, H_q²(R+, Lq(Rⁿ⁻¹, X))). Here γ0,n denotes the classical trace operator on H_q²(R+, Lq(Rⁿ⁻¹, X))associated withxn= 0, i.e.γ0,nu=u(xn= 0).

(iii) The operator E^W[h]has the following mapping property

E^W[h]∈L



X_M(O,0)∩X_M−1(O,1),

M

\

j=0

X_M−j(O, j)



. Proof. (i) This can be proved in the same way as in the proof of Lemma 6.10.

(ii) Letg ∈Lp,%(R+, H_q²(R+, Lq(Rⁿ⁻¹, X)))and (gj)_j∈N⊆D(R+, H_q²(R+, Lq(Rⁿ⁻¹, X))) withgj →g in Lp,%(R+, H_q²(R+, Lq(Rⁿ⁻¹, X))). Due to the representation by Fourier multipliers we obtain

−(E^W[h]gj)(t) = (T1(D^W_t )gj+T2(D_t^W)∂ngj)(t) = (F⁻¹(T1(iτ)(Fgj) +T2(iτ)(F∂ngj)))(t)

= (2π)^−1/2 Z

R

e^itτ(T₁(iτ)(Fg_j)(τ) +T₂(iτ)(∂_nFg_j)(τ))dτ_[Lq(R+,Lq(Rn−1,X))]

for almost allt >0, where

T1(λ) := (G^N[λ, ∂nh])⁺, T2(λ) := (G^N[λ, h])⁺, λ∈Sθ, N :=Lq(Rⁿ⁻¹, X).

6.2: Extension symbols 131

For almost allxn >0 andτ∈Rwe obtain (T1(iτ)(Fgj)(τ) +T2(iτ)(∂nFgj)(τ))(xn) =

Z ∞ 0

(∂nh)(iτ,∇^N, xn+yn)[(Fgj)(τ)](yn)dyn[Lq(Rⁿ⁻¹,X)]

+ Z ∞

0

h(iτ,∇^N, xn+yn)[∂n(Fgj)(τ)](yn)dyn[L_q(Rⁿ⁻¹,X)]

= Z ∞

0

∂_yn

h(iτ,∇^N, x_n+y_n)[(Fg_j)(τ)](y_n) dy_n[L

q(Rⁿ⁻¹,X)]

= −h(iτ,∇^N, x_n)γ_0,n((Fg_j)(τ))

= −h(iτ,∇^N, xn)[(F(γ_0,n⁺ gj))(τ)]

by the classical fundamental theorem of calculus. Altogether, we get [(E[h]gj)(t)](xn) = (2π)^−1/2

Z

R

e^itτh(iτ,∇^N, xn)[(F(γ_0,n⁺ gj))(τ)]dτ_[Lq(Rn−1,X)]

= (F⁻¹(h(iτ,∇^N, xn)F(γ⁺_0,ngj)))(t)

= (h(∇^W₊⁰⁰, xn)(γ_0,n⁺ gj))(t), a.a. t, xn>0

by Lemma 2.44. Due toG^W[∂nh], G^W[h]∈L(Lp,%(R⁺, Lq(R⁺, Lq(Rⁿ⁻¹, X))))we have ((E[h]g_j)(t))(x_n)→((E[h]g)(t))(x_n), j→ ∞

in Lq(Rⁿ⁻¹, X) for almost all t, xn > 0. With γ_0,n⁺ gj → γ_0,n⁺ g in Lp,%(R+, Lq(Rⁿ⁻¹, X)) and the boundedness ofh(∇^W₊⁰⁰, x_n)we obtain

h(∇^W₊⁰⁰, xn)(γ_0,n⁺ gj)→h(∇^W₊⁰⁰, xn)(γ_0,n⁺ g), j→ ∞ inLp,%(R+, Lq(Rⁿ⁻¹, X)).

(iii) Remark 6.15, Proposition 6.12, and the trivial embedding X_M(O,0)∩X_M−1(O,1) ,→X_M(O,0) already yield

G^W[∂_nh]∈L



X_M(O,0)∩X_M−1(O,1),

M

\

j=0

X_M−j(O, j)



. It is easy to see thatUΦ⁻¹_N(O)(∇^W₊⁰)U⁻¹∂n ∈L(XM−1(O,1), XM(O,0))and therefore

G^W[Φ_N(O)h](UΦ⁻¹_N(O)(∇^W₊⁰)U⁻¹∂n∈L



XM(O,0)∩X_M−1(O,1),

M

\

j=0

X_M−j(O, j)



 by Remark 6.15, Proposition 6.12, andX_M(O,0)∩X_M−1(O,1),→X_M(O,0). Using (i) we derive the assertion.

Corollary 6.18. Let ω(λ, z) := (ρλ+µ|z|²₋)^1/2,|z|− :=

−Pn−1 k=1z_k^21/2

,(ρ, µ >0), and h(λ, z, x_n) := exp(−ω(λ, z)xn)

for(λ, z)∈Sθ×Σⁿ⁻¹_δ ,θ > π/2,xn >0. We also define

W := Lp,%(R+, Lq(R+, Lq(Rⁿ⁻¹))), W⁰ := L_p,%(R+, L_q(Rⁿ⁻¹, L_q(R+))), W⁰⁰ := Lp,%(R+, Lq(Rⁿ⁻¹)).

(i) We have h∈E2(Sθ×Σⁿ⁻¹_δ ,O)whereO(γ) := max{1, γ/2} forγ >0.

(ii) The symbolhgives rise to a bounded operator

E^W[h]∈L(X).

where

X:=0H_p,%¹ (R+, Lq(Rⁿ+))∩Lp,%(R+, H_q²(Rⁿ+)).

(iii) Leten be the extension operator associated withγ0,n. Then we have E^W[h]e_n∈L(γ_0,nX,X) andE^W[h]enϕadditionally fulfills

γ0,nE^W[h]enϕ = ϕ,

γ0,n∂nE^W[h]enϕ = −ω(∇^W₊⁰⁰)ϕ

for all ϕ∈γ0,nX:=0Fpq,%^1−1/(2q)(R⁺, Lq(Rⁿ⁻¹))∩Lp,%(R⁺, B^2−1/qq (Rⁿ⁻¹)). Note that the represen-tation of the trace space as well as the existence of an extension operator follow from the discussion in Section 5.2, cf. Remark 5.22 and Proposition 5.23.

(iv) We have

∂_n²E^W[h]enϕ = −U ω²(∇^W₊⁰)U⁻¹E^W[h]enϕ

= −(∂t−∆⁰)E^W[h]enϕ for all ϕ∈γ_0,nX.

Proof. (i) This is easy to see.

(ii) According to Remark 6.15 and Proposition 6.17 (iii) we already know E^W[h]∈L



X2(O,0)∩X1(O,1),

2

\

j=0

X_2−j(O, j)



, (6.6)

where

X₀(O,2) = L_p,%(R+, H_q²(R+, L_q(Rⁿ⁻¹))),

X1(O,1) = 0H_p,%^1/2(R+, H_q¹(R+, Lq(Rⁿ⁻¹)))∩Lp,%(R+, H_q¹(R+, H_q¹(Rⁿ⁻¹))), X2(O,0) = 0H_p,%¹ (R+, Lq(R+, Lq(Rⁿ⁻¹)))∩Lp,%(R+, Lq(R+, H_q²(Rⁿ⁻¹))).

With

Lp,%(R+, H_q²(Rⁿ+)) = Lp,%(R+, H_q²(R+, Lq(Rⁿ⁻¹)))∩Lp,%(R+, Lq(R+, H_q²(Rⁿ⁻¹)))

∩Lp,%(R+, H_q¹(R+, H_q¹(Rⁿ⁻¹))), and the embedding

X,→0H_p,%^1/2(R+, H_q¹(Rⁿ+)),→0H_p,%^1/2(R+, H_q¹(R+, Lq(Rⁿ⁻¹))) (6.7) (cf. Lemma 4.5 and Remark 4.19) we get

2

\

j=0

X2−j(O, j) =X∩0H_p,%^1/2(R+, H_q¹(R+, Lq(Rⁿ⁻¹))) =X.

From (6.6) we therefore derive E^W[h] ∈ L(X₂(O,0)∩X₁(O,1),X). Again with (6.7) we obtain X,→X2(O,0)∩X1(O,1)and therefore the assertion follows from (6.6).

6.2: Extension symbols 133 (iii) Due to (ii) and en ∈ L(γ0,nX,X) we trivially have E^W[h]en ∈ L(γ0,nX,X). Due to (ii) we have

(E^W[h]e_nϕ)(t)∈C([0,∞), Lq(Rⁿ⁻¹))and we can also show h

xn7→(h(∇^W₊⁰⁰, xn)(γ⁺_0,ng))(t)i

∈C([0,∞), Lq(Rⁿ⁻¹))

withW⁰⁰:=Lp,%(R⁺, Lq(Rⁿ⁻¹, X)). According to Proposition 6.17 we have for almost allt >0 γ0,n(E^W[h]enϕ)(t) = lim

j→∞[L_q(Rⁿ⁻¹)]

[(E^W[h]enϕ)(t)](1/j)

= lim

j→∞[L_q(Rⁿ⁻¹)]

(h(∇^W₊⁰⁰,1/j)ϕ)(t)

= (h(∇^W₊⁰⁰,0)ϕ)(t) =ϕ(t).

Forf ∈Xwe obtain

∂nE^W[h]f = −G^W[Φ⁻¹_N(O)∂_n²h](UΦ_N(O)(∇^W₊⁰)U⁻¹f)−G^W[Φ⁻¹_N(O)∂nh](UΦ_N(O)(∇^W₊⁰)U⁻¹∂nf)

= G^W[ωΦ⁻¹_N(O)∂_nh](UΦ_N(O)(∇^W₊⁰)U⁻¹f) +G^W[ωΦ⁻¹_N(O)h](UΦ_N(O)(∇^W₊⁰)U⁻¹∂_nf) according to Lemma 6.11. Due to the boundedness ofω⁻¹Φ_N(O)one can show

∂nE^W[h]f = G^W[∂nh](U ω(∇^W₊⁰)U⁻¹f) +G^W[h](U ω(∇^W₊⁰)U⁻¹∂nf)

= −E^W[h](U ω(∇^W₊⁰)U⁻¹f)

as in the proof of Lemma 6.11. With γ0,nU ω(∇^W₊⁰)U⁻¹f = ω(∇^W₊⁰)γ0,nf, Proposition 6.17, and the same arguments as before we get γ⁺_0,n∂_nE^W[h]e_nϕ=−ω(∇^W₊⁰)ϕfor allϕ∈γ_0,nX. Note that the mapping properties ofω(∇^W₊⁰)are characterized by Theorem 4.6.

(iv) This can be proved by the same techniques used in the proof of part (iii).

Remark 6.19. By minor modifications the theory of this chapter can also be established for xn < 0.

Hence, we can also construct operatorsE^W−[h] on the ground spaceW− :=Lp,%(R+, Lq(Rⁿ−))) with the same properties mutatis mutandis as in Proposition 6.17 and Corollary 6.18 with

h(λ, z, xn) := exp(ω(λ, z)xn), xn<0.

Remark 6.20. For p =q the result of Corollary 6.18 can also be obtained in a vector-valued version whereX is of classHT with property(α).

Using the concepts developed in this chapter we can give a representation of the solution of the Dirichlet heat equation in Lp-Lq. One can find Lp-Lq maximal regularity results for parabolic equa-tions with inhomogeneous boundary condiequa-tions in [Wei02] and [DHP07], for instance. In [Wei02] the author considers Dirichlet and conormal boundary conditions under the restrictionq≤p. The boundary conditions in [DHP07] are more general but exhibit no time derivatives on the boundary.

The theory developed in Chapter 6 and Chapter 5 enables us to give a solution of the linearized two-phase Stefan problem in anLp-Lq-setting. The treatment of this problem can be found in Section 7.8.

Example 6.21. Let 1< p, q <∞. Consider the Lp-Lq Dirichlet heat equation







(∂t−∆)u = 0 in R+×Rⁿ+, γ_0,nu = ϕ on R+×Rⁿ⁻¹, u(t= 0) = 0 in Rⁿ+

whereϕ∈0Fpq,%^1−1/(2q)(R+, Lq(Rⁿ⁻¹))∩Lp,%(R+, Bq^2−1/q(Rⁿ⁻¹)),%≥0. Then a solution in

0H_p,%¹ (R+, Lq(Rⁿ+))∩Lp,%(R+, H_q²(Rⁿ+))is given by

u=E^W[h]enϕ, W := Lp,%(R+, Lq(R+, Lq(Rⁿ⁻¹))) wherehis given as in Corollary 6.18.

Chapter7

Application to parabolic differential equations

In this chapter we present a selection of applications of our main result on mixed order systems stated in Theorem 4.12, Theorem 5.40, Corollary 4.22, and Corollary 5.43. The applications split into two subgroups of parabolic partial differential equations. In the first part we consider problems on the whole space and in the second part we consider boundary value problems. This seems to be one of the first work which treats parabolic problems on the whole space by a direct approach without semigroup theory and a reduction to a first order system.

The applications to free boundary problems in the Sections 7.6-7.8 will show the full potential of the developed theory. For example, one can find works on N-parabolic mixed order systems associated with free boundary problems in [DSS08] and [DV08]. In [DV08] the authors present theL₂-theory of boundary value problems where the boundary operators can also depend on∂_t. They treat the associated Lopatinskii matrix as a mixed order system inL2. In [DSS08] the authors consider symbols onLp employing a joint H^∞-calculus with some restrictive conditions on the structure of the symbols. With this approach they are able to derive a solution of the problem which occurs as the reduction of the Stefan problem. Our work is a generalization and linking of both papers cited above.

Most of the considered problems have been solved in the literature before but our approach allows a unified and systematic treatment of all these problems. Furthermore, our approach yields proofs which are shorter and more direct than in the literature.

In this chapter we always search solutions of partial differential equations with vanishing time trace (i.e. u(t = 0) = 0) and vanishing right-hand side for the equations on the half space. For a suitable treatment of the associated non-linear problem the fully inhomogeneous system with non-vanishing time trace is required. In [DSS08, Theorem 4.5] R. Denk, J. Saal, and J. Seiler established the existence of an extension operator for the time trace, where the ground space is given by intersections of Sobolev-Slobodeckij spaces. Using [DSS08, Theorem 4.5] it is possible to give suitable extensions of the time traces appearing in the treatment of fully inhomogeneous systems, see the treatment of the Stefan problem in [DSS08, Section 5] for example. In general the solution of the fully inhomogeneous system can then be derived by the superposition principle.

Im Dokument General parabolic mixed order systems in Lp and applications (Seite 139-145)