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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 15, Issue 1

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Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces

Valerii Los
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  • National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Prospect Peremohy 37, 03056, Kyiv-56, Ukraine
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/ Aleksandr Murach
  • Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01004, Ukraine
  • Chernihiv National Pedagogical University Het’mana Polubotka str. 53, 14013 Chernihiv, Ukraine
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Published Online: 2017-02-15 | DOI: https://doi.org/10.1515/math-2017-0008

Abstract

In Hörmander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate Hörmander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the Hörmander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.

Keywords: Parabolic initial-boundary value problem; Hörmander space; Slowly varying function; Isomorphism property; Interpolation with a function parameter

MSC 2010: 35K35; 46B70; 46E35

1 Introduction

The modern theory of general parabolic initial-boundary problems has been developed for the classical scales of Hölder-Zygmund and Sobolev function spaces [1-9]. The central result of this theory are the theorems on well-posedness by Hadamard of these problems on appropriate pairs of these spaces. For applications, especially to the spectral theory of differential operators, inner product Sobolev spaces play a special role.

In 1963 Hörmander [10] proposed a broad and meaningful generalization of the Sobolev spaces in the framework of Hilbert spaces. He introduced the spaces B2,μ:={wS(Rk):μ(ξ)w^(ξ)L2(Rk,dξ)},

for which a general Borei measurable weight function μ : ℝk → (0, ∞) serves as an index of regularity of a distribution w. (Here, w^ denotes the Fourier transform of w.) These spaces and their versions within the category of normed spaces (so called spaces of generalized smoothness) have found various applications to analysis and partial differential equations [11-19].

Recently Mikhailets and Murach [20-24] have built a theory of solvability of general elliptic systems and elliptic boundary-value problems on Hilbert scales of spaces Ηs;φ ≔ 𝓑2,μ for which the index of regularity is of the form μ(ξ):=(1+|ξ|2)s/2φ((1+|ξ|2)1/2).

Here, s is a real number, and φ is a function varying slowly at infinity in the sense of Karamata [26]. This theory is based on the method of interpolation with a function parameter between Hilbert spaces, specifically between Sobolev spaces. This allows Mikhailets and Murach to deduce theorems about solvability of elliptic systems and elliptic problems from the known results on the solvability of elliptic equations in Sobolev spaces. This theory is set force in [18, 25].

Generally, the method of interpolation between normed spaces proved to be very useful in the theory of elliptic [27-29] and parabolic [4, 8] partial differential equations. Specifically, Lions and Magenes [4] systematically used the interpolation with a number (power) parameter between Hilbert spaces in their theory of solvability of parabolic initial-boundary value problems on a complete scale of anisotropic Sobolev spaces. Using the more flexible method of interpolation with a function parameter between Hilbert spaces, Los, Mikhailets, and Murach [30, 31] proved theorems on solvability of semi-homogeneous parabolic problems in 2b-anisotropic Hörmander spaces Hs,s/(2b);φ, where 2b is a parabolic weight and where the parameters s and φ are the same as those in the above mentioned elliptic theory. These problems were considered in the case of homogeneous initial conditions (Cauchy data).

The purpose of this paper is to establish the well-posedness of inhomogeneous parabolic problems on appropriate pairs of the Hörmander spaces, i.e. to prove new isomorphism theorems for these problems. We consider the problems that consist of a general second order parabolic partial differential equation, the Dirichlet boundary condition or a general first order boundary condition, and the Cauchy datum. We deduce these isomorphism theorems from Lions and Magenes’ result [4] with the help of the interpolation with a function parameter between anisotropic Sobolev spaces. The use of this method in the case of inhomogeneous parabolic problems meets additional difficulties connected with the necessity to take into account quite complex compatibility conditions imposed on the right-hand sides of the problem. The model case of initial boundary-value problems for heat equation is investigated in [32].

2 Statement of the problem

We arbitrarily choose an integer n ≥ 2 and a real number τ > 0. Let G be a bounded domain in ℝn with an infinitely smooth boundary Γ ≔ ∂G. We put Ω ≔ G × (0, τ) and S ≔ Γ × (0, τ); so, Ω is an open cylinder in ℝn+1, and S is its lateral boundary. Then Ω¯:=G¯×[0,τ]andS¯:=rτ×[0,τ] are the closures of Ω and S respectively. In Ω, we consider a parabolic second order partial differential equation Au(x,t)tu(x,t)+|α|2aα(x,t)Dxαu(x,t)=f(x,t)for allxGandt(0,τ).(1)

Here and below, we use the following notation for partial derivatives: t ≔ ∂/∂t and Dxα:=D1α1Dnαn, where

Dj i∂/∂xj, x = (x1,..., xn)∈ℝn, and a ≔ (α1 ,..., αn) with 0 ≤ α1,..., αn ∈ℤ and |α|≔ α1 +…+αn. We suppose that all the coefficients aα of A belong to the space C(Ω¯)In the paper, all functions and distributions are supposed to be complex-valued, so we consider complex function spaces.

We suppose that the partial differential operator A is Petrovskii parabolic on Ω¯,, i.e. it satisfies the following condition (see, e.g. [1, Section 9, Subsection 1]):

Condition 2.1

For arbitrary xG¯,t[0,τ],ξ=(ξ1,,ξn)Rn, and p ∈ℂ with Re p ≥ 0, the inequality p+|α|=2aα(x,t)ξ1α1ξnαn0holdswhenever|ξ|+|p|0.

In the paper, we investigate the initial-boundary value problem that consists of the parabolic equation (1), the initial condition u(x,0)=h(x)for allxG,(2)

and the zero-order (Dirichlet) boundary condition ux(x,t)=g(x,t)for allxΓandt(0,τ)(3)

or the first order boundary condition Bu(x,t)j=1nbj(x,t)Dju(x,t)+b0(x,t)u(x,t)=g(x,t)for allxΓandt(0,τ).(4)

As to (4), we assume that all the coefficients b0, b1, ..., bn of Β belong to C(S̅) and that Β covers A on [1, Section 9, Subsection 1]. The latter assumption means the fulfilment of the following:

Condition 2.2

Choose arbitrarily x ∈ Γ, t ∈ [0, τ], vector η = (η1,..., ηn)n tangent to the boundary Γ at the point x, and number p ∈ ℂ with Re p ≥ 0 so that |η| + |p| ≠ 0. Let ν (x) = (ν1(x),... ,vn(x)) be the unit vector of the inward normal to Γ at x. Then:

a) the inequality j=1nbj(x,t)vj(x)0 holds true;

b) the number ζ=j=1nbj(x,t)ηjj=1nbj(x,t)vj(x)1

is not a root of the polynomial p+|α|=2aα(x,t)(η1+ζv1(x))α1(ηn+ζvn(x))αnofζC.

It is useful to note that if all the coefficients b1,...,bn are real-valued, then part b) of Condition 2.2 is satisfied. This follows directly from Condition 2.1.

Thus, we examine both the parabolic problem (1), (2), (3) and the parabolic problem (1), (2), (4). We investigate them in appropriate Hörmander inner product spaces considered in the next section.

3 Hörmander spaces

Among the normed function spaces 𝓑p,μ introduced by Hörmander in [10, Section 2.2], we use the inner product spaces Hμ(ℝk) ≔ 𝓑2,μ defined over ℝk, with 1 ≤ k ∈ ℤ. Here, μ : ℝk (0, ∞) is an arbitrary Borei measurable function that satisfies the following condition: there exist positive numbers c and l such that μ(ξ)μ(η)c(1+|ξη|)lfor allξ,ηRk.

By definition, the (complex) linear space Hμ (ℝk)consists of all tempered distributions wS′(ℝk)whose Fourier transform w^ is a locally Lebesgue integrable function subject to the condition Rkμ2(ξ)|w^(ξ)|2dξ<.

The inner product in Hμ (ℝk)is defined by the formula (w1,w2)Hμ(Rk)=Rkμ2(ξ)w1^(ξ)w2^(ξ)¯dξ,

where w1,w2Hμ(ℝk). This inner product induces the norm wHμ(Rk):=(w,w)Hμ(Rk)1/2.

According to [ 10, Section 2.2], the space Hμ(ℝk)is Hilbert and separable with respect to this inner product. Besides that, this space is continuously embedded in the linear topological space S′ (ℝk) of tempered distributions on ℝk, and the set C0(Rk)of test functions on ℝk is dense in Hμ(ℝk)(see also Hörmander’s monograph [33, Section 10.1]). We will say that the function parameter μ is the regularity index for the space Hμ(ℝk)and its versions Hμ(⋅).

A version of Hμ(ℝk) for an arbitrary nonempty open set V ⊂ ℝk is introduced in the standard way. Namely, Hμ(V):={wV:wHμ(Rk)},uHμ(V):=inf{wHμ(Rk):wHμ(Rk),u=wV},(5)

where uHμ(V). Here, as usual, wV stands for the restriction of the distribution wHμ(ℝk) to the open set V. In other words, Hμ(V)is the factor space of the space Hμ(ℝk) by its subspace HQμ(Rk):={wHμ(Rk):suppwQ}withQ:=RkV.(6)

Thus, Hμ(V)is a separable Hilbert space. The norm (5) is induced by the inner product (u1,u2)Hμ(V):=(w1Υw1,w2Υw2)Hμ(Rk),

where wjHμ(ℝk), wj = uj in V for each j ∈ {1, 2}, and Υ is the orthogonal projector of the space Hμ(ℝk) onto its subspace (6). The spaces Hμ(V)and HQμ(Rk) were introduced and investigated by Volevich and Paneah [11, Section 3].

It follows directly from the definition of Hμ(V)and properties of Hμ(ℝk) that the space Hμ(V)is continuously embedded in the linear topological space 𝓓′(V)of all distributions on V and that the set C0(V¯):={wV¯:wC0(Rk)}

is dense in Hμ(V).

Suppose that the integer k ≥ 2. Dealing with the above-stated parabolic problems, we need the Hörmander spaces Hμ(ℝk) and their versions in the case where the regularity index μ takes the form μ(ξ,ξk)=(1+|ξ|2+|ξk|)s/2φ((1+|ξ|2+|ξk|)1/2)for allξRk1andξkR.(7)

Here, the number parameter s is real, whereas the function parameter φ runs over a certain class 𝓜. By definition, the class 𝓜 consists of all Borel measurable functions φ : [1, ∞) →(0, ∞) such that

a) both the functions φ and 1/φ are bounded on each compact interval [1,b], with 1 < b < ∞;

b) the function φ varies slowly at infinity in the sense of Karamata [26], i.e. φ (λr) / φ (r)→1 as r →∞for each λ >0.

The theory of slowly varying functions (at infinity) is expounded, e.g., in [34, 35]. Their standard examples are the functions φ(r):=(logr)θ1(loglogr)θ2(loglogktimesr)θkofr1,

where the parameters k ∈ ℕ and θ1, θ2,..., θk∈ R are arbitrary.

Let s ∈ ℝ and φ ∈ 𝓜. We put Hs,s/2;φ (ℝk)≔ Hμ(ℝk)in the case where μ is of the form (7). Specifically, if φ(r)≡1, then Hs,s/2;φ (ℝk) becomes the anisotropic Sobolev inner product space Hs,s/2 (ℝk) of order (s, s/2). Generally, if φ ∈ 𝓜 is arbitrary, then the following continuous and dense embeddings hold: Hs1,s1/2(Rk)Hs,s/2;φ(Rk)Hs0,s0/2(Rk)whenevers0<s<s1.(8)

Indeed, let s0 < s < s1; since φ ∈ 𝓜, there exist positive numbers c0 and c1 such that c0rs0s,φ(r)c1rs1s for every r ≥ 1 (see e.g., [35, Section 1.5, Property 1°]). Then c0(1+|ξ|2+|ξk|)s0/2<_(1+|ξ|2+|ξk|)s/2φ((1+|ξ|2+|ξk|)1/2)c1(1+|ξ|2+|ξk|)s1/2

for arbitrary ξ′ ∈ ℝk1 and ξk ∈ℝ. This directly entails the continuous embeddings (8). They are dense because the set C0(Rk) is dense in all the spaces from (8).

Consider the class of Hörmander inner product spaces {Hs,s/2;φ(Rk):sR,φM}.(9)

The embeddings (8) show, that in (9) the function parameter φ defines additional regularity with respect to the basic anisotropic (s, s/2)-regularity. Specifically, if φ(r)→∞[or φ(r)→0] as r →∞, then φ defines additional positive [or negative] regularity. In other words, φ refines the basic smoothness (s,s/2).

We need versions of the function spaces (9) for the cylinder Ω = G × (0, τ) and its lateral boundary S = Γ × (0, τ).We put Hs,s/2;φ(Ω)≔ Hμ(Ω) in the case where μ is of the form (7) with k ≔ n + 1. For the function space Hs,s/2;φ(Ω), the numbers s and s/2 serve as the regularity indices of distributions u(x,t) with respect to the spatial variable x G and to the time variable t ∈(0, τ) respectively.

Following [36, Section 1], we will define the function space Hs,s/2;φ(S)with the help of special local charts on S. Let s > 0 and φ ∈ 𝓜. We put Hs,s/2;φ(Π)≔ Hμ(Π)for the strip Π ≔ ℝn-1 × (0, τ) in the case where μ is defined by formula (7) with k ≔ n. Recall that, according to our assumption Γ = ∂ Ω is an infinitely smooth closed manifold of dimension η – 1, the C-structure on Γ being induced by ℝn. From this structure we arbitrarily choose a finite atlas formed by local charts θj : ℝn–1 ↔ Γj· with j = 1,..., λ. Here, the open sets Γ1,..., Γλ make up a covering of Γ. We also arbitrarily choose functions χj C∞(Γ), with j = 1,..., λ, so that supp χj ⊂ Γj and χ1 +… χλ = 1 on Γ.

By definition, the linear space HS,S/,2;φ(S)consists of all square integrable functions g : S →·ℂ that the function gj(x,t):=χj(θj(x))g(θj(x),t)ofxRn1andt(0,τ)

belongs to HS,S/,2;φ(Π) for each number j ∈ {1,... ,λ}. The inner product in HS,S/,2;φ(S)is defined by the formula (g,g)Hs,s/2;φ(s):=j=1λ(gj,gj)Hs,s/2;φ(Π),

where g,g’ Hs,s/2;φ(S). This inner product naturally induces the norm gHs,s/2;φ(s):=(g,g)Hs,s/2;φ(s)1/2.

The space Hs,s/2;φ(S) is complete (i. e. Hilbert) and does not depend up to equivalence of norms on the choice of local charts and partition of unity on Γ [36, Theorem 1]. Note that this space is actually defined with the help of the following special local charts on S : θj:Π=Rn1×(0,τ)Γj×(0,τ),j=1,,λ,(10)

where θj(x,t):=(θj(x),t)for allxRn1 and t ∈ (0, τ).

We also need isotropic Hörmander spaces HS;φ(V)over an arbitrary open nonempty set V ⊆ ℝk with k ≥ 1. Let s ∈ ℝ and φ ∈ 𝓜. We put Hs(V)≔ Hμ(V)in the case where the regularity index μ takes the form μ(ξ)=(1+|ξ|2)s/2φ((1+|ξ|2)1/2)for arbitraryξRk.(11)

Since the function (11) is radial (i.e., depends only on |ξ|), the space Hs;φ(V)is isotropic. We will use the spaces Ηs(V)given over the whole Euclidean space V ≔k or over the domain V G in ℝ.

Besides, we will use Hörmander spaces Hs;φ(Γ)over Γ = ∂Ω. The are defined with the help of the above-mentioned collection of local charts {θj} and partition of unity {χj}on Γ similarly to the spaces over S. Let s ∈ ℝ and φ ∈𝓜. By definition, the linear space Hs;φ(r)consists of all distributions ω ∈ 𝓓′(Γ) on Γ that for each number; e {1,..., λ} the distribution ωj(x)≔χj (θj(xj) ω(θj(x)) of x ∈ ℝn-1 belongs to HS;φ(ℝn-1). The inner product in HS;φ(Γ)is defined by the formula (ω,ω)Hs;φ(Γ):=j=1λ(ωj,ωj)Hs;φ(Rn1),

where ω, ω’ Hs;φ(Γ). It induces the norm ωHs;φ(Γ):=(ω,ω)Hs;φ(Γ)1/2.

The space HS;φ(Γ)is Hilbert separable and does not depend up to equivalence of norms on our choice of local charts and partition of unity on Γ [37, Theorem 3.6(i)]. Note that the classes of isotropic inner product spaces {Hs;φ(V):sR,φM}and{Hs;φ(Γ):sR,φM}

were selected, investigated, and systematically applied to elliptic differential operators and elliptic boundary-value problems by Mikhailets and Murach [18, 25].

If φ ≡1, then the considered spaces HS,S/2;φ(·)and HS;φ(·) become the Sobolev spaces HS,S/2(·) and Hs(·) respectively. It follows directly from (8) that Hs1,s1/2()Hs,s/2;φ()Hs0,s0/2()whenevers0<s<s1.(12)

Analogously, Hs1()Hs;φ()Hs0()whenevers0<s<s1;(13)

see [18, Theorems 2.3(iii) and 3.3(iii)]. These embeddings are continuous and dense. Of course, if s = 0, then Hs (·) = Hs,s/2(·)is the Hilbert space L2(·) of all square integrable functions given on the corresponding measurable set.

In the Sobolev case of φ = 1, we will omit the index φ in designations of function spaces that will be introduced on the base of the Hörmander spaces Hs,s/2;φ(·) and Hs;φ(·).

4 Main results

Consider first the parabolic problem (l)-(3), which corresponds to the Dirichlet boundary condition on S. In order that a regular enough solution u to this problem exist, the right-hand sides of the problem should satisfy certain compatibility conditions (see, e.g., [1, Section 11] or [3, Chapter 4, Section 5]). These conditions consist in that the partial derivatives tku(x,t)|t=0,which could be found from the parabolic equation (1) and initial condition (2), should satisfy the boundary condition (3) and some relations that are obtained by means of the differentiation of the boundary condition with respect to t. To write these compatibility conditions we use Sobolev inner product spaces. We associate the linear mapping Λ0:u(Au,uS¯,u(,0)),whereuC(Ω¯),(14)

with the problem (l)-(3). Let real s ≥ 2; the mapping (14) extends uniquely (by continuity) to a bounded linear operator Λ0:Hs,s/2(Ω)Hs2,s/21(Ω)Hs1/2,s/21/4(S)Hs1(G).(15)

This follows directly from [38, Chapter I, Lemma 4, and Chapter II, Theorems 3 and 7]. Choosing any function u(x,t)from the space HS,S/2(Ω), we define the right-hand sides fHs2,s/21(Ω),gHs1/2,s/21/4(S),andhHs1(G)(16)

of the problem by the formula (f,g,h)≔ Λ0u with the help of this bounded operator.

According to [38, Chapter II, Theorem 7], the traces tku(,0)Hs12k(G)are well defined by closure for all k e Ζ such that 0 ≤ k < s/2 — 1/2 (and only for these k). Using (1) and (2), we express these traces in terms of the functions ƒ (x, t) and h(x) by the recurrent formula u(x,0)=h(x),tku(x,O)=|α|2q=0k1k1qtk1qaα(x,0)Dxαtqu(x,0)+tk1f(x,0)(17)

for each k ∈ ℤ such that 1 ≤ k < s/2 – 1/2,

the equalities holding for almost all x G.

Besides, the traces tkg(,0)Hs3/22k(Γ)are well defined by closure for all k ∈ ℤ such that 0 ≤ k < s/2 — 3/4 (and only for these k). Therefore, owing to the Dirichlet boundary condition (3), the equality tkg(x,0)=tku(x,0)for almost allxΓ(18)

holds for these integers k. The right-hand part of this equality is well defined because the function tku(,0)Hs12k(G) has the trace tku(,0)ΓHs3/22k(Γ)in view of s - 3/2- 2k > 0. Now, substituting (17) into (18), we obtain the compatibility conditions tkgΓ=vkΓwithkZand0k<s/23/4.(19)

Here, the functions vk are defined by the recurrent formula v0(x,0):=h(x),vk(x,0):=|α|2q=0k1k1qtk1qaα(x,0)Dxαvq(x)+tk1f(x,0)(20)

for each k ∈ ℤ such that 1 ≤ k < s/2 – 1/2,

these relations holding for almost all x G. Since vkHs12k(G)for eachkZ[0,s/21/2)(21)

due to (16), the trace vk ↾ Γ ∈ Hs–3/2–2k(Γ)is defined by closure whenever s — 3/2 — 2k > 0. Thus, the compatibility conditions (19) are well posed.

For instance, if 2 < s ≤ 7/2, then formula (19) gives one compatibility condition g ↾ Γ = h ↾ Γ. Next, if 7/2 < s ≤ 11/2, then (19) gives two compatibility conditions g ↾ Γ = h ↾ Γ and tgΓ=(|α|2aα(x,0)Dxαh(x)+f(x,0))Γ,

and so on.

We put E0≔ {2r + 3/2 : 1 ≤ r ∈ ℤ}. Note that E0is the set of all discontinuities of the function that assigns the number of compatibility conditions (19) to s ≥ 2.

Our main result on the parabolic problem (l)-(3) consists in that the linear mapping (14) extends uniquely to an isomorphism between appropriate pairs of Hörmander spaces introduced in the previous section. Let us indicate these spaces. We arbitrarily choose a real number s > 2 and function parameter φ ∈ 𝓜. We take ΗS,S/2;φ(Ώ) as the source space of this isomorphism; otherwise speaking, ΗS,S/2;φ(Ώ)serves as a space of solutions u to the problem. To introduce the target space of the isomorphism, consider the Hilbert space H0s2,s/21;φ:=Hs2,s/21;φ(Ω)Hs1/2,s/21/4;φ(S)Hs1;φ(G).

In the Sobolev case of φ≡ 1 this space coincides with the target space of the bounded operator (15). The target space of the isomorphism is imbedded in H0s2,s/21;φand is denoted by Q0s2,s/21;φ. We separately define this space in the s E0case and s ∈ E0case.

Suppose first that s ∉ E0. By definition, the linear space Q0s2,s/21;φ consists of all vectors (f,g,h)H0s2,s/21;φthat satisfy the compatibility conditions (19). As we have noted, these conditions are well defined for every (f,g,h)H0s2ϵ,s/21ϵ/2 for sufficiently small ε > 0. Hence, they are also well defined for every (f,g,h)H0s2,s/21;φdue to the continuous embedding H0s2,s/21;φH0s2ϵ,s/21ϵ/2.(22)

The latter follows directly from (12) and (13). Thus, our definition is reasonable.

We endow the linear space Q0s2,s/21;φwith the inner product and norm in the Hilbert space H0s2,s/21;φ.

The space Q0s2,s/21;φis complete, i.e. a Hilbert one. Indeed, if the number ε > 0 is sufficiently small, then Q0s2,s/21;φ=H0s2,s/21;φQ0s2ϵ,s/21ϵ/2.

Here, the spaceQ0s2ϵ,s/21ϵ/2 is complete because the differential operators and traces operators used in the

compatibility conditions are bounded on the corresponding pairs of Sobolev spaces. Therefore the right-hand side of this equality is complete with respect to the sum of the norms in the components of the intersection, this sum being equivalent to the norm in H0s2,s/21;φdue to (22). Thus, the space Q0s2,s/21;φis complete (with respect to the latter norm).

If s ∈ E0, then we define the Hilbert space Q0s2,s/21;φby means of the interpolation between its analogs just introduced. Namely, we put Q0s2,s/21;φ:=[Q0;s2ϵ,s/21ϵ/2;φQ0s2+ϵ,s/21+ϵ/2;φ]1/2.(23)

Here, the number ε ∈ (0,1/2) is arbitrarily chosen, and the right-hand side of the equality is the result of the interpolation of the written pair of Hilbert spaces with the parameter 1/2. We will recall the definition of the interpolation between Hilbert spaces in Section 5. The Hilbert space Q0s2,s/21;φ defined by formula (23) does not depend on the choice of ε up to equivalence of norms and is continuously embedded in H0s2,s/21;φ This will be shown in Remark 6.3 at the end of Section 6.

Now we can formulate our main result concerning the parabolic initial-boundary value problem (l)-(3).

Theorem 4.1

For arbitrary s > 2 and φ ∈ 𝓜 the mapping (14) extends uniquely (by continuity) to an isomorphism Λ0:Hs,s/2;φ(Ω)Q0s2,s/21;φ.(24)

Otherwise speaking, the parabolic problem (l)-(3) is well posed (in the sense of Hadamard) on the pair of Hilbert spaces HS,S;φ(Ω)and Q0s2,s/21;φ whenever s > 2 and φ 𝓜,the right-hand side (f,g,h)Q0s2,s/21;φ of the problem being defined by closer for an arbitrary function u ∈ HS,S/2;φ(Ω).

Note that the necessity to define the target space Q0s2,s/21;φseparately in the s ∈ E0case is caused by the following: if we defined this space for s ∈ E0in the way used in the s E0case, then the isomorphism (24) would not hold at least for φ ≡ 1. This follows from a result by Solonnikov [39, Section 6].

Consider now the parabolic problem (1), (2), (4), which corresponds to the first order boundary condition on S. Let us write the compatibility conditions for the right-hand sides of this problem.

We associate the linear mapping Λ1:u(Au,Bu,u(,0)),whereuC(Ω¯),(25)

with the problem (1), (2), (4). For arbitrary real s ≥ 2, this mapping extends uniquely (by continuity) to a bounded linear operator Λ1:Hs,s/2(Ω)Hs2,s/21(Ω)Hs3/2,s/23/4(S)Hs1(G).(26)

Choosing any function u(x, t) from HS,S/2(Ω), we define the right-hand sides fHs2,s/21(Ω),gHs3/2,s/23/4(S),andhHs1(G)

of the problem by the formula (f,g,h) ≔ Λ1u with the help of this bounded operator. Here, unlike (16), the inclusion uHs,s/2(Ω)impliesg=BuHs3/2,s/23/4(S) dueto [38, Chapter II, Theorem 7]. According to this theorem, the traces tkg(,0)Hs5/22k(Γ) are defined by closure for all k ∈ ℤ such that 0 ≤ k < s/2—5/4 (and only for these k). We can express these traces in terms of the function u(x,t)and its time derivatives; namely, tkg(x,0)=(tkBu(x,t))|r=0=q=0kkqj=1ntkqbj(x,0)Djtqu(x,0)+tkqb0(x,0)tqu(x,0)(27)

for almost all x ∈ Γ. Here, all the functions u(x,0),tu(x,0),,tku(x,0) are expressed in terms of the functions f(x,t)and h(x)by the recurrent formula (17).

Substituting (17) in the right-hand side of formula (27), we obtain the compatibility conditions tkgΓ=Bk[v0,,vk]Γ,withkZand0k<s/25/4.(28)

Here, the functions v0, v1,..., vk are defined on G by the recurrent formula (20), and we put Bk[v0,,vk](x):=q=0kkqj=1ntkqbj(x,0)Djvq(x)+tkqb0(x,0)vq(x)

for all x G. The right-hand side of the equality (28) is well defined because the function Bk [v0 ,...,vk]belongs to Hs-2-2k(G)due to (21) and therefore the trace Bk[v0,,vk]ΓHs5/22k(Γ)

is defined by closure whenever s – 5/2– 2k > 0. Note that if s ≤ 5/2, then there are no compatibility conditions.

We set E1 {2r + 1/2 : 1 ≤ r ∈ ℤ}. Observe that E1is the set of all discontinuities of the function that assigns the number of compatibility conditions (28) to s ≥ 2.

To formulate our isomorphism theorem for the parabolic problem (1), (2), (4), we introduce the source and target spaces of this isomorphism. Let s > 2 and φ ∈ 𝓜. As above, we take HS,S/2;φ(Ω)as the source space. The target space denoted by Q1s2,s/21;φ is embedded in the Hilbert space H1s2,s/21;φ:=Hs2,s/21;φ(Ω)Hs3/2,s/23/4;φ(S)Hs1;φ(G).

In the Sobolev case of φ ≡ 1 this space coincides with the target space of the bounded operator (26).

If s ∉ Ε1, then the linear space Q1s2,s/21;φ is defined to consist of all vectors (f,g,h)H1s2,s/21;φ that satisfy the compatibility conditions (28). The definition is reasonable because these conditions are well defined for every (f,g,h)H1s2ϵ,s/21ϵ/2for sufficiently small ε > 0 and because H0s2,s/21;φH1s2ϵ,s/21ϵ/2.(29)

This continuous embedding follows immediately from (12) and (13). The linear space Q1s2,s/21;φis endowed with the inner product and the norm in the Hilbert space H1s2,s/21;φ.The space Q1s2,s/21;φis complete, i.e. a Hilbert one. This is justified by the same reasoning as we have used to prove the completeness of Q0s2,s/21;φ.. Note that if 2 < s < 5/2, then the spaces H1s2,s/21;φandQ1s2,s/21;φ coincide because the compatibility conditions (28) are absent.

If s ∈ Ε1, then we define the Hilbert space Q1s2,s/21;φ by the interpolation, namely Q1s2,s/21;φ:=[Q1s2ϵ,s/21ϵ/2;φQ1s2+ϵ,s/21+ϵ/2;φ]1/2,(30)

with the number ε ∈ (0,1/2) chosen arbitrarily. This Hilbert space does not depend on the choice of ε up to equivalence of norms and is embedded continuously in H1s2,s/21;φ, which will be shown in Remark 6.3. Now we can formulate our main result concerning the parabolic initial-boundary value problem (1), (2), (4).

Theorem 4.2

For arbitrary s > 2 and φ ∈ 𝓜 the mapping (25) extends uniquely (by continuity) to an isomorphism Λ1:Hs,s/2;φ(Ω)Q1s2,s/21;φ.(31)

In other words, the parabolic problem (1), (2), (4) is well posed on the pair of Hilbert spaces HS,S/2;φ(Ω)and Q1s2,s/21;φ whenever s > 2 < and φ ∈ 𝓜, the right-hand side (f,g,h)Q1s2,s/21;φ of this problem being defined by closer for an arbitrary function u ∈ HS,S/2;φ(Ω).

Note that the necessity to define the target space Q1s2,s/21;φseparately in the s ∈ E1case is stipulated by a similar cause as that indicated for the space Q0s2,s/21;φ.Namely, if we defined this space for s ∈ Ε1in the way used in the sE1case, then the isomorphism (31) would not hold at least when φ ≡ 1and (4) is the Neumann boundary condition (see [39, Section 6]).

Theorems 4.1 and 4.2 are known in the Sobolev case where φ1and neither s nors/2 is half-integer. Namely, they are contained in Agranovich and Vishik’s result [1, Theorem 12.1] in the case of s, s/2 ∈ℤand are covered by Lions and Magenes’ result [4, Theorem 6.2]. Solonnikov [39, Theorem 17] proved the corresponding a priory estimates for anisotropic Sobolev norms of solutions to the problem (l)-(3) and to the problem (1), (2), (4) provided that (4) is the Neumann boundary condition. Note that these results include the limiting case of s = 2.

In Section 6 we will deduce Theorems 4.1 and 4.2 from the above-mentioned results with the help of the method of interpolation with a function parameter between Hilbert spaces, specifically between Sobolev inner product spaces. Therefore we devote the next section to this method and its applications to Sobolev and Hörmander spaces.

5 Interpolation with a function parameter between Hilbert spaces

This method of interpolation is a natural generalization of the classical interpolation method by S. Krein and J.-L. Lions to the case when a general enough function is used instead of a number as an interpolation parameter; see, e.g., monographs [40, Chapter IV, Section 1, Subsection 10] and [28, Chapter 1, Sections 2 and 5]. For our purposes, it is sufficient to restrict the discussion of the interpolation with a function parameter to the case of separable complex Hilbert spaces. We mainly follow the monograph [18, Section 1.1], which systematically expounds this interpolation (see also [37, Section 2]).

Let X ≔ [X0,X1] be an ordered pair of separable complex Hilbert spaces such that X1X0and this embedding is continuous and dense. This pair is said to be admissible. For X, there is a positive-definite self-adjoint operator J on X0 with the domain X1such that ||Jv||X0 = ||v||X1for every v X1. This operator is uniquely determined by the pair X and is called a generating operator for X; see, e.g., [40, Chapter IV, Theorem 1.12]. The operator defines an isometric isomorphism J : X1X0.

Let 𝓑 denote the set of all Borei measurable functions ψ : (0, ∞) (0, ∞) such that ψ is bounded on each compact interval [a, b], with 0 <a < b <∞, and that 1 is bounded on every semiaxis [a, ∞), with a >0.

Choosing a function ψ ∈ 𝓑 arbitrarily, we consider the (generally, unbounded) operator ψ(J)defined on X0 as the Borei function ψ ofJ. This operator is built with the help of Spectral Theorem applied to the self-adjoint operator J. Let [X0,X1]ψ or, simply, Χψ denote the domain of ψ (J) endowed with the inner product (v1, v2)Xψ := (ψ(J)v1, ψ(J)v2)X0and the corresponding norm ||v||Xψ≔ ||ψ(J)v||X0. The linear space Χψis Hilbert and separable with respect to this norm.

A function ψ ∈ 𝓑 is called an interpolation parameter if the following condition is satisfied for all admissible pairs X = [X0,X1]and Y = [Y0,Y1]of Hilbert spaces and for an arbitrary linear mapping Τ given on X0: if the restriction of Τ to Xjis a bounded operator Τ : Xj→ Yjfor each j ∈ {0, 1}, then the restriction of Τ to Χψis also a bounded operator Τ : Χψ→ Υψ.

If ψ is an interpolation parameter, then we say that the Hilbert space Χψis obtained by the interpolation with the function parameter ψ of the pair X = [X0,X1]or, otherwise speaking, between the spaces X0 and X1. In this case, the dense and continuous embeddingsX1Χψ X0 hold.

The class of all interpolation parameters (in the sense of the given definition) admits a constructive description. Namely, a function ψ ∈ 𝓑 is an interpolation parameter if and only if ψ is pseudoconcave in a neighbourhood of infinity. The latter property means that there exists a concave positive function ψ1(r) of r ≫ 1 that both the functions ψ/ψ1and ψ1 are bounded in some neighbourhood of infinity. This criterion follows from Peetre’s description of all interpolation functions for the weighted Lebesgue spaces [41, 42] (this result of Peetre is set forth in the monograph [43, Theorem 5.4.4]). The proof of the criterion is given in [18, Section 1.1.9].

An application of this criterion to power functions gives the classical result by Lions and S. Krein. Namely, the function ψ(r) ≡ rθis an interpolation parameter whenever 0 ≤θ ≤ 1. In this case, the exponent θ serves as a number parameter of the interpolation, and the interpolation space Χψis also denoted by X0.This interpolation was used in formulas (23) and (30) in the special case of θ = 1/2.

Let us formulate some general properties of interpolation with a function parameter; they will be used in our proofs. The first of these properties enables us to reduce the interpolation of subspaces to the interpolation of the whole spaces (see [18, Theorem 1.6] or [29, Section 1.17.1, Theorem 1]). As usual, subspaces of normed spaces are assumed to be closed. Generally, we consider nonorthogonal projectors onto subspaces of a Hilbert space.

Proposition 5.1

Let X = [Χ0,Χ1] be an admissible pair of Hilbert spaces, and let Y0be a subspace of X0. Then Y1 := X1Y0is a subspace of X1. Suppose that there exists a linear mapping Ρ : X0 X0 such that Ρ is a projector of the space Xj onto its subspace Yj for each j ∈ {0, 1}. Then the pair [Y0, Y1] is admissible, and [Y0, Υ1]ψ = ΧψY0with equivalence of norms for an arbitrary interpolation parameter ψ ∈ 𝓑. Here, ΧψY0 is a subspace of Xψ.

The second property reduces the interpolation of orthogonal sums of Hilbert spaces to the interpolation of their summands (see [18, Theorem 1.8].

Proposition 5.2

Let [X0(j),X1(j)] with j = 1, . . . , q, be a finite collection of admissible pairs of Hilbert spaces. Then j=1qX0(j),j=1qX1(j)ψ=j=1q[X0(j)X1(j)]ψ

with equality of norms for every function ψ𝓑.

The third property shows that the interpolation with a function parameter is stable with respect to its repeated fulfillment [18, Theorem 1.3].

Proposition 5.3

Let α, β,ψΒ, and suppose that the function α/ β is bounded in a neighbourhood of infinity. Define the function ω ∈ 𝓑by the formula ω(r):= α(r)ψ(ß(r)(r))forr >0. Then ω∈ 𝓑, and [Xα, Χβ]ψ = Χω with equality of norms for every admissible pair X of Hilbert spaces. Besides, if α, β, ψ are interpolation parameters, then ω is also an interpolation parameter.

Our proof of Theorems 4.1 and 4.2 is based on the key fact that the interpolation with an appropriate function parameter between margin Sobolev spaces in (12) and (13) gives the intermediate Hörmander spaces HS,S/,2(·) and Hs;φ(·)respectively. Let us formulate this property separately for isotropic and for anisotropic spaces.

Proposition 5.4

Let real numbers s0, s, and s1satisfy the inequalities s0<s < s1, and let φ ∈ 𝓜. Put ψ(r):=r(ss0)/(s1s0),(r1/s1s0))ifr1,(1)if0<r<1.(32)

Then the function ψ𝓑 is an interpolation parameter, and the equality of spaces Hsλ;φ(W)=[Hs0λ(W),Hs1λ(W)]ψ(33)

holds true up to equivalence of norms for arbitrary λ ∈ ℝ provided that W = G or W = Γ. If W = R with 1 <k ∈ℤ, then (33) holds true with equality of norms in spaces.

This result is due to [21, Theorems 3.1 and 3.5]; see also monograph [18, Theorems 1.14, 2.2, and 3.2] for the cases where W =k, W = Γ, and W = G respectively.

Proposition 5.5

Let real numbers s0, s, and s1satisfy the inequalities 0 s0<s <s1, and let φ ∈ 𝓜. Define an interpolation parameter ψ𝓑 by formula (32). Then the equality of spaces Hsλ,sλ)/2;φ(W)=[Hs0λ,s0λ)/2(W),Hs1λ,s1λ)/2(W)]ψ(34)

holds true up to equivalence of norms for arbitrary real λs0provided that W = Ω or W = S. If W =k with 2k ∈ ℤ, then (34) holds true with equality of norms in spaces without the assumption that 0 ≤ s0.

This result is due to [36, Theorem 2 and Lemma 1] for the cases where W = S and W =krespectively. In the W = Ω case, the proof of the result is the same as the proof of its analog for a strip [36, Lemma 2].

6 Proofs

To deduce Theorems 4.1 and 4.2 from their known counterparts in the Sobolev case, we need to prove a version of Proposition 5.5 (with λ = 0) for the target spaces of isomorphisms (24) and (31). This proof will be based on the following lemma about properties of the operator that assigns the Cauchy data to an arbitrary function gHs,s/2;φ(S).

lemma 6.1

Choose an integer r≥ 1, and consider the linear mapping R:g(gΓ,tgΓ,,Γtr1gΓ),withgC(S¯).(35)

This mapping extends uniquely (by continuity) to a bounded linear operator R:Hs,s/2;φ(S)k=0r1Hs2k1,φ(Γ)=:Hs;φ(Γ)(36)

for arbitrary s > 2r — 1 and φ ∈ 𝓜. This operator is right invertible; moreover, there exists a linear mapping Τ : (L2(Γ))rL2(S) that for arbitrary s > 2r — 1 and φ ∈ 𝓜the restriction of Τ to the spacesφ(Γ)is a bounded linear operator T:Hs;φ(Γ)Hs,s/2;φ(S)(37)

and that RTv = ν for every ν ∈ ℍs;φ(Γ).

Proof

We first prove an analog of this lemma for Hörmander spaces defined on ℝn and ℝn-1 instead of S and Γ. Then we deduce the lemma with the help of the special local charts on S. Consider the linear mapping R0:w(w|t=0,tw|t=0,,tr1w|t=0),withwC0(Rn).(38)

Here, we interpret w as a function w(x,t)of x∈ ℝn-1andt∈ ℝ so thatR0w(C0(Rn1))r.Choose s >2r — 1 and φ∈ 𝓜 arbitrarily, and prove that the mapping (38) extends uniquely (by continuity) to a bounded linear operator R0:Hs,s/2;φ(Rn)k=0r1Hs2k1;φ(Rn1)=:Hs;φ(Rn1)(39)

This fact is known in the Sobolev case of φ ≡ 1 due to [38, Chapter II, Theorem 7]. Using the interpolation with a function parameter between Sobolev spaces, we can deduce this fact in the general situation of arbitrary φ ∈ 𝓜. Namely, chooses0, s1∈ ℝ such that 2r — 1<s0<s < s1and consider the bounded linear operators R0:Hsj,sj/2(Rn)Hsj(Rn1),withj{0,1}.(40)

Let ψ be the interpolation parameter (32). Then the restriction of the mapping (40) with j = 0 to the space [Hs0,s0/2(Rn),Hs1,s1/2(Rn)]ψ=Hs,s/2;φ(Rn)(41)

is a bounded operator R0:Hs,s/2;φ(Rn)[Hs0(Rn1),Hs1(Rn1)]ψ.(42)

The latter equality is due to Proposition 5.5. This operator is an extension by continuity of the mapping (38) because the set C0(Rn)is dense in Hs,s/2;φ(ℝn). Owing to Propositions 5.2 and 5.4, we get [Hs0(Rn1),Hs1(Rn1)]ψ=k=0r1[Hs02k1(Rn1),Hs12k1(Rn1)]ψ=k=0r1Hs2k1;φ(Rn1)=Hs;φ(Rn1).(43)

Hence, the linear bounded operator (42) is the required operator (39). Let us now build a linear mapping T0:(L2(Rn1))rL2(Rn)(44)

that its restriction to each space ℍs;φ(ℝn-1 ) with s > 2r — 1 and φ ∈ 𝓜 is a bounded operator between the spaces ℍs;φ(ℝn-1 )and Hs;s/2;φn)and that this operator is right inverse to (39).

Similarly to Hörmander [10, Proof of Theorem 2.5.7] we define the linear mapping T0:vFξx1[β(ξ2t)k=0r11k!vk^(ξ)×tk](x,t)(45)

on the linear topological space of vectors v:=(v0,,vr1)(S(Rn1))r.

We consider T0υ as a distribution on the Euclidean space ℝn of points (x, t), with x= {x1,..., xn-1)∈ ℝn-1 and t ∈ ℝ. In (45), the function βC0(R) is chosen so that β = 1 in a certain neighbourhood of zero. As usual, Fξx1denotes the inverse Fourier transform with respect to ξ =1,... ,ξn-1) ∈ ℝn-1,and (ξ) := (1 + |ξ|2)1//2. The variable ξ is dual to xrelative to the direct Fourier transform w^(ξ)=(Fw)(ξ) of a function w(x).

Obviously, the mapping (45) is well defined and acts continuously between (S′(ℝn-1)r and S′(ℝn). It is also evident that the restriction of this mapping to the space (L2(ℝn-1))r is a bounded operator between (L2(ℝn-1))r and (L2(ℝn-1))r.

We assert that R0T0v=vfor everyv(S(Rn1))r.(46)

Here, as usual, S(ℝn-1)denotes the linear topological space of all rapidly decreasing infinitely smooth functions on ℝn-1. Since vS(ℝn-1) implies T0vS(ℝn-1), the left-hand side of the equality (46) is well defined. Let us prove this equality.

Choosing j ∈ {0,..., r — 1} and v= (v0, ..., vr-1) ∈ S(ℝn-1)rarbitrarily, we get F[tjT0v|t=0](ξ)=tjFxξ[T0v](ξ,t)|t=0=tj(β(ξ2t)k=0r11k!vk^(ξ)tk)|t=0=β(0)(tjk=0r11k!vk^(ξ)tk)|t=0=β(0)j!1j!vj^(ξ)=vj^(ξ)

for every ξ ∈ ℝn-1. In the fourth equality, we have used the fact that β = 1 in a neighbourhood of zero. Thus, the Fourier transforms of all components of the vectors R0 T0υ and υ coincide, which is equivalent to (46). Let us now prove that the restriction of the mapping (45) to each space H2m(Rn1)=k=0r1H2m2k1(Rn1)(47)

with 0 ≤ m ∈ ℤis a bounded operator between ℍ2m(ℝn-1) and H2m, m(ℝn). Note that the integers 2m-2k-1 may be negative in (47).

Let an integer m ≥ 0. We make use of the fact that the norm in the space H2m,m(ℝn)is equivalent to the norm w2m,m:=w2+j=1n1xj2mw2+tmw21/2

(see, e.g., [44, Section 9.1]). Here and below in this proof, || · || stands for the norm in the Hilbert space L2(ℝn). Of course, xjuand ∂tdenote the operators of generalized partial derivatives with respect to xjand t respectively. Choosing v= (v0, ..., vr-1)∈(S(ℝn-1))rarbitrarily and using the Parseval equality, we obtain the following: T0v2m,m2=T0v2+j=1n1xj˙2mT0v2+tmT0v2=T0v^2+j=1n1ξj2mT0v^2+tmT0v^2k=0r11k!Rn|β(ξ2t)vk^(ξ)tk|2dξdt+j=1n1k=0r11k!Rn|ξj2mβ(ξ2t)vk^(ξ)tk|2dξdt+k=0r11k!Rn|tm(β(ξ2t)tk)vk^(ξ)|2dξdt.

Let us estimate each of these three integrals separately. We begin with the third integral. Changing the variable τ = 〈ξ〉2tin the interior integral with respect to t, we get the equalities Rn|tm(β({ξ}2t)tk)vk^(ξ)|2dξdt=Rn1|vk^(ξ)|2dξR|tm(β(ξ2t)tk)|2dt=Rn1ξ4m4k2|vk^(ξ)|2dξR|τm(β(τ)τk)|2dτ.

Hence, Rn|tm(β(ξ2t)tk)vk^(ξ)|2dξdt=c1vkH2m2k1Rn1)2,

with c1:=R|τm(β(τ)τk)|2dτ<.

Using the same changing of the variable t in the second integral, we obtain the following: Rn|ξj2mβ(ξ2t)vk^(ξ)tk|2dξdt=Rn1|ξj|4m|vk^(ξ)|2dξR|tkβ({ξ}2t)|2dt=Rn1|ξj|4mξ4k2|vk^(ξ)|2dξR|τkβ(τ)|2dτRn1ξ4m4k2|vk^(ξ)|2dξR|τkβ(τ)|2dτ,

Hence, Rn|ξj2mβ(ξ2t)vk^(ξ)tk|2dξdtc2vkH2m2k1(Rn1)2.

with c2:=R|τkβ(τ)|2dτ<.

Finally, replacing the symbol ξjwith 1 in the previous reasoning, we obtain the following estimate for the first integral: Rn|β(ξ2t)vk^(ξ)tk|2dξdtc2vkH2k1(Rn1)2c2vkH2m2k1(Rn1)2.

Thus, we conclude that T0vH2m,m(Rn)2ck=0r1vkH2m2k1(Rn1)2=cvH2m(Rn1)2

for any v∈ (S(ℝn-1))r, with the number c >0 being independent of υ. Since the set (S(ℝn-1))r is dense in ℍ2m (ℝn-1), it follows from the latter estimate that the mapping (45) sets a bounded linear operator T0:H2m(Rn1)H2m,m(Rn)whenever0mZ.

Let us deduce from this fact that the mapping (45) acts continuously between the spaces ℍs;φ(ℝn-1) and Hs,s/2;φ(ℝn-1) s > 2r — 1 and φ ∈ 𝓜 . Put s0= 0, choose an even integer s1> s, and consider the linear bounded operators T0:Hsj(Rn1)Hsj,sj/2(Rn),withj{0,1}.(48)

Let, as above, ψ be the interpolation parameter (32). Then the restriction of the mapping (48) with j = 0 to the space [Hs0(Rn1),Hs1(Rn1)]ψ=Hs;φ(Rn1)

is a bounded operator T0:Hs;φ(Rn1)Hs,s/2;φ(Rn).(49)

Here, we have used formulas (41) and (43), which remain true for the considered s0and s1.

Now the equality (46) extends by continuity over all vectors v∈ℍs;φ(ℝn-1) . Hence, the operator (49) is right inverse to (39). Thus, the required mapping (44) is built.

We need to introduce analogs of the operators (39) and (49) for the strip Π={(x,t):xRn1,0<t<τ}.

Let s > 2r — 1 and φ ∈ 𝓜. Given uHs,s/2;φ(Π), we put R1u := R0w, where a function wHs,s/2;φ(ℝn) satisfies the condition w ↾Π = u. Evidently, this definition does not depend on the choice of w. The linear mapping u R1u is a bounded operator R1:Hs,s/2;φ(Π)Hs;φ(Rn1).(50)

This follows immediately from the boundedness of the operator (39) and from the definition of the norm in Hs,s/2;φ(Π)

Let us introduce a right-inverse of (50) on the base of the mapping (45). We put T1 v := (Τ0v) ↾Π for arbitrary v∈(L2(ℝn-1))r. The restriction of the linear mapping v T1v over vectors v∈ ℍs;φ(ℝn-1)is a bounded operator T1:Hs;φ(Rn1)Hs,s/2;φ(Π).(51)

This follows directly from the boundedness of the operator (49). Observe that R1T1v=R1((T0v)Π)=R0T0v=vfor everyvHs;φ(Rn1).

Thus, the operator (51) is right inverse to (50).

Using operators (50) and (51), we can now prove our lemma with the help of the special local charts (10) on S. As above, lets>2r — 1 and φ ∈ 𝓜. Choosing k ∈{0,..., r— 1} and g∈ C(S̅) arbitrarily, we get the following: tkgΓHs2k1;φ(Γ)2=j=1λ(χj(tkgΓ))θjHs2k1;φ(Rn1)2=j=1λtk((χjg)θj)(Rn1)Hs2k1;φ(Rn1)2c2j=1λ(xjg)θjHs,s/2;φ)2=c2gHs,s/2;φS)2.

Here, c denotes the norm of the bounded operator (50), and, as usual, symbol "o" designates a composition of functions. Recall that {θj·} is a collection of local charts on Γ and that {xj} is an infinitely smooth partition of unity on Γ. Thus, RgHs;φcrgHs,s/2;φ)(S)for everygC(S¯).

This implies that the mapping (35) extends by continuity to the bounded linear operator (36).

Let us build the linear mapping T: (L2(Γ))r L2(S)whose restriction to ℍs;φ(Γ) is a right-inverse of (36). Consider the linear mapping of flattening of Γ L:v((χ1v)θ1,,(χλv)θλ),withvL2(Γ).

Its restriction to Ησ;φ(Γ)is an isometric operator L:Hσ;φ(Γ)(Hσ;φ(Rn1))λwheneverσ>0.(52)

Besides, consider the linear mapping of sewing of Γ K:(h1,,hλ)j=1λOj((ηjhj)θj1),withh1,,hλL2(Rn1).

Here, each function ηjC0(Rn1)is chosen so that ηj= 1 on the set θj1 (supp Xj) whereas Oj denotes the operator of the extension by zero to Γ of a function given on Γj·. The restriction of this mapping to (Ησ;φ(ℝn-1))λ is a bounded operator (Hσ;φ(Rn1))λHσ;φ(rτ)wheneverσ>0,

and this operator is left inverse to (52) (see [18, the proof of Theorem 2.2]).

The mapping Κ induces the operator K1of the sewing of the manifold S = Γ x(0, τ) by the formula (K1(g1,,gλ))(x,t):=(K(g1(,t),,gλ(,t)))(x)

for arbitrary functions g1, . . . ,gλL2(Π) and almost all x∈Γ and t ∈(0, τ). The restriction of the mapping K1 to (Ησ,σ/2;φΠ)λis a bounded operator K1:(Hσ,σ/2;φ(Π))λHσ,σ/2;φ(S)wheneverσ>0(53)

(see [36, the proof of Theorem 2]).

Given v:= (v0, v1, ..., vr-1)(L2(Γ))r, we set Tv:=K1(T1(v0,1,,vr1,1),,T1(v0,λ,,vr1,λ)),

where (vk,1,,vk,λ):=Lvk(L2(Rn1))λ

for each integer k ∈ {0,..., r — 1}. The linear mapping vTv acts continuously between (L2(Γ))rand L2(S), which follows directly from the definitions of L, T1, and K1. The restriction of this mapping to Ηs(Γ)is the bounded operator (37). This follows immediately from the boundedness of the operators (51), (52), and (53). The operator (37) is right inverse to (36). Indeed, choosing a vector v= (v0, v1,..., vr-1) ∈Ηs(Γ)arbitrarily, we obtain the following equalities: (RTv)k=(RK1(T1(v0,1,,vr1,1),,T1(v0,λ,,vr1,λ)))k=K((R1T1(v0,1,,vr1,1))k,,(R1T1(v0,λ,,vr1,λ))k)=K(vk,1,,vk,λ)=KLvk=vk.

Here, the index k runs over the set {0,..., r — 1} and denotes the k-th component of a vector. Hence, RTv = v.

Using this lemma, we will now prove a version of Proposition 5.5 for the target spaces of isomorphisms (24) and (31). Note that the number of the compatibility conditions (19) and (28) are constant respectively on the intervals J0,1:=(2,7/2),J0,r:=(2r1/2,2r+3/2),with2rZ,

and J1,0:=(2,5/2),J1,r:=(2r+1/2,2r+5/2),with1rZ,

of the varying of s. Namely, if s ranges over some Jl, r,then this number equals r.

lemma 6.2

Let l ∈ {0, 1} and 1 ≤ r ∈ ℤ. Suppose that real number s0,s,s1Jl,r satisfy the inequality s0<s <s1and that φ ∈ 𝓜. Define an interpolation parameter ψ ∈ 𝓑by formula (32). Then the equality of spaces Qls2,s/21;φ=[Qls02,s0/21,Qls12,s1/21]ψ(54)

holds true up to equivalence of norms.

Proof

Recall that Qls2,s/21;φandQlsj2,sj/21, with j ∈ {0,1} are subspaces of the Hilbert spaces Hls2,s/21;φandHlsj2,sj/21respectively. According to Propositions 5.2, 5.4, and 5.5 we obtain the following: [Hls02,s0/21,Hls12,s1/21]ψ=[Hs02,s0/21(Ω)Hs02l+1)/2,s0/22l+1)/4(S)Hs01(G),Hs12,S1/21(Ω)Hs12l+1)/2s;1/22l+1)/4(S)Hs11(G)]ψ=[Hs02,s0/21(Ω),Hs12,s1/21(Ω)]ψ[Hs02l+1)/2,s0/22l+1)/4(S),Hs12l+1)/2,s1/22l+1)/4(S)]ψ[Hs01(G),Hs11(G)]ψ=Hs2,s/21;φ(Ω)Hs2l+1)/2,s/22l+1)/4;φ(S)Hs1;φ(G)=Hls2,s/21;φ.

Thus, [Hls02,s0/21,Hls12,s1/21]ψ=Hls2,s/21;φ(55)

up to equivalence of norms.

We will deduce the required formula (54) from (55) with the help of Proposition 5.1. To this end, we need to present a linear mapping Ρ on Hls02,s0/21suchthat Ρ is a projector of the space Hlsj2,sj/21 onto its subspace Qlsj2,sj/21 for each j ∈{0, 1}. If we have this mapping, we will get [Qls02,s0/21,Qls12,s1/21]ψ=[Hls02,s0/21,Hls12,s1/21]ψQls02,s0/21=Hls2,s/21;φQls02,s0/21=Qls2,s/21;φ

due to Proposition 5.1, formula (55), and the conditions s0,sJι, rand s0< s. Note that these conditions imply the last equality because the elements of the spaces Qls02,s0/21andQls2,s/21;φ satisfy the same compatibility conditions and because Hls2,s/21;φ is embedded continuously in Hls02,s0/21.

We will build the above-mentioned mapping Ρ with the help of Lemma 6.1. Consider first the case of / = 0. Given (f,g,h)H0s02,s0/2,1 we put g:=g+T(v0ΓgΓ,,vr1Γtr1gΓ).

Here, the functions vkHs012k(G), with k = 0, . . ., r — 1, are defined by the recurrent formula (20), and the mapping Τ is taken from Lemma 6.1. The linear mapping Ρ : (f, g, h)↦(f,g*,h)defined on all vectors (f,g,h)H0s02,s0/2,1 is required. Indeed, its restriction to each space H0sj˙2,sj/21, with j ∈{0, 1}, is a bounded operator on this space. This follows directly from Lemma 6.1 in which we take s := Sj 1/2. Moreover, if (f,g,h)Q0sj2,j0/2,1 then P(f,g,h) = (ƒ, g, h) due to the compatibility conditions (19).

Consider now the case of l = 1. Given (f,g,h)H1s02,s0/21, we put g:=g+T(B0[v0]ΓgΓ,,Br1[v0,,vr1]Γtr1gΓ).

Here, the functions v0,..., vr-1 and mapping Τ are the same as in the l = 0 case. The linear mapping Ρ : (ƒ, g, h) ↦(f, g*, h) defined on all vectors (f,g,h)H0s02,00/2,1is required. Indeed, its restriction to each space H0sj2,sj/12, with j∈{0, 1}, is a bounded operator on this space due to Lemma 6.1 in which s := sj—3/2. Moreover, if (f,g,h)Q0sj2,sj/21, then P(f, g, h) = (ƒ, g, h)by the compatibility conditions (28).

Remark 6.3

If l = 1 and r = 0, then the conclusion of Lemma 6.2 remains true. Indeed, in this case Q1s2,s/21;φ=H1s2,s/21;andQ1sj2,sj/21=H1sj2,sj/21 for each j ∈{0, 1}so that (54) coincides with the equality (55). The latter is valid in the case considered as well.

Now we are in position to prove the main results of the paper.

Proofs of Theorems 4.1 and 4.2. Let s >2, φ ∈ 𝓜, and l∈ {0, 1}. If l = 0 [or l = 1], then our reasoning relates to Theorem 4.1 [or Theorem 4.2]. We first consider the case where sΕl. Then sJl, rfor a certain integer r. Choose numbers s0,s1Jl, rsuch that s0<s < s1. According to Lions and Magenes [4, Theorem 6.2], the mapping uΛlu,withuC(Ω¯),(56)

extends uniquely (by continuity) to an isomorphism Λl:Hsj;sj/2(Ω)Qlsj2,sj/21for eachj{0,1}.(57)

Let ψ be the interpolation parameter from Proposition 5.4. Then the restriction of the operator (57) with j = 0 to the space [Hs0,s0/2(Ω),Hs1,s1/2(Ω)]ψ=Hs0,s0/2;φ(Ω)

is an isomorphism Λl:Hs,s/2;φ(Ω)[Qls02,s0/21,Qls12,s1/21]ψ=Qls2,s/21;φ.(58)

Here, the equalities of spaces hold true up to equivalence of norms due to Proposition 5.5 and Lemma 6.2 (see also Remark 6.3). The operator (58) is an extension by continuity of the mapping (56) because C(Ω¯)is dense in Hs,s/2;(Ω)Thus, Theorems 4.1 and 4.2 are proved in the case considered.

Consider now the case where sEl. Choose ε ∈(0, 1/2) arbitrarily. Since s ±εEland s — ε >2, we have the isomorphisms Λl:Hs±ϵ,s±ϵ)/2;φ(Ω)Qls±ϵ2,s±ϵ)/21;φ(59)

They imply that the mapping (56) extends uniquely (by continuity) to an isomorphism Λl:[Hs,s)/2;φ(Ω),Hs+,s+)/2;φ(Ω)]1/2[Ql;s2,s)/21;Qls+2,s+)/21;φ]1/2=Qls2,s/21;φ.(60)

Recall that the last equality is the definition of the space Qls2,s/21;φ. It remains to prove that Hs,s/2;φ(Ω)=[Hsϵ,sϵ)/2;φ(Ω),Hs+ϵ,s+ϵ)/2;φ(Ω)]1/2(61)

up to equivalence of norms. We reduce the interpolation of Hörmander spaces to an interpolation of Sobolev spaces with the help of Proposition 5.3. Let us choose real S >0 such that s — ε — S >0. According to Proposition 5.5 we have the equalities Hsϵ,sϵ)/2;φ(Ω)=[Hsϵδ,(sϵδ)/2(Ω),Hs+ϵ+δ,(s+ϵ+δ)/2(Ω)]α

and Hs+ϵ,s+ϵ)/2;φ(Ω)=[Hsϵδ,(sϵδ)/2(Ω),Hs+ϵ+δ,(s+ϵ+δ)/2(Ω)]β.

Here, the interpolation parameters α;and β are defined by the formulas α(r):=rδ/(2ϵ+2δ),(r1/(2ϵ+2δ)),β(r):=r(2ϵ+δ)/(2ϵ+2δ),r(1/(2ϵ+2δ))ifr1

and α(r) = ß(r):= 1 if 0 <r <1. Therefore, owing to Propositions 5.3 and 5.5, we get [Hsϵ,(sϵ)/2;φ(Ω),Hs+ϵ,(s+ϵ)/2;φ(Ω)]1/2 =[[Hsδ,(sδ)/2(Ω),Hs++δ,(s++δ)/2(Ω)]α,[Hsδ,(sδ)/2(Ω),Hs++δ,(s++δ)/2(Ω)]β]1/2=[Hsδ,(sδ)/2(Ω),Hs++δ,(s++δ)/2(Ω)]ω=Hs,s/2;φ(Ω).

Here, the interpolation parameter ω is defined by the formulas ω(r):=α(r)(β(r)/α(r))1/2=r1/2,(r1/(2ϵ+2δ))ifr1

and w(r): = 1 if 0 <r <1. Thus, (61) is valid.

Remark 6.4

The spaces defined by formulas (23) and (30) are independent of the choice of the number ε ∈ (0, 1/2) up to equivalence of norms. Indeed, let l ∈{0, 1}, sEl; then according to Theorems 4.1 and 4.2 we have the isomorphisms Λl:Hs,s/2;φ(Ω)[Qls2ϵ,s/21ϵ/2;φQls2+ϵ,s/21+ϵ/2;φ]1/2.(62)

whenever 0 <ε <1/2. This means the required independence.

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About the article

Received: 2016-11-29

Accepted: 2017-01-16

Published Online: 2017-02-15


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 57–76, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0008.

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