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

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

for which a general Borei measurable weight function μ : ℝ^k → (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 Η^s;φ ≔ 𝓑₂,_μ for which the index of regularity is of the form

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 H^s,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

Here and below, we use the following notation for partial derivatives: ∂_t ≔ ∂/∂_t and Dxα:=D1α1…Dnαn, where

D_j ≔ i∂/∂x_j, x = (x₁,..., x_n)∈ℝⁿ, and a ≔ (α₁ ,..., α_n) with 0 ≤ α₁,..., α_n ∈ℤ and |α|≔ α₁ +…+α_n. We suppose that all the coefficients a_α 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]):

3 Hörmander spaces

Among the normed function spaces 𝓑_p,μ introduced by Hörmander in [10, Section 2.2], we use the inner product spaces H^μ(ℝ^k) ≔ 𝓑_2,μ defined over ℝ^k, with 1 ≤ k ∈ ℤ. Here, μ : ℝ^k → (0, ∞) is an arbitrary Borei measurable function that satisfies the following condition: there exist positive numbers c and l such that

By definition, the (complex) linear space H^μ (ℝ^k)consists of all tempered distributions w ∈ S′(ℝ^k)whose Fourier transform w^ is a locally Lebesgue integrable function subject to the condition

The inner product in H^μ (ℝ^k)is defined by the formula

where w₁,w₂ ∈ H^μ(ℝ^k). This inner product induces the norm

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

A version of H^μ(ℝ^k) for an arbitrary nonempty open set V ⊂ ℝ^k is introduced in the standard way. Namely,

where u ∈ H^μ(V). Here, as usual, w ↾ V stands for the restriction of the distribution w ∈ H^μ(ℝ^k) to the open set V. In other words, H^μ(V)is the factor space of the space H^μ(ℝ^k) by its subspace

Thus, H^μ(V)is a separable Hilbert space. The norm (5) is induced by the inner product

where w_j ∈ H^μ(ℝ^k), w_j = u_j in V for each j ∈ {1, 2}, and Υ is the orthogonal projector of the space H^μ(ℝ^k) onto its subspace (6). The spaces H^μ(V)and HQμ(Rk) were introduced and investigated by Volevich and Paneah [11, Section 3].

It follows directly from the definition of H^μ(V)and properties of H^μ(ℝ^k) that the space H^μ(V)is continuously embedded in the linear topological space 𝓓′(V)of all distributions on V and that the set

is dense in H^μ(V).

Suppose that the integer k ≥ 2. Dealing with the above-stated parabolic problems, we need the Hörmander spaces H^μ(ℝ^k) and their versions in the case where the regularity index μ takes the form

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

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

Let s ∈ ℝ and φ ∈ 𝓜. We put H^s,s/2;φ (ℝ^k)≔ H^μ(ℝ^k)in the case where μ is of the form (7). Specifically, if φ(r)≡1, then H^s,s/2;φ (ℝ^k) becomes the anisotropic Sobolev inner product space H^s,s/2 (ℝ^k) of order (s, s/2). Generally, if φ ∈ 𝓜 is arbitrary, then the following continuous and dense embeddings hold:

Indeed, let s₀ < s < s₁; since φ ∈ 𝓜, there exist positive numbers c₀ and c₁ such that c0rs0−s≤,φ(r)≤c1rs1−s for every r ≥ 1 (see e.g., [35, Section 1.5, Property 1°]). Then

for arbitrary ξ′ ∈ ℝ^k–1 and ξ_k ∈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

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 H^s,s/2;φ(Ω)≔ H^μ(Ω) in the case where μ is of the form (7) with k ≔ n + 1. For the function space H^s,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 H^s,s/2;φ(S)with the help of special local charts on S. Let s > 0 and φ ∈ 𝓜. We put H^s,s/2;φ(Π)≔ H^μ(Π)for the strip Π ≔ ℝ^n-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 θ_j : ℝ^n–1 ↔ Γ_j· with j = 1,..., λ. Here, the open sets Γ₁,..., Γ_λ make up a covering of Γ. We also arbitrarily choose functions χ_j ∈ C∞(Γ), with j = 1,..., λ, so that supp χ_j ⊂ Γ_j and χ₁ +… χ_λ = 1 on Γ.

By definition, the linear space H^S,S/,2;φ(S)consists of all square integrable functions g : S →·ℂ that the function

belongs to H^S,S/,2;φ(Π) for each number j ∈ {1,... ,λ}. The inner product in H^S,S/,2;φ(S)is defined by the formula

where g,g’ ∈ H^s,s/2;φ(S). This inner product naturally induces the norm

The space H^s,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 :

where θj∗(x,t):=(θj(x),t)for allx∈Rn−1 and t ∈ (0, τ).

We also need isotropic Hörmander spaces H^S;φ(V)over an arbitrary open nonempty set V ⊆ ℝ^k with k ≥ 1. Let s ∈ ℝ and φ ∈ 𝓜. We put H^s;φ(V)≔ H^μ(V)in the case where the regularity index μ takes the form

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

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

where ω, ω’ ∈ H^s;φ(Γ). It induces the norm

The space H^S;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

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 H^S,S/2;φ(·)and H^S;φ(·) become the Sobolev spaces H^S,S/2(·) and H^s(·) respectively. It follows directly from (8) that

Analogously,

see [18, Theorems 2.3(iii) and 3.3(iii)]. These embeddings are continuous and dense. Of course, if s = 0, then H^s (·) = H^s,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 H^s,s/2;φ(·) and H^s;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

with the problem (l)-(3). Let real s ≥ 2; the mapping (14) extends uniquely (by continuity) to a bounded linear operator

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 H^S,S/2(Ω), we define the right-hand sides

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)∈Hs−1−2k(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

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)∈Hs−3/2−2k(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

holds for these integers k. The right-hand part of this equality is well defined because the function ∂tku(⋅,0)∈Hs−1−2k(G) has the trace ∂tku(⋅,0)↾Γ∈Hs−3/2−2k(Γ) in view of s - 3/2- 2k > 0. Now, substituting (17) into (18), we obtain the compatibility conditions

Here, the functions v_k are defined by the recurrent formula

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

these relations holding for almost all x ∈ G. Since

due to (16), the trace v_k ↾ Γ ∈ H^s–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

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 Η^S,S/2;φ(Ώ) as the source space of this isomorphism; otherwise speaking, Η^S,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

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 H0s−2,s/2−1;φ and is denoted by Q0s−2,s/2−1;φ. We separately define this space in the s ∉ E₀case and s ∈ E₀case.

Suppose first that s ∉ E₀. By definition, the linear space Q0s−2,s/2−1;φ consists of all vectors (f,g,h)∈H0s−2,s/2−1;φ that satisfy the compatibility conditions (19). As we have noted, these conditions are well defined for every (f,g,h)∈H0s−2−ϵ,s/2−1−ϵ/2 for sufficiently small ε > 0. Hence, they are also well defined for every (f,g,h)∈H0s−2,s/2−1;φ due to the continuous embedding

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

We endow the linear space Q0s−2,s/2−1;φ with the inner product and norm in the Hilbert space H0s−2,s/2−1;φ.

The space Q0s−2,s/2−1;φ is complete, i.e. a Hilbert one. Indeed, if the number ε > 0 is sufficiently small, then

Here, the space Q0s−2−ϵ,s/2−1−ϵ/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 H0s−2,s/2−1;φ due to (22). Thus, the space Q0s−2,s/2−1;φ is complete (with respect to the latter norm).

If s ∈ E₀, then we define the Hilbert space Q0s−2,s/2−1;φ by means of the interpolation between its analogs just introduced. Namely, we put

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 Q0s−2,s/2−1;φ defined by formula (23) does not depend on the choice of ε up to equivalence of norms and is continuously embedded in H0s−2,s/2−1;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).

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 X₁⊆X₀and 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 ||J_v||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 : X₁↔ X₀.

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/ψ 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, Χ_ψ denote the domain of ψ (J) endowed with the inner product (v₁, v₂)X_ψ := (ψ(J)v₁, ψ(J)v₂)X₀and the corresponding norm ||v||X_ψ≔ ||ψ(J)v||X₀. The linear space Χ_ψis 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 X_jis a bounded operator Τ : X_j→ Y_jfor each j ∈ {0, 1}, then the restriction of Τ to Χ_ψis also a bounded operator Τ : Χ_ψ→ Υ_ψ.

If ψ is an interpolation parameter, then we say that the Hilbert space Χ_ψis 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 embeddingsX₁↪ Χ_ψ↪ 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/ψ 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) ≡ r^θis 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 Χ_ψis 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.

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 g ∈ H^s,s/2;φ(S).