The regular part of symmetric forms

associated with second order elliptic differential expressions

Hendrik Vogt

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

Abstract

Let τ be a (not necessarily closable) positive symmetric form associated with a second order elliptic differential expression. We show that the regular part of τ (in the sense of B. Simon) can be obtained by modifying the coefficients of τ suitably; in particular, the regular part is again associated with a second order elliptic differential expression.

MSC 2000: 35J15, 47A07

Selfadjoint second order elliptic differential operators in divergence form are commonly defined by the form method: One starts with a positive symmetric form determined by the coefficients of the operator and the boundary conditions under consideration. If this form is closable, one defines the elliptic differential operator via Kato’s representation theorem as the selfadjoint operator associated with the closure of the form.

In [ERSZ07], this approach is generalised as follows: Instead of assuming the form to be closable, the regular part of the form is taken; since the latter is always closable, one can then proceed as before. The aim of the present note is to demonstrate that by this approach one does not obtain new types of elliptic differential operators. Indeed, taking the regular part amounts to a modification of the coefficients of the form; see Theorem 1 below.

Before specifying our assumptions and notation, we recall the definition of the regular and singular parts of a positive symmetric form, which is due to Simon [Sim78; Sec. 2]. Let τ be a positive symmetric form in a Hilbert space H, and endow D(τ) with the norm || · ||_{τ} given by ||u||_{τ}^{2} = τ(u) + ||u||_{2}^{2}. Let { \atop D(τ)} be the completion of D(τ) and i_{τ} the continuous extension of the embedding D(τ) ↪ H to { \atop D(τ)}. Denote by Q_{τ} the orthogonal projection from { \atop D(τ)} onto the kernel of i_{τ}. Then the regular part τ_{r} and the singular part τ_{s} of τ are the symmetric forms in H defined by D(τ_{r}) := D(τ_{s}) := D(τ),

τ_{s}(u, v) := ⟨Q_{τ} u, v⟩{ \atop D(τ)},  τ_{r}(u, v) := τ(u, v) − τ_{s}(u, v).  (1)

By [Sim78; Thms. 2.1 and 2.2], τ_{r} is the largest closable symmetric form less than τ, and τ_{r} is positive. (In [Sim78; Sec. 2], it is assumed that H is a complex Hilbert space and that τ is densely defined, but these assumptions are not needed for the definition and properties of τ_{r} and τ_{s}.)

Let now Ω ⊆ ℝ^{n} be an open set and μ a positive Radon measure on Ω with spt μ = Ω. Let a: Ω → ℝ^{n×n} be a locally integrable function with values in the symmetric, positive semidefinite matrices. We define the symmetric form τ_{max} in L_{2}(μ) by

D(τ_{max}) := {u ∈ L_{2}(μ) ∩ W_{1,loc}^{1}(Ω); a∇u · ∇u ∈ L_{1}(Ω)},
τ_{max}(u, v) := ∫_{Ω} a∇u · ∇v.

(Throughout we assume the function spaces to be real vector spaces.) Let τ be a restriction of τ_{max} with the following two properties:

(i) D(τ) ∩ L_{∞}(μ) is invariant under multiplication with C_{c}^{∞}(Ω)-functions, (ii) there exists ψ ∈ C_{b}^{1}(ℝ) with ψ(0) = 0 and ψ′(0) = 1 such that ψ ∘ u ∈ D(τ) for all

u ∈ D(τ). One easily checks that these properties are satisfied, e.g., if τ = τ_{max}, D(τ) = D(τ_{max}) ∩ C^{∞}(Ω) or D(τ) = C_{c}^{∞}(Ω), but also periodic type boundary conditions are covered.

Since a takes its values in the symmetric, positive semidefinite matrices, one can pointwise take the square root of a. The function a^{1/2} thus obtained is measurable: For f ∈ C(ℝ) one sees by approximation with polynomials that the mapping B ↦ f(B), in the space of all selfadjoint operators B ∈ L(ℝ^{N}), is continuous. For f(t) := (t^{+})^{1/2} it follows that the function x ↦ a(x)^{1/2} on Ω is measurable.

The following is the main theorem of this note; its proof will be given below.

Theorem 1. Under the above assumptions, the regular part τ_{r} of τ is given by

τ_{r}(u, v) = ∫ a_{r}∇u · ∇v (u, v ∈ D(τ_{r}) = D(τ)),

where a_{r} = a^{1/2}pa^{1/2} for some measurable function p: Ω → ℝ^{n×n} with values in the orthogonal projection matrices.

Corollary 2. Assume that n = 1 and D(τ) = C_{c}^{∞}(Ω). Then the assertion of Theorem 1 holds with a_{r} = a11_{Ω_{0}}, where

Ω_{0} = {x ∈ Ω; ∃ ε > 0: ∫_{x−ε}^{x+ε} a^{−1} < ∞}.

Proof. For A ⊆ Ω let τ_{A} be the symmetric form in L_{2}(μ) defined by D(τ_{A}) = C_{c}^{∞}(Ω), τ_{A}(u, v) = ∫_{Ω} a11_{A}u′v′. It follows from [RöWi85; Thm. 1.1] that A = Ω_{0} is the largest set A ⊆ Ω such that τ_{A} is closable. By Theorem 1, this implies the assertion.

For the proof of Theorem 1 we need the following criterion for an orthogonal projection in a vector-valued L_{2}-space to be a multiplication operator.

Proposition 3. Let Ω ⊆ ℝ^{n} be an open set and H a separable Hilbert space. Let G be a closed subspace of L_{2}(Ω; H) satisfying φf ∈ G for all φ ∈ C_{c}^{∞}(Ω), f ∈ G. Then the orthogonal projection Q from L_{2}(Ω; H) onto G is given by

(Qf)(x) = q(x)f(x)  (f ∈ L_{2}(Ω; H), x ∈ Ω),  (2)

for some strongly measurable function q: Ω → L(H) with values in the orthogonal projections.

Proof. Below we will show that Qf = 0 a.e. on [f = 0] := {x ∈ Ω; f(x) = 0}, for all f ∈ L_{2}(Ω; H). Then from [ArTh05; Thm. 2.3 and Cor. 2.4] (see also [Eva76; Thm. 5.7]) it follows that Q is given by (2), for some strongly measurable function q: Ω → L(H) with ||q||_{L_{∞}(Ω;L_{s}(H))} = ||Q|| ⩽ 1. For all f ∈ L_{2}(Ω; H) we obtain that q^{2}f = Q^{2}f = Qf = qf, so q^{2} = q a.e. Recalling that projections with norm less than or equal to 1 are orthogonal, we can thus assume without loss of generality that all the operators q(x) are orthogonal projections, and the assertion is proved.

Let now f ∈ L_{2}(Ω; H); we show that Qf = 0 a.e. on [f = 0]. For A := [f ≠ 0] we find by convolution a sequence (φ_{k}) ⊆ C_{c}^{∞}(Ω) such that φ_{k} → 11_{A} a.e. and ||φ_{k}||_{∞} ⩽ 1 for all k ∈ ℕ. Then φ_{k}Qf → 11_{A}Qf in L_{2}(Ω; H), so the assumption implies that 11_{A}Qf ∈ G. Since

||f − 11_{A}Qf||_{2} = ||11_{A}(f − Qf)||_{2} ⩽ ||f − Qf||_{2}

and Qf is the best approximation of f in G, we conclude that 11_{A}Qf = Qf, i.e., Qf = 0 a.e. on Ω \ A = [f = 0].

Proof of Theorem 1. We endow D(τ) with the norm || · ||_{τ} given by ||u||_{τ}^{2} = τ(u) + ||u||_{2}^{2}. Then the embedding

D(τ) ↪ L_{2}(μ) × L_{2}(Ω)^{n}, u ↦ (u, a^{1/2}∇u)

is isometric, so we can consider the completion { \atop D(τ)} of D(τ) as a subspace of L_{2}(μ)×L_{2}(Ω)^{n}. The continuous extension of the embedding D(τ) ↪ L_{2}(μ) to { \atop D(τ)} has the kernel \{0\} × H_{s}, where

H_{s} := \{w ∈ L_{2}(Ω)^{n}; (0, w) ∈ { \atop D(τ)}\}.

Let Q be the orthogonal projection from L_{2}(Ω)^{n} onto H_{s}. Then the orthogonal projection from { \atop D(τ)} onto \{0\} × H_{s} is given by (u, w) ↦ (0, Qw). Thus, by (1) the singular part τ_{s} of τ is given by

τ_{s}(u, v) = ⟨(0, Q(a^{1/2}∇u)), (v, a^{1/2}∇v)⟩{ \atop D(τ)} = ∫_{Ω} Q(a^{1/2}∇u) · a^{1/2}∇v

for all u, v ∈ D(τ_{s}) = D(τ).

Below we will show that φf ∈ H_{s} for all φ ∈ C_{c}^{∞}(Ω), f ∈ H_{s}. Then by Proposition 3 we obtain that the projection Q is given by

(Qf)(x) = q(x)f(x) (f ∈ L_{2}(Ω)^{n}, x ∈ Ω),

for some measurable function q: Ω → ℝ^{n×n} with values in the orthogonal projection matrices. We conclude that τ_{s}(u, v) = ∫_{Ω} a^{1/2}qa^{1/2}∇u · ∇v for all u, v ∈ D(τ_{s}), and the assertion follows (with p = id −q) since τ_{r} = τ − τ_{s}.

Let f ∈ H_{s}. Then there exists a sequence (ũ_{k}) ⊆ D(τ) such that

ũ_{k} → 0 in L_{2}(μ),  a^{1/2}∇ũ_{k} → f in L_{2}(Ω)^{n}  (k → ∞),

without loss of generality ũ_{k} → 0 a.e. Let ψ ∈ C_{b}^{1}(ℝ) be as in property (ii) of τ. Then u_{k} := ψ ∘ ũ_{k} ∈ D(τ) for all k ∈ ℕ. Moreover, the sequence (ψ′ ∘ ũ_{k})_{k} is bounded in L_{∞}(Ω) and ψ′ ∘ ũ_{k} → ψ′(0) = 1 a.e., so we obtain that

u_{k} → 0 in L_{2}(μ),  a^{1/2}∇u_{k} = (ψ′ ∘ ũ_{k})a^{1/2}∇ũ_{k} → f in L_{2}(Ω)^{n}  (k → ∞).

Let now φ ∈ C_{c}^{∞}(Ω). Then

||u_{k}a^{1/2}∇φ||_{L_{2}(Ω)^{n}}^{2} = ∫_{Ω} |u_{k}|^{2}a∇φ · ∇φ → 0 (k → ∞)

by the dominated convergence theorem (use ||ψ||_{∞}^{2}a∇φ · ∇φ as a dominating function), and hence

a^{1/2}∇(φu_{k}) = φa^{1/2}∇u_{k} + u_{k}a^{1/2}∇φ → φf in L_{2}(Ω)^{n} (k → ∞).

Moreover, φu_{k} ∈ D(τ) for all k ∈ ℕ by property (i) of τ and φu_{k} → 0 in L_{2}(μ) as k → ∞, so we conclude that φf ∈ H_{s}.

Acknowledgement. The author thanks Jürgen Voigt for useful discussions.

References

[ArTh05] W. Arendt and S. Thomaschewski, Local operators and forms, Positivity 9 (2005), no. 3, 357–367.

[ERSZ07] A. F. M. ter Elst, D. W. Robinson, A. Sikora and Y. Zhu, Second-order operators with degenerate coefficients, Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 299–328.

[Eva76] D. E. Evans, Time dependent perturbations and scattering of strongly continuous groups on Banach spaces, Math. Ann. 221 (1976), no. 3, 275–290.

[RöWi85] M. Röckner and N. Wielens, Dirichlet forms—closability and change of speed measure, in Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), pp. 119–144, Res. Notes in Math. 124, Pitman, Boston, 1985.

[Sim78] B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), no. 3, 377–385.