Equivalence of pointwise and global ellipticity estimates

Hendrik Vogt

Abstract Defining an elliptic operator −∇ · (a∇) via the form method one normally imposes pointwise conditions on the matrix valued function a in order to get positivity, ellipticity and sectoriality of the form. In this note we show that the pointwise conditions on a are equivalent to the corresponding global ones on the form.

MSC 1991: 35J20 (primary)

Let Ω be an arbitrary open subset of ℝ^{d}, a: Ω → ℂ^{d×d} a locally integrable, hermitian matrix valued function. Define the symmetric form τ in L_{2}(Ω) by τ(u) := ∫ a∇u·∇\overline{u} on D(τ) := C_{c}^{∞}(Ω). If τ is positive and closable then, by the form representation theorem, τ̄ is associated with a positive selfadjoint operator in L_{2}(Ω) (which corresponds to Dirichlet boundary conditions). The main aim of this note is to show that the positivity of the form τ is equivalent to the positivity of the function a, i.e., a ⩾ 0 in the matrix sense a.e.

A case of particular interest is the following: Let a_{1}: Ω → ℝ^{d×d} be locally integrable, symmetric matrix valued and locally strictly elliptic, i.e., for every compact set K ⊆ Ω there exists σ > 0 such that a_{1}(x) ⩾ σ in the matrix sense for almost all x ∈ K. Then it is known that

D(τ_{N}) := \{u ∈ L_{2}(Ω) ∩ W_{2,loc}^{1}(Ω); τ_{N}(u) := ∫ a_{1}∇u · ∇\overline{u} < ∞\}

defines a symmetric Dirichlet form in L_{2}(Ω) (cf. [1, Thm. 1.3.9]; one can show the closedness of τ_{N} like the completeness of the Sobolev space W_{2}^{1}(Ω) because of the local strict ellipticity of a_{1}). The associated selfadjoint operator in L_{2}(Ω) corresponds to Neumann boundary conditions.

Notation. Let |M| denote the Lebesgue measure of a measurable set M ⊆ ℝ^{d}, χ_{M} the characteristic function of M. S_{d−1} is the unit sphere of ℝ^{d}, for the spectral radius of a hermitian matrix A ∈ ℂ^{d×d} we write |A|(= sup_{ξ∈S_{d−1}} |Aξ · ξ|). For a function f: Ω → ℝ we use the shorthand [f > 0] for the set {x ∈ Ω; f(x) > 0} (and similarly [f < g] etc). Q(f) denotes the form domain of the multiplication operator f in L_{2}(Ω).

Theorem. Let a: Ω → ℂ^{d×d} be a locally integrable hermitian matrix valued function, τ(u) := ∫ a∇u · ∇\overline{u} for u ∈ C_{c}^{∞}(Ω).

(a) The following are equivalent:

(i) a ⩾ 0 a.e.,

(ii) τ(u) ⩾ 0 for all u ∈ C_{c}^{∞}(Ω),

(iii) τ is bounded from below, i.e., there exists c > 0 such that τ(u) ⩾ −c||u||_{2}^{2} for all u ∈ C_{c}^{∞}(Ω).

(b) Let D be a sublattice of W_{1,loc}^{1}(Ω)∩L_{2}(Ω) with C_{c}^{∞}(Ω) ⊆ D ⊆ {u; a∇u· ∇\overline{u} ∈ L_{1}(Ω)\}. Let V: Ω → [0, ∞) be a measurable function such that D ∩ Q(V) is dense in L_{2}(Ω). Define the form τ_{0} in L_{2}(Ω) by τ_{0}(u) := ∫ a∇u · ∇\overline{u} + ∫ V |u|^{2} on D(τ_{0}) := D ∩ Q(V). Then τ_{0} is positive if and only if a ⩾ 0 in the matrix sense a.e.

Remark. Let a_{1} and τ_{N} be as in the introduction. Assume that a_{1}ξ · ξ ⩾ |aξ · ξ| a.e. for all ξ ∈ ℝ^{d} (e.g. a_{1} = (|a| + 1)I). Then D := D(τ_{N}) is an example of a sublattice D satisfying the assumption of part (b) of the theorem (or any sublattice D of D(τ_{N}) with C_{c}^{∞}(Ω) ⊆ D). A particular example is D = W_{∞,c}^{1}(Ω), the space of Lipschitz continuous functions with compact support.

Proof of the Theorem. The only nontrivial part of (a) is showing that (iii) implies (i). For this purpose we assume that [a ⩾̸ 0] has positive measure. In the following we construct a function violating (iii), later on we modify this function in order to show that the form τ_{0} defined in (b) is not positive.

Let (ξ_{k}) be a dense sequence in S_{d−1}. If a(x) ⩾̸ 0 for some x ∈ Ω then there exists k ⩾ |a(x)| such that a(x)ξ_{k} · ξ_{k} ⩽ −1/k. We obtain

[a ⩾̸ 0] = ⋃_{k∈ℕ}([aξ_{k} · ξ_{k} ⩽ −1/k] ∩ [|a| ⩽ k]),

so we can fix k ∈ ℕ such that F := [aξ_{k} · ξ_{k} ⩽ −1/k] ∩ [|a| ⩽ k] has positive measure.

Now we localize F: Almost all x ∈ Ω are Lebesgue points of the function χ_{F}, so since F has positive measure there exists x ∈ Ω satisfying

\frac{1}{|B_{δ}(x)|} ∫_{B_{δ}(x)} χ_{F} → χ_{F} (x) = 1 for δ → 0.

Let ε ∈ (0, 1) be given. Fix δ > 0 in such a way that B := B_{δ}(x) ⊆ \overline{B_{δ}(x)} ⊆ Ω and |B ∩F | = ∫_{B} χ_{F} ⩾ (1−ε)|B|. Since the function |a| restricted to B is integrable, there exists α > 0 such that for all measurable sets M ⊆ B with |M| ⩽ α we have ∫_{M} |a| ⩽ ε|B|. By the regularity of the Lebesgue measure there exists an open set Ω_{0}, B ∩ F ⊆ Ω_{0} ⊆ B, with |Ω_{0} \ F | ⩽ α. It follows that

∫_{Ω_{0}\F} |a| ⩽ ε|B| and |Ω_{0}| ⩾ |B ∩ F | ⩾ (1 − ε)|B|.  (1)

We can choose functions u_{n} ∈ C_{c}^{∞}(Ω_{0}) satisfying

0 ⩽ u_{n} ⩽ \frac{1}{n}, ||∇u_{n}||_{∞} ⩽ 1, and |[∇u_{n} ≠ ±ξ_{k}] ∩ Ω_{0}| ⩽ ε|B|

for all n ∈ ℕ. (The reader should think of pieces of hyperplanes orthogonal to ξ_{k} which have distance \frac{2}{n} to each other and to the boundary of Ω_{0}. On these hyperplanes set u_{n} := \frac{1}{n} and extend like a wash board.) The important point is that on the set G := F ∩ [∇u_{n} = ±ξ_{k}] the function a∇u_{n} · ∇u_{n} is less or equal −1/k.

Let H := Ω_{0} \ G. Then |H| ⩽ |Ω_{0} \ F | + |Ω_{0} ∩ [∇u_{n} ≠ ±ξ_{k}]| ⩽ 2ε|B| since |Ω_{0} \ F | ⩽ |B \ F | ⩽ ε|B| by (1). We can estimate

τ(u_{n}) = ∫_{G} aξ_{k} · ξ_{k} + ∫_{H} a∇u_{n} · ∇u_{n} ⩽ −\frac{1}{k}|G| + ∫_{H∩F} |a| + ∫_{H\F} |a|.

By (1) we obtain |G| = |Ω_{0}| − |H| ⩾ (1 − 3ε)|B| and (since |a| ⩽ k on F)

τ(u_{n}) ⩽ −\frac{1}{k}|G| + k|H ∩ F | + ∫_{Ω_{0}\F} |a| ⩽ (−\frac{1}{k}(1 − 3ε) + 2kε + ε) |B|. (2)

Note that ||u_{n}||_{2} ⩽ \frac{1}{n}||χ_{B}||_{2} since 0 ⩽ u_{n} ⩽ \frac{1}{n}. Therefore, given c > 0, it is easy to choose first ε and then n in such a way that τ(u_{n})+c||u_{n}||_{2}^{2} < 0. This completes the proof of (a).

Now we use the sequence (u_{n}) in order to construct a function u ∈ D(τ_{0}) with τ_{0}(u) < 0, thus proving (b). Let ε := 1/(2k^{2} + 5k + 3). Since D(τ_{0}) is a dense sublattice of L_{2}(Ω), there exists a function 0 ⩽ φ ∈ D(τ_{0}) satisfying |[φ < χ_{B}]| < α. Hence

∫_{[φ<χ_{B}]} |a| ⩽ ε|B|  (3)

according to the choice of α. Since φ ∈ Q(V) and 0 ⩽ u_{n} ⩽ \frac{1}{n} (n ∈ ℕ) there exists n_{0} ∈ ℕ with

∫ V |φ ∧ u_{n}|^{2} < ε|B| for all n ⩾ n_{0}.  (4)

By the assumptions on D we have a∇φ · ∇φ ∈ L_{1}(Ω), so there exists α_{2} > 0 such that for all measurable sets M ⊆ Ω with |M| ⩽ α_{2} we have ∫_{M} |a∇φ · ∇φ| ⩽ ε|B|. Obviously ⋂_{n∈ℕ}[0 < φ < 1/n] = ∅, so since u_{n} ⩽ 1/n there exists n ⩾ n_{0} with |[0 < φ < u_{n}]| ⩽ α_{2}. Noting ∇φ = 0 a.e. on [φ = 0], we obtain

∫_{[φ<u_{n}]} |a∇φ · ∇φ| = ∫_{[0<φ<u_{n}]} |a∇φ · ∇φ| ⩽ ε|B|.  (5)

For u := φ ∧ u_{n} ∈ D(τ_{0}) we estimate (noting ∇u = ∇u_{n} a.e. on [u = u_{n}] and ∇u = ∇φ a.e. on [u < u_{n}] = [φ < u_{n}])

τ(u) = ∫_{[u=u_{n}]} a∇u · ∇u + ∫_{[u<u_{n}]} a∇u · ∇u
⩽ ∫ a∇u_{n} · ∇u_{n} + ∫_{[φ<u_{n}]} |a∇u_{n} · ∇u_{n}| + ∫_{[φ<u_{n}]} |a∇φ · ∇φ|.

By (3) and (5) we conclude τ(u) ⩽ τ(u_{n}) + 2ε|B|. From (4) recall ∫ V |u|^{2} ⩽ ε|B|. According to (2) and the choice of ε it follows that

τ_{0}(u) ⩽ τ(u_{n}) + 3ε|B| ⩽ −ε|B| < 0.

This completes the proof of (b).

Now let a: Ω → ℂ^{d×d} be a locally integrable matrix valued function (no longer assumed to be hermitian). Define the sesquilinear form τ_{a} in L_{2}(Ω) by τ_{a}(u) := ∫ a∇u · ∇\overline{u} on C_{c}^{∞}(Ω). We call the function a sectorial if | Im(aξ · ξ̄)| ⩽ α Re(aξ · ξ̄) a.e. for all ξ ∈ ℂ^{d} and some α ⩾ 0 and strictly elliptic if Re(aξ · ξ̄) ⩾ σ|ξ|^{2} a.e. for all ξ ∈ ℂ^{d} and some σ > 0. The form τ_{a} is called sectorial if | Im τ_{a}(u)| ⩽ α Re τ_{a}(u) + c||u||_{2}^{2} and strictly elliptic if Re τ_{a}(u) ⩾ σ||∇u||_{2}^{2} − c||u||_{2}^{2} for all u ∈ C_{c}^{∞}(Ω) and some c ∈ ℝ, α ⩾ 0, σ > 0.

Corollary. The form τ_{a} is sectorial (strictly elliptic) if and only if a is sectorial (strictly elliptic).

Proof. Let a(x)^{∗} denote the adjoint matrix of a(x). Then a_{R} := (a + a^{∗})/2 and a_{I} := (a − a^{∗})/2i are hermitian matrix valued functions. One calculates

Re(aξ · ξ̄) = a_{R}ξ · ξ̄ and Im(aξ · ξ̄) = a_{I}ξ · ξ̄  (6)

for all ξ ∈ ℂ^{d}, so Re τ_{a} = τ_{a_{R}} and Im τ_{a} = τ_{a_{I}}. The form τ_{a} is sectorial with α as above if and only if the forms α Re τ_{a} ± Im τ_{a} = τ_{αa_{R}±a_{I}} are bounded from below. By the above theorem this is equivalent to αa_{R} ± a_{I} ⩾ 0 a.e., i.e. to the sectoriality of a according to (6).

For the second part of the Corollary just apply the theorem to the function a_{R} − σ.

Acknowledgements. The research of the author was supported by the Deutsche Forschungsgemeinschaft. The support is gratefully acknowledged. The author would like to thank Jürgen Voigt for a number of valuable remarks.

References

[1] E. B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, 1989.

Fachrichtung Mathematik Institut für Analysis Technische Universität Dresden D-01062 Dresden Germany vogt@math.tu-dresden.de