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Decay Rate Estimates for the Wave Equation with Subcritical Semilinearities and Locally Distributed Nonlinear Dissipation

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Abstract

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.

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Correspondence to V. H. Gonzalez Martinez.

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Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant 304895/2003-2. Research of Victor Hugo Gonzalez Martinez is partially supported by CAPES Grant 88882.449176/2019-01. Research of Türker Özsarı is supported by the Science Academy’s Young Scientist Award in Mathematics (BAGEP 2020).

Appendix A: Preliminaries on Microlocal Analysis

Appendix A: Preliminaries on Microlocal Analysis

In this section, we supplement details to proofs of some of the theorems on pseudo-differential operators and microlocal defect measures, whose original versions in French can be found in the elegant lecture notes of Burq and Gérard [7].

1.1 A.1. Pseudo-differential Operators

Let \(\Omega \) be an open and nonempty subset of \({\mathbb {R}}^d\), \(d\ge 1\). A differential operator on \(\Omega \) is a linear map \(P:{\mathcal {D}}'(\Omega ) \rightarrow {\mathcal {D}}'(\Omega )\) of the form

$$\begin{aligned} P u(x) := \sum _{|\alpha |\le m} a_\alpha (x) \partial ^\alpha _x u(x),~(\partial ^\alpha :=\partial _{x_1}^{\alpha _1} \cdots \partial _{x_d}^{\alpha _d}) \end{aligned}$$
(A.1)

where \(a_\alpha \in C^\infty (\Omega )\) are complex valued functions. The greatest integer m such that the functions \(a_\alpha \),  \(|\alpha |=m\) are not all zero is called the order of P. The map \(p:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\), defined by

$$\begin{aligned} p(x,\xi ):= \sum _{|\alpha |\le m}a_\alpha (x)\, (i\xi )^{\alpha }, \end{aligned}$$

is called the symbol of P.

We observe that P is characterized by the identity

$$\begin{aligned} P(e_\xi )(x)=p(x,\xi )\,e_\xi (x),~~\hbox { where }~e_\xi (\cdot ) =e^{ i\,\left<(\cdot ),\xi \right>}= e^{ i\,(\cdot )\cdot \xi }. \end{aligned}$$
(A.2)

Adopting to the notation

$$\begin{aligned} D=\frac{1}{i} \partial , \ D_j=\frac{1}{ i}\partial _j \hbox { and }D^\alpha = \frac{1}{i^{|\alpha |}}\partial ^\alpha , \end{aligned}$$

the operator P can be rewritten as

$$\begin{aligned} P= \sum _{|\alpha |\le m} a_\alpha (x) i^{|\alpha |}D^\alpha = p(x,D). \end{aligned}$$
(A.3)

The formula (A.2) can be generalized as follows: for all \((x,\xi )\in \Omega \times {\mathbb {R}}^d\) and for all \(u\in {\mathcal {D}}(\Omega )\),

$$\begin{aligned} P(u e_\xi ) = p(x,D)(u e_\xi )= e_\xi \,p(x,\xi +D)(u) =e_\xi \, \sum _{| \alpha | \le m} \frac{\partial _\xi ^\alpha p(x,\xi )}{\alpha !} D^\alpha u,\nonumber \\ \end{aligned}$$
(A.4)

where the previous sum is finite once p is a polynomial in the variable \(\xi \).

If P is a differential operator of order m and symbol p, then the principal symbol of order m, denoted by \(\sigma _m(P)\), is the homogeneous part of degree m in \(\xi \) of the polynomial function \(p(x,\xi )\), namely

$$\begin{aligned} \sigma _m(P)(x, \xi ) = \sum _{|\alpha |=m}a_\alpha (x) (i\xi )^\alpha . \end{aligned}$$
(A.5)

Definition A.1

Let \(m\in {\mathbb {R}}\). Then a symbol of order at most m in \(\Omega \) is said to be a function \(a:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\) of class \(C^\infty \), with support in \(K \times {\mathbb {R}}^d\), where K is a compact subset of \(\Omega \), such that for all \(\alpha \in {\mathbb {N}}^d\), \(\beta \in {\mathbb {N}}^d\), there exists a constant \(C_{\alpha ,\beta }\) with

$$\begin{aligned} \left| \partial _x^\alpha \partial _\xi ^\beta a(x,\xi ) \right| \le C_{\alpha ,\beta ,K} \left( 1+ |\xi |\right) ^{m-|\beta |} . \end{aligned}$$
(A.6)

We shall denote the vectorial space of all symbols of order at most m in \(\Omega \) by \({\mathcal {S}}_c^m(\Omega )\) .

Proposition A.1

If \(a\in {\mathcal {S}}_c^m(\Omega )\), the formula

$$\begin{aligned} Au(x) = \frac{1}{(2 \pi )^d}\int _{{\mathbb {R}}^d} e^{i x\cdot \xi } a(x,\xi )\hat{u}(\xi )\,d\xi \end{aligned}$$
(A.7)

defines, for all \(u\in C_0^\infty (\Omega )\), an element Au of \(C_0^\infty (\Omega )\).

The formula (A.7) defines a linear map \(A:C_0^\infty (\Omega ) \longrightarrow C_0^\infty (\Omega )\), which is called the pseudo-differential operator of order m and symbol a. We will often denote the map A by a(xD).

The set of all pseudo-differential operators of order m on \(\Omega \) will be denoted by \(\Psi ^m_c(\Omega )\).

Definition A.2

An operator \(A\in \Psi ^m_c(\Omega )\) is essentially homogeneous if there exists a function \(a_m=a_m(x,\xi )\) with \(\mathrm{supp}a_m \subset K \times ({\mathbb {R}}^{d}\setminus \{0\})\), homogeneous of order m in \(\xi \) and smooth except at \(\xi =0\) and a function \(\chi \in C^\infty ({\mathbb {R}}^d)\) being zero near 0 and 1 in the infinity such that

$$\begin{aligned} a(x,\xi )=a_m(x,\xi )\chi (\xi )+ r(x,\xi ), \end{aligned}$$
(A.8)

for some \(r\in {\mathcal {S}}^{m-1}_{c}(\Omega )\).

Proposition A.2

Let \(A\in \Psi ^m_c(\Omega )\) be essentially homogeneous. Then, for all \(u\in C_0^\infty (\Omega )\), \(\xi \in {\mathbb {R}}^d\backslash \{0\}\), and \(x\in \Omega \),

$$\begin{aligned} t^{-m} e^{- i (t x)\cdot \xi } A(u e_{t\xi })(x) \rightarrow a_m(x,\xi )u(x),~\hbox { as }t\rightarrow +\infty . \end{aligned}$$
(A.9)

Definition A.3

Under the conditions of Proposition A.2, we say that A admits a principal symbol of order m. The function \(a_m\) characterized by (A.9) is called the principal symbol of order m of A and is denoted by \(\sigma _m(A)\).

Theorem A.1

Let A be a pseudo-differential operator of symbol \(a\in {\mathcal {S}}_{c}^{m}(\Omega )\) with \(\mathrm{supp}a \subset K \times {\mathbb {R}}^{d}\), and let \(\chi \in C_0^\infty (\Omega )\) satisfy \(\chi =1\) in a neighborhood of K. Then, there exists a pseudo-differential operator \(A_{\chi }^*\) on \(\Omega \) such that, for all \(u,v \in C_0^\infty (\Omega )\),

$$\begin{aligned} \left( A(\chi u),v\right) _{L^2(\Omega )} =\left( u, A_{\chi }^*v \right) _{L^2(\Omega )}. \end{aligned}$$

In addition, \(A_{\chi }^*\) admits a symbol \(a_\chi ^*\in {\mathcal {S}}_{c}^{m}(\Omega )\) verifying, for all \(N\in {\mathbb {N}}\),

$$\begin{aligned} a_\chi ^*- \sum _{|\alpha |\le N} \frac{1}{\alpha !} D_x^\alpha \partial _\xi ^\alpha \overline{a}\in {\mathcal {S}}_{c}^{m-N-1}(\Omega ). \end{aligned}$$
(A.10)

In particular, if A admits a principal symbol of order m, it is the same of \(A^*\), and

$$\begin{aligned} \sigma _m(A_\chi ^*)= \overline{\sigma _m(A)}. \end{aligned}$$
(A.11)

Theorem A.2

Let A and B be pseudo-differential operators of symbols \(a\in {\mathcal {S}}^{m}_{c}(\Omega )\), \(b\in {\mathcal {S}}_{c}^{n}(\Omega )\), respectively. Then, the composition AB is a pseudo-differential operator admitting a symbol \(a\#b \in S_c^{m+n}(\Omega )\) which satisfies

$$\begin{aligned} a\# b -\sum _{|\alpha |\le N} \frac{1}{\alpha !} \partial _\xi ^\alpha a D_x^\alpha b \in {\mathcal {S}}_{c}^{m+n-N-1}(\Omega ), \end{aligned}$$
(A.12)

for all \(N \in {\mathbb {N}}\).

In addition, if A admits a principal symbol of order m and B admits a principal symbol of order n, then AB admits a principal symbol of order \(m+n\) and [AB] admits a principal symbol of order \(m+n-1\), given by

$$\begin{aligned}&\sigma _{m+n}(AB) =\sigma _m(A) \sigma _n(B), \end{aligned}$$
(A.13)
$$\begin{aligned}&\sigma _{m+n-1}([A,B])=\frac{1}{i}\left\{ \sigma _m (A), \sigma _n(B)\right\} . \end{aligned}$$
(A.14)

Definition A.4

For any compact set K contained in \(\Omega \) and for all \(s\in {\mathbb {R}}\), \(H_K^s(\Omega )\) shall denote the space of distributions with compact support in K whose extensions by zero belong to \(H^s({\mathbb {R}}^d)\), i.e.,

$$\begin{aligned} H_K^s(\Omega ) \equiv \{u \in {\mathcal {E}}'(\Omega ): \tilde{u} \in H^s({\mathbb {R}}^d) \hbox { and } supp\, u \subset K \} \end{aligned}$$

We set \(H_{comp}^s(\Omega )=\underset{K}{\bigcup }H_K^s(\Omega )\) where K ranges over all compact subsets of \(\Omega \). We equip \(H_{comp}^s(\Omega )\) with the finest locally convex topology such that all the inclusion maps

$$\begin{aligned} H_K^s(\Omega ) \hookrightarrow H_{comp}^s(\Omega ) \end{aligned}$$

are continuous.

Theorem A.3

Let \(a\in {\mathcal {S}}_{c}^{m} (\Omega )\) and let K be the projection on \(\Omega \) of \(\mathrm{supp}(a)\). Thus, for all \(s\in {\mathbb {R}}\), the operator defined in (A.7) admits a unique extension to a linear and continuous map from \(H_{comp}^s(\Omega )\) in \(H_K^{s-m}(\Omega )\).

Remark A.1

If \(A \in \Psi _{c}^{m}(\Omega )\) is a pseudo-differential operator with \(m < 0\) then \(A: L_{comp}^{2}(\Omega ) \rightarrow L^{2}(\Omega )\) is a compact operator. Indeed, from Rellich’s Theorem the inclusion \(H_{comp}^{-m} \hookrightarrow L^2(\Omega )\) is compact.

Remark A.2

Let \(P \in \Psi ^m(\Omega )\) be a compactly supported operator, then P extends continuously from \(H_{loc}^{s}(\Omega )\) to \(H^{s-m}_{comp}(\Omega ).\)

Theorem A.4

(A Gårding type inequality) Let A be a pseudo-differential operator of order 0 on \(\Omega \) whose principal symbol \(\sigma _0(A)\) exists and is a positive function in \({\mathcal {S}}_{c}^{0}(\Omega )\). Then, for all \(\delta >0\) there exists \(C_\delta \) such that

$$\begin{aligned} Re\, (Av,v)_{L^2}\ge -\delta \Vert v\Vert _{L^2}^2 - C_\delta \Vert v\Vert _{H^{-1/2}}^2,~\hbox { for all }v\in L_{comp}^2. \end{aligned}$$
(A.15)

Furthermore, there exists \(C>0\) such that

$$\begin{aligned} |Im (Av,v)_{L^2}| \le C \Vert v\Vert _{H^{-1/2}}^2, ~\hbox { for all }v\in L_{comp}^2. \end{aligned}$$
(A.16)

1.2 A.2. Microlocal Defect Measures

Let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L^2_{loc}(\Omega )\), i.e.,

$$\begin{aligned} \sup _{n\in {\mathbb {N}}} \int _K |u_n(x)|^2\,dx<+\infty , \end{aligned}$$

for any compact set K contained in \(\Omega \).

We shall say that \(u_n\) converges weakly to \(u\in L^2_{loc}(\Omega )\) if one has

$$\begin{aligned} \int _\Omega u_n(x)f(x)\,dx \underset{n\rightarrow \infty }{\longrightarrow }\int _\Omega u(x)f(x)\,dx \end{aligned}$$

for each \(f\in L_{comp}^2(\Omega )\).

We describe the loss of strong convergence of the sequence \(u_n\) to 0 in \(L^2_{loc}(\Omega )\), by means of a positive Radon measure on \(\Omega \times S^{d-1}\), that is, \(u_n \rightarrow 0\) in \(L^2_{loc}(\Omega )\) if, and only if, \(\mu =0\).

Lemma A.1

Let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Let A be a pseudo-differential operator of order 0 on \(\Omega \) that admits a principal symbol \(\sigma _0(A)\) of order 0. If \(\sigma _0(A) \ge 0\), then one has

$$\begin{aligned} Im (A(\chi u_n), \chi u_n)_{L^2}\underset{n\rightarrow +\infty }{\longrightarrow }0~\hbox { and }~\liminf _{n\rightarrow +\infty } Re(A(\chi u_n),\chi u_n)_{L^2}\ge 0. \end{aligned}$$

Proof

First, note that since \(u_n\) is bounded in \(L_{loc}^2(\Omega )\) and converges weakly to 0, we have

$$\begin{aligned} \Vert \chi u_n\Vert _{H^{-1/2}}^2 \underset{n\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$
(A.17)

Indeed, recall that

$$\begin{aligned} \Vert \chi u_{n}\Vert _{H^{-1/2}}^2 = (2\pi )^{-d}\int _{{\mathbb {R}}^d} (1+|\xi |^2)^{-1/2}|\widehat{\chi u_n}(\xi )|^2\,d\xi , \end{aligned}$$

where

$$\begin{aligned} \widehat{\chi u_n}(\xi ) =\int _{{\mathbb {R}}^d} \chi (x) u_n(x) e^{- ix\cdot \xi }\,dx \end{aligned}$$

tends to 0 for all \(\xi \in {\mathbb {R}}^d\) and remains uniformly bounded in \(\xi \) and n. Indeed, from the Cauchy–Schwarz inequality we have

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{d}}{\chi (x)u_{n}(x)e^{- i x \xi }}dx \right|= & {} \left| \int _{\mathrm{supp}( \chi )}{\chi (x)u_{n}(x)e^{- i x \xi }}dx \right| \\\le & {} \limsup _{n \rightarrow \infty } \Vert \chi u_{n} \Vert _{L^{2}} \int _{\mathrm{supp}(\chi )} 1 dx < \infty . \end{aligned}$$

Applying the dominated convergence theorem, we have, for all \(R>0\),

$$\begin{aligned} (2\pi )^{-d}\int _{|\xi |\le R} (1+|\xi |^2)^{-1/2}|\widehat{\chi u_n} (\xi )|^2\,d\xi \le (2\pi )^{-d}\int _{|\xi |\le R} |\widehat{\chi u_n} (\xi )|^2\,d\xi \underset{n\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$

On the other hand, the Plancherel theorem yields

$$\begin{aligned} (2\pi )^{-d}\int _{|\xi |>R} |\widehat{\chi u_n}(\xi )|^2(1+|\xi |^2)^{-1/2} \,d\xi \le \frac{1}{R}\Vert \chi u_n\Vert _{L^2}^2, \end{aligned}$$

so,

$$\begin{aligned} \limsup _{n\rightarrow +\infty }\Vert \chi u_n\Vert _{H^{-1/2}}^2 \le \frac{1}{R}\limsup _{n\rightarrow +\infty }\Vert \chi u_n\Vert _{L^2}^2, \end{aligned}$$

which proves (A.17), since R is arbitrary. Applying (A.15) and (A.16) to \(v=\chi u_n\), letting \(n\rightarrow +\infty \), and using (A.17), we obtain

$$\begin{aligned} \liminf _{n\rightarrow +\infty }Re (A(\chi u_n),\chi u_n)_{L^2}\ge & {} \liminf _{n\rightarrow +\infty } [-\delta \Vert \chi u_n\Vert _{L^2}^2] + \liminf _{n\rightarrow 0}[-C_\delta \Vert \chi u_n \Vert _{H^{-1/2}}] \\\ge & {} -\delta \limsup _{n\rightarrow +\infty }\Vert \chi u_n\Vert _{L^2}^2. \end{aligned}$$

In addition,

$$\begin{aligned} Im (A(\chi u_n), \chi u_n)_{L^2} \underset{n\rightarrow +\infty }{\longrightarrow }0, \end{aligned}$$

which proves the lemma since \(\delta >0\) is arbitrary. \(\square \)

Theorem A.5

Let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Then, there exists a subsequence \((u_n)_{n \in {\mathbb {N}}'}\) and a positive Radon measure \(\mu \) on \(\Omega \times S^{d-1}\) such that for any essentially homogeneous pseudo-differential operators \(A \in \Psi ^{0}_{c}(\Omega )\) with principal symbol \(\sigma _{0}(A)\), one has

$$\begin{aligned} \left( A(\chi u_n), \chi u_n\right) _{n \in {\mathbb {N}}'} \underset{n\rightarrow +\infty }{\longrightarrow }\int _{\Omega \times S^{n-1}}\sigma _0(A)(x,\xi )\,d\mu \end{aligned}$$
(A.18)

for all \(\chi \in C_{0}^{\infty }(\Omega )\) such that \(\chi =1\) in \(\pi _x(\mathrm{supp}(\sigma _0(A)))\).

Proof

From the assumption of the theorem, one has

$$\begin{aligned} \limsup _{n \rightarrow +\infty }\Vert A(\chi u_{n}) \Vert _{L^{2}(\Omega )}\le C(K,\chi )\,\underset{(x,\xi ) \in \Omega \times S^{d-1}}{\max }|\sigma _{0}(A)|. \end{aligned}$$
(A.19)

Indeed, first observe that since \(A\in \Psi _c^0(\Omega )\), it maps \(H_{comp}^s(\Omega )\) into \(H_K^s(\Omega )\) for all \(s\in {\mathbb {R}}\) and some compact set \(K\subset \Omega \). In particular, A maps \(L_{comp}^2(\Omega )\) continuously into \(L_K^2(\Omega )\). Since \((u_n)_{n\in {\mathbb {N}}}\) is bounded in \(L^2_{loc}(\Omega )\), it follows that \((A (\chi u_n))\) is bounded in \(L_{K}^2(\Omega )\).

Now, from the Cauchy–Schwarz inequality,

$$\begin{aligned} |(A(\chi u_n), \chi u_n)_{L^2(\Omega )}|\le & {} \Vert A(\chi u_n) \Vert _{L^2(\Omega )}\Vert \chi u_n\Vert _{L^2(\Omega )} \nonumber \\= & {} (A(\chi u_{n}), A(\chi u_{n}))_{L^{2}(\Omega )}\Vert \chi u_n\Vert _{L^2(\Omega )} \nonumber \\= & {} (A_{\chi }^{*} A (\chi u_n), u_n)\Vert \chi u_n\Vert _{L^2(\Omega )}. \end{aligned}$$
(A.20)

The principal symbol of the operator \(A_{\chi }^{*}A\) is \(|\sigma _0(A)|^2\). Hence, \(A_{\chi }^{*}A= b(x,D) +r(x,D)\), where \(r(x,D)\in \Psi _{c}^{-1}(\Omega )\) and b(xD) is an operator with symbol \(\gamma (\xi )|\sigma _{0}(A)|^{2}\), in which \(\gamma \) is 0 near the origin and 1 at infinity.

Then,

$$\begin{aligned} \gamma (\xi ) \sigma _{0}(A_{\chi }^{*}A) = \gamma (\xi ) \chi |\sigma _{0}(A)|^{2} \le \chi \left( \displaystyle \max _{(x,\xi ) \in \Omega \times S^{d-1}}|\sigma _{0}(A)| \right) ^{2} = \chi M^{2}. \end{aligned}$$

Observe that \(\chi u_{n} \rightharpoonup 0\) weakly in \(L_{comp}^{2}(\Omega )\). Indeed, let \(g \in L_{loc}^{2}(\Omega )\). Then, \(\chi g \in L_{comp}^{2}(\Omega )\). Thus

$$\begin{aligned} \int _{\Omega }(\chi u_n)gdx= & {} \int _{\Omega } u_n(\chi g)dx \rightarrow 0, \end{aligned}$$

since \(u_n \rightharpoonup 0\) in \(L_{loc}^{2}(\Omega )\).

From Remark A.1, \(r(x, D)(\chi u_{n}) \rightarrow 0\) strongly in \(L^{2}(\Omega )\), and hence

$$\begin{aligned} (r(x, D)(\chi u_{n}), u_{n})_{L^2(\Omega )} \rightarrow 0 \hbox { as } n \rightarrow \infty . \end{aligned}$$

On the other hand, since \(\chi =1\) in \(K = \pi _{x}(\mathrm{supp}(\sigma _{0}(A)))\),

$$\begin{aligned} (b(x,D)(\chi u_{n}), u_{n})= & {} \int _{K} b(x,D)(\chi u_{n})(x) u_{n}(x)dx \\= & {} (b(x,D)(\chi u_{n}), \chi u_{n}) \\= & {} - ((M^{2}\chi I - b(x,D))(\chi u_{n}), \chi u_{n}) +(( M^{2}\chi I)\chi u_{n}, \chi u_{n}). \end{aligned}$$

We estimate the term \(\displaystyle \limsup _{n \rightarrow \infty } \Vert A(\chi u_{n}) \Vert _{L^{2}(\Omega )}^{2}\) as follows:

$$\begin{aligned} \limsup _{n \rightarrow \infty } \Vert A(\chi u_{n}) \Vert _{L^{2}(\Omega )}^{2}= & {} \limsup _{n \rightarrow \infty } ( A_{\chi }^{*}A(\chi u_{n}), u_{n}) \\= & {} \limsup _{n \rightarrow \infty } Re ( A_{\chi }^{*}A(\chi u_{n}), u_{n})\\\le & {} \limsup _{n \rightarrow \infty } Re ( b(x,D)(\chi u_{n}), u_{n}) +\limsup _{n \rightarrow \infty }Re(r(x,D)(\chi u_n),u_n)\\\le & {} \limsup _{n \rightarrow \infty } Re -(M^2 \chi I - b(x,D)(\chi u_n),\chi u_n)\\&+ \limsup _{n \rightarrow \infty }(( M^{2}\chi I)\chi u_{n}, \chi u_{n}) \\\le & {} - \liminf _{n \rightarrow \infty }Re((M^{2} \chi I - b(x,D))(\chi u_{n}), \chi u_{n})\\&+ M^{2} \max _{x \in \Omega }\chi (x) \limsup _{n \rightarrow \infty }\Vert \chi u_n\Vert ^{2}\\\le & {} M^{2} \max _{x \in \Omega }\chi (x) \limsup _{n \rightarrow \infty }\Vert \chi u_{n}\Vert ^{2}, \end{aligned}$$

where the last inequality follows from Lemma A.1.

Therefore, from the properties of \(\limsup \), it follows that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \Vert A(\chi u_{n})\Vert \le C(\chi , K)\underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|\sigma _{0}(A)|. \end{aligned}$$

Thus, from (A.20) we obtain

$$\begin{aligned} \limsup _{n \rightarrow \infty }|(A(\chi u_{n}), \chi u_{n})| \le C \underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|\sigma _{0}(A)|. \end{aligned}$$
(A.21)

For any compact set K of \(\Omega \), \(C_K^\infty (\Omega \times S^{d-1})\) will denote the vectorial space of functions \(C^\infty \) on \(\Omega \times S^{d-1}\), with compact support in \(K\times S^{d-1}\), endowed with the \(L^\infty \) norm. The space \(C_K^\infty (\Omega \times S^{d-1})\) is separable, since it is isometric to a subspace of the separable space \(C(K \times S^{d - 1})\).

So, let \(D := \mathrm{span}\{ a_i : i\in {\mathbb {N}}\}\) be a countable and dense subset of \(C_K^\infty (\Omega \times S^{d-1})\). For all \(i \in {\mathbb {N}}\), let \(A_i\) be a pseudo-differential operator such that \(\sigma _0(A_i)=a_i\).

Taking (A.21) into account, for each fixed i there exists a constant

$$\begin{aligned} C_i:=C(\chi ,K)\underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|\sigma _{0}(A_i)| \end{aligned}$$

such that

$$\begin{aligned} \limsup _{n\rightarrow +\infty }|(A_i(\chi u_n),\chi u_n)|\le C_i \hbox { for all }n\in {\mathbb {N}}. \end{aligned}$$

By virtue of Cantor’s diagonal argument there exists a infinite subset \({\mathbb {N}}'\) of \({\mathbb {N}}\) such that the quantity \((A_i(\chi u_{n}),\chi u_n)_{n \in {\mathbb {N}}'}\) has a limit for all i.

In fact, the sequence \(\{(A_1(\chi u_n),\chi u_n)\}_{n \in {\mathbb {N}}}\), being bounded, has a convergent subsequence. Thus there exist an infinite subset \({\mathbb {N}}_{1} \subset {\mathbb {N}}\) and a number \(\alpha _1\) for which \(\underset{n \in {\mathbb {N}}_1}{\lim } (A_1(\chi u_n),\chi u_n)=\alpha _1\). The sequence \((A_2(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}_1}\) is also bounded. So there exist an infinite subset \({\mathbb {N}}_2 \subset {\mathbb {N}}_1\) and a number \(\alpha _2\) such that \(\underset{n \in {\mathbb {N}}_2}{\lim } (A_1(\chi u_n),\chi u_n)=\alpha _2\). Proceeding in the same fashion, we obtain, for each \(i \in {\mathbb {N}}\), an infinite subset \({\mathbb {N}}_i \subset {\mathbb {N}}\), such that \({\mathbb {N}}_1 \supset {\mathbb {N}}_2 \supset \cdots \supset {\mathbb {N}}_i \supset \cdots \) and a number \(\alpha _i\) such that \(\underset{n \in {\mathbb {N}}_i}{\lim } (A_i(\chi u_n),\chi u_n)=\alpha _i\). Let us then define an infinite subset \({\mathbb {N}}' \subset {\mathbb {N}}\), taking the i-th element of \({\mathbb {N}}'\) as the i-th element of \({\mathbb {N}}_i\). For each \(i \in {\mathbb {N}}\), the sequence \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'}\) is, from its i-th element, a subsequence of \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}_i}\) and therefore converges. Thus, \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'} \rightarrow \alpha _i\) for all \(i \in {\mathbb {N}}\).

Define \(L: D \subset C_{K}^{\infty }(\Omega \times S^{d-1}) \longrightarrow {\mathbb {C}}\) by setting \(L(a_k)=\alpha _k\). From (A.21), it follows that

$$\begin{aligned} |L(a_k)|= & {} |\alpha _ k| \nonumber \\= & {} \lim _{n \in {\mathbb {N}}'} |(A_k(\chi u_n),\chi u_n)|\nonumber \\\le & {} C \underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|\sigma _{0}(A_k)|\nonumber \\= & {} C \underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|a_k|. \end{aligned}$$
(A.22)

Note that \(\alpha _k\) does not depend on the choice of the operator \(A_k\) satisfying \(\sigma _0(A_k)=a_k\). In fact, let \(B_k\) be another operator with \(\sigma _0(B_k)=a_k\), then from (A.21), it follows that

$$\begin{aligned} \limsup _{n \in {\mathbb {N}}'} |((B_k-A_k)(\chi u_n),\chi u_n)| \le C \underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|\sigma _{0}(B_k-A_k)|=0. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{n \in {\mathbb {N}}'}(B_k(\chi u_n),\chi u_n)= & {} \lim _{n \in {\mathbb {N}}'}((B_k-A_k)(\chi u_n),\chi u_n) +\lim _{n \in {\mathbb {N}}'}(A_k(\chi u_n),\chi u_n)\\= & {} \alpha _k. \end{aligned}$$

Thus, the mapping L is well defined and densely defined. Hence, it uniquely extends to a bounded linear functional \(\tilde{L}: C_{K}^{\infty }(\Omega \times S^{d-1}) \rightarrow {\mathbb {C}}\) satisfying the estimate

$$\begin{aligned} |\tilde{L}(a)|\le C \underset{(x,\xi )\in \Omega \times S^{d-1}}{\max }|a|. \end{aligned}$$

By construction, given \(\sigma _0(A) \in C_{K}^{\infty }(\Omega \times S^{d-1})\), we have

$$\begin{aligned} (A(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'} \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}(\sigma _0(A)). \end{aligned}$$

In fact, let \((a_i) \subset D\) with \(a_i \rightarrow \sigma _0(A)\) as \(i\rightarrow +\infty \); that is, given \(\varepsilon > 0,\) there exists \(i_{0} \in {\mathbb {N}}\) such that

$$\begin{aligned} \Vert \sigma _{0}(A) - a_{i}\Vert _{\infty } < \frac{\varepsilon }{3C}, \hbox { for all } i \ge i_{0}. \end{aligned}$$
(A.23)

Moreover, there is \(n_{0} \in {\mathbb {N}},\) such that for all \(n \ge n_{0}\),

$$\begin{aligned} |(A(\chi u_{n}), \chi u_{n}) - (A_{i_{0}}(\chi u_{n}), \chi u_{n})|\le & {} C \Vert \sigma _{0}(A) - a_{i_{0}}\Vert _{\infty } \nonumber \\\le & {} C \frac{\varepsilon }{3 C} =\frac{\varepsilon }{3}. \end{aligned}$$
(A.24)

On the other hand, there exists \(n_{1} \in {\mathbb {N}}\) such that for all \(n \ge n_{1}\),

$$\begin{aligned} |(A_{i_{0}}(\chi u_{n}), \chi u_{n}) - \tilde{L}(a_{i_{0}})| < \frac{\varepsilon }{3}. \end{aligned}$$
(A.25)

So, taking \(n_{2} = \max \{n_{0}, n_{1}\}\), from (A.23), (A.24) and (A.25), it follows that for all \(n \ge n_{2},\)

$$\begin{aligned} |(A(\chi u_{n}), \chi u_{n}) - \tilde{L}(\sigma _{0}(A))|\le & {} |(A(\chi u_{n}), \chi u_{n}) - (A_{i_{0}}(\chi u_{n}), \chi u_{n})| \\&+ |(A_{i_{0}}(\chi u_{n}), \chi u_{n}) - \tilde{L}(a_{i_{0}})|\\&+|\tilde{L}(a_{i_{0}}) - \tilde{L}(\sigma _{0}(A))|\\\le & {} \frac{\varepsilon }{3} + \frac{\varepsilon }{3} +C \Vert \sigma _{0}(A) - a_{i}\Vert _{\infty }\\\le & {} \frac{\varepsilon }{3} + \frac{\varepsilon }{3} +\frac{\varepsilon }{3} = \varepsilon . \end{aligned}$$

Then,

$$\begin{aligned} (A(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'} \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}(\sigma _0(A)). \end{aligned}$$

Note that L does not depend on the choice of \(\chi \). Indeed, let \(\chi , \tilde{\chi } \in C_{0}^{\infty }(\Omega )\) with \(\chi \sigma _{0}(A) = \sigma _{0}(A) = \tilde{ \chi }\sigma _{0}(A)\). For \(\eta \in C_{0}^{\infty }(\Omega )\) such that \(\eta =1\) in a neighborhood of \(\mathrm{supp}\chi \cup \mathrm{supp}\tilde{\chi }\), we have

$$\begin{aligned}&\left( A(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( A(\chi u_{n}), \chi u_{n} \right) \\&\quad = \left( b(x,D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( b(x,D)(\chi u_{n}), \chi u_{n} \right) \\&\qquad + \left( r(x, D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( r(x, D)(\chi u_{n}), \chi u_{n} \right) \\&\quad = \left( b(x,D)((\tilde{\chi }-\chi )u_{n}), \tilde{\chi } u_{n} \right) +\left( b(x,D)(\chi u_{n}), (\tilde{\chi }-\chi ) u_{n} \right) \\&\qquad + \left( r(x, D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( r(x, D)(\chi u_{n}), \chi u_{n} \right) \\&\quad = \left( b(x,D)(\eta (\tilde{\chi }-\chi )u_{n}), \tilde{\chi } u_{n} \right) +\left( b(x,D)(\chi u_{n}), (\tilde{\chi }-\chi ) u_{n} \right) \\&\qquad + \left( r(x, D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( r(x, D)(\chi u_{n}), \chi u_{n} \right) \\&\quad = \left( (\tilde{\chi }-\chi )u_{n}, b(x,D)_{\eta }^{*}(\tilde{\chi } u_{n}) \right) + \left( b(x,D)(\chi u_{n}), (\tilde{\chi }-\chi ) u_{n} \right) \\&\qquad + \left( r(x, D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( r(x, D)(\chi u_{n}), \chi u_{n} \right) \\&\quad = \left( (\tilde{\chi }-\chi )u_{n}, \overline{b(x,D)}(\tilde{\chi } u_{n}) \right) + \left( b(x,D)(\chi u_{n}), (\tilde{\chi }-\chi ) u_{n} \right) \\&\qquad + \left( r(x, D)(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) -\left( r(x, D)(\chi u_{n}), \chi u_{n} \right) \\&\qquad + \left( (\tilde{\chi }-\chi )u_{n}, r_1(\tilde{\chi } u_{n}) \right) , \end{aligned}$$

since \(A=b(x,D) + r(x,D)\), where \(b_{\eta }^{*}(x,\xi )=\overline{b(x,\xi )}+r_1\) with \(r_1 \in {\mathcal {S}}_{c}^{-1}(\Omega )\). Since \(\chi - \tilde{\chi } =0\) in \(\pi _{x}(\mathrm{supp}\sigma _{0}(A))\), we have

$$\begin{aligned} \left( A(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) _{n \in {\mathbb {N}}'} - \left( A(\chi u_{n}), \chi u_{n} \right) _{n \in {\mathbb {N}}'} \rightarrow 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( A(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) _{n \in {\mathbb {N}}'}&=\left( A(\tilde{\chi }u_{n}), \tilde{\chi } u_{n} \right) _{n \in {\mathbb {N}}'} -\left( A(\chi u_{n}), \chi u_{n} \right) _{n \in {\mathbb {N}}'} +\left( A(\chi u_{n}), \chi u_{n} \right) _{n \in {\mathbb {N}}'}\\&\rightarrow L(\sigma _0(A)). \end{aligned}$$

So far we have extracted a subsequence of \((u_n)\) such that

$$\begin{aligned} (A(\chi u_n),\chi u_n) \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}(\sigma _0(A)) \end{aligned}$$

for all \(A \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A) \subset K \times {\mathbb {R}}^{d}\) in some fixed compact K. We need to remove the dependence on K. To this end, let \(K_i \subset \overset{\circ }{K}_{i+1}\) be a monotone sequence of compact subsets of \(\Omega \) such that \(\Omega = \overset{\infty }{\underset{n=1}{\bigcup }}K_n\). From the construction, there exists an infinite subset \({\mathbb {N}}_1 \subset {\mathbb {N}}\) and a continuous linear form \(\tilde{L}_1\) in \(C_{K_1}^{\infty }(\Omega \times S^{d-1})\) such that

$$\begin{aligned} (A_1(\chi u_n),\chi u_n)_{{\mathbb {N}}_{1}} \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}_1(\sigma _0(A_1)) \end{aligned}$$

for all \(A_1 \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_1) \subset K_1 \times \Omega \). The sequence \((u_n)_{n \in {\mathbb {N}}_1}\) is still a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Thus we can obtain an infinite subset \({\mathbb {N}}_{2} \subset {\mathbb {N}}_1\) and a continuous linear form \(\tilde{L}_2\) in \(C_{K_2}^{\infty }(\Omega \times S^{d-1})\) such that

$$\begin{aligned} (A_2(\chi u_n),\chi u_n)_{{\mathbb {N}}_{2}} \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}_2(\sigma _0(A_2)), \end{aligned}$$

for all \(A_2 \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_2)\subset K_2 \times \Omega \).

Proceeding in the same fashion, we obtain, for each \(i \in {\mathbb {N}}\), an infinite subset \({\mathbb {N}}_i \subset {\mathbb {N}}\) and a continuous linear form \(\tilde{L}_i\) in \(C_{K_i}^{\infty }(\Omega \times S^{d-1})\) such that \({\mathbb {N}}_1 \supset {\mathbb {N}}_2 \supset \cdots \supset {\mathbb {N}}_i \supset \cdots \) and, for all \(i\in {\mathbb {N}}\),

$$\begin{aligned} (A_i(\chi u_n),\chi u_n)_{{\mathbb {N}}_{i}} \underset{n \rightarrow \infty }{\longrightarrow } \tilde{L}_i(\sigma _0(A_i)) \end{aligned}$$

for all \(A_i \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_i)\subset K_i \times \Omega \). Let us then define an infinite subset \({\mathbb {N}}' \subset {\mathbb {N}}\), whose i-th element is the i-th element of \({\mathbb {N}}_i\). In this way, for each \(i \in {\mathbb {N}}\), the sequence \((u_n)_{n \in {\mathbb {N}}'}\) is, starting from its i-th element, a subsequence of \((u_n)_{n \in {\mathbb {N}}_i}\).

We want to define a linear functional \(L:C_{0}^{\infty }(\Omega \times S^{d-1}) \longrightarrow {\mathbb {C}}\) satisfying the limit

$$\begin{aligned} (A(\chi u_n),\chi u_n)_{{\mathbb {N}}'} \underset{n \rightarrow \infty }{\longrightarrow } L(\sigma _0(A)) \end{aligned}$$

for all \(A \in \Psi _{c}^{0}(\Omega )\) with \(\sigma _0(A)\) having compact support in the variable x.

Now, if \(a \in C_{0}^{\infty }(\Omega \times S^{d-1})\), \(\mathrm{supp}a \subset K \times S^{d-1}\), and \(a=\sigma _0(A)\), with \(K \subset K_{i_0}\), the same argument used to show the independence of the cut-off functions implies

$$\begin{aligned} \lim _{n \in {\mathbb {N}}'}(A(\chi u_n),\chi u_n)= \lim _{n \in {\mathbb {N}}'}(A(\chi _{i_0} u_n),\chi _{i_0} u_n)=\tilde{L}_{i_0}(\sigma _0(A)), \end{aligned}$$

where \(\chi =1\) in a neighborhood of K, \(\chi _{i_0}=1\) in a neighborhood of \(K_{i_0}\) and that the following inequality holds:

$$\begin{aligned} |\tilde{L}_{i_0}(\sigma _0(A))|\le C(\chi _{i_0})\Vert \sigma _0(A)\Vert . \end{aligned}$$

Defining \(L(\sigma _0(A))=\tilde{L}_{i_0}(\sigma _0(A))\), we obtain

$$\begin{aligned} |L(\sigma _0(A))|\le C(\chi _{i_0})\Vert \sigma _0(A)\Vert , \hbox { if } \mathrm{supp}\sigma _0(A) \subset K \times S^{d-1}. \end{aligned}$$

From Lemma A.1 it follows that, L extends to a Radon measure \(\mu \ge 0\) on \(\Omega \times S^{d-1}\), which finishes the proof. \(\square \)

Definition A.5

\(\mu \) is said to be the microlocal defect measure of the sequence \(\{u_n\}_{n\in {\mathbb {N}}'}\) in Theorem A.5.

Remark A.3

Theorem A.5 assures that any bounded sequence \((u_n)_{n\in {\mathbb {N}}}\) in \(L_{loc}^2({\mathcal {O}})\) that weakly converges to zero has a subsequence associated with a microlocal defect measure (in short, m.d.m.). We observe from (A.18) that if \(A=f\in C_0^\infty ({\mathcal {O}})\), then in particular

$$\begin{aligned} \int _\Omega f(x) |u_{\varphi (n)}(x)|^2\,dx \rightarrow \int _{{\mathcal {O}} \times S^{d-1}}f(x)\,d\mu (x,\xi ), \end{aligned}$$
(A.26)

so that \((u_n)_{{\mathbb {N}}'}\) strongly converges to 0 if and only if \(\mu =0\).

Theorem A.6

Let P be a differential operator of order m on \(\Omega \) with \(P^*=P\), and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0 associated with a microlocal defect measure \(\mu \). Let us assume that \(P u_n \underset{n\rightarrow +\infty }{\longrightarrow }0\) strongly in \(H_{loc}^{1-m}\). Then, for any \(a \in C^\infty (\Omega \times ({\mathbb {R}}^d)\backslash \{0\})\) homogeneous of degree \(1-m\) in the second variable and with compact support in the first variable,

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\{a,p\}(x,\xi )\,d\mu (x,\xi )=0. \end{aligned}$$
(A.27)

Proof

Let \(\chi , \chi _1 \in C_{0}^{\infty }(\Omega )\) such that \(\chi =1\) in a neighborhood of \(\pi _1(\mathrm{supp}(a))\) and \(\chi _1=1\) in a neighborhood of the support of the function \(\chi \). Consider \(\gamma \in C^\infty (\Omega )\) with \(\gamma =0\) in a neighborhood of 0 and \(\gamma =1\) at infinity. Let A be the operator defined by the symbol \(b(x,\xi )=\gamma (\xi )a(x,\xi )\). The symbol of the adjoint operator \(A_{\chi _1}^{*}\) is given by \(b^*(x,\xi )=\overline{b(x,\xi )}+r(x,\xi )\), where \(r \in S_{c}^{-m}(\Omega )\). For a sufficiently regular function u, we have

$$\begin{aligned} \left( [A,P](\chi u),\chi u\right) _{L^2(\Omega )}= & {} \left( (AP-PA)(\chi u), \chi u\right) _{L^2(\Omega )}\nonumber \\= & {} \left( AP(\chi u), \chi u\right) _{L^2(\Omega )}-\left( PA(\chi u),\chi u\right) _{L^2(\Omega )}\nonumber \\= & {} \left( A(P(\chi u)), \chi u \right) _{L^2(\Omega )}-\left( A(\chi u), P^{*}(\chi u) \right) _{L^2(\Omega )}\nonumber \\= & {} \left( A(\chi _1P(\chi u)), \chi u \right) _{L^2(\Omega )}-\left( A(\chi u), P(\chi u) \right) _{L^2(\Omega )}\nonumber \\= & {} \left( P(\chi u)),A_{\chi _1}^{*} (\chi u) \right) _{L^2(\Omega )}-\left( A(\chi u), P(\chi u) \right) _{L^2(\Omega )}\nonumber \\= & {} \left( P(\chi u)),A_{\chi _1}^{*} (\chi u) \right) _{L^2(\Omega )}-\left( A(\chi u), \chi Pu \right) _{L^2(\Omega )}\nonumber \\&-(A(\chi u), [P,\chi ]u)_{L^2(\Omega )}. \end{aligned}$$
(A.28)

Note that

$$\begin{aligned} P(\chi u)= & {} \sum _{|\alpha |\le m}a_\alpha \partial _{x}^{\alpha }(\chi u)\\= & {} \sum _{|\alpha |\le m}a_\alpha \sum _{\beta \le \alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \partial _{x}^{\beta }\chi \partial _{x}^{\alpha -\beta }u\\= & {} \chi Pu+ \sum _{|\alpha |\le m} a_\alpha \sum _{0<\beta \le \alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \partial _{x}^{\beta } \chi \partial _{x}^{\alpha -\beta }u\\= & {} \chi Pu+ [P,\chi ]u. \end{aligned}$$

Therefore, \([P,\chi ]\) is a differential operator of order \(m-1\) involving the partial derivatives of \(\chi \). We note that \(A(\chi u)(x)=0\) for \(x \not \in \pi _1(\mathrm{supp}(a))\). Thus, we can write

$$\begin{aligned} (A(\chi u),[P,\chi ]u)_{L^2(\Omega )} = (A(\chi u),[P,\chi ]u)_{L^2(\pi _1(\mathrm{supp}(a)))}= 0 \end{aligned}$$
(A.29)

since \(\chi =1\) in a neighborhood of this set. Therefore, \(\partial _{x}^{\beta } \chi =0\) for \(\beta >0\).

Regarding the term \((P(\chi u),A_{\chi _1}^{*}(\chi u)_{L^2(\Omega )}\), we have

$$\begin{aligned} (P(\chi u),A_{\chi _1}^{*}(\chi u))_{L^2(\Omega )} =(P(\chi u),\overline{b}(x,D)(\chi u))_{L^2(\Omega )} +(P(\chi u),r(x,D)(\chi u))_{L^2(\Omega )}. \end{aligned}$$
(A.30)

By the same argument above,

$$\begin{aligned} (P(\chi u),\overline{b}(x,D)(\chi u))_{L^2(\Omega )}= & {} (\chi Pu+[P,\chi ]u,\overline{b}(x,D)(\chi u))_{L^2(\Omega )}\nonumber \\= & {} (\chi Pu,\overline{b}(x,D)(\chi u))_{L^2(\Omega )}\nonumber \\&+([P,\chi ]u,\overline{b}(x,D)(\chi u))_{L^2(\pi _1(\mathrm{supp}(a)))}\nonumber \\= & {} (\chi Pu,\overline{b}(x,D)(\chi u))_{L^2(\Omega )}. \end{aligned}$$
(A.31)

With respect to the term \((P(\chi u),r(x,D)(\chi u))_{L^2(\Omega )}\), we have,

$$\begin{aligned} (P(\chi u),r(x,D)(\chi u))_{L^2(\Omega )}= & {} (P(\chi u),r(x,D) (\chi _1 \chi u))_{L^2(\Omega )}\\= & {} (r(x,D)_{\chi _1}^{*}P(\chi u),\chi u)_{L^2(\Omega )}\\= & {} (r(x,D)_{\chi _1}^{*}(\chi Pu),\chi u)_{L^2(\Omega )}\nonumber \\&\qquad +(r(x,D)_{\chi _1}^{*}([P,\chi ]u),\chi u)_{L^2(\Omega )}. \end{aligned}$$

From the hypothesis, \(\chi P u_n \rightarrow 0\) in \(H_{comp}^{1-m}(\Omega )\). Then, considering \(r(x,D)_{\chi _1}^{*}:H_{comp}^{1-m}(\Omega )\longrightarrow H_{K_1}^{1}\), \(K_1=\mathrm{supp}\chi _1\), we have \(r(x,D)_{\chi _1}^{*}(\chi P u_n) \rightarrow 0\) in \(L^2(\Omega )\). Thus,

$$\begin{aligned}&|(r(x,D)_{\chi _1}^{*}(\chi Pu_n),\chi u_n)_{L^2(\Omega )}|\nonumber \\&\quad \le \Vert r(x,D)_{\chi _1}^{*}(\chi Pu_n)\Vert _{L^2(\Omega )}\Vert \chi u_n\Vert _{L^2(\Omega )} \rightarrow 0, \hbox { as } n \rightarrow \infty . \end{aligned}$$

Note that \([P,\chi ]\) has order \(m-1\) and is compactly supported, that is, has compact kernel. Thus, \([P,\chi ]:L_{loc}^{2}(\Omega )\longrightarrow H_{comp}^{1-m}\). By Rellich’s theorem,

$$\begin{aligned} r(x,D)_{\chi _1}^{*} \circ [P,\chi ]: L_{loc}^{2} (\Omega ) \longrightarrow L^2(\Omega ) \end{aligned}$$

is a compact operator. Therefore, \(r(x,D)_{\chi _1}^{*} \circ [P,\chi ]\) maps weakly convergent sequences to strongly convergent sequences. Hence,

$$\begin{aligned} |(r(x,D)_{\chi _1}^{*} [P,\chi ](u_n),\chi u_n)_{L^2(\Omega )}|&\le \Vert (r(x,D)_{\chi _1}^{*} [P,\chi ](u_n)\Vert _{L^2(\Omega )}\Vert \chi u_n\Vert _{L^2(\Omega )}\\&\rightarrow 0, \hbox { as } n \rightarrow \infty . \end{aligned}$$

Hence,

$$\begin{aligned} |(P(\chi u_n),r(x,D)(\chi u_n))_{L^2(\Omega )}| \rightarrow 0, \hbox { as } n \rightarrow \infty . \end{aligned}$$
(A.32)

From (A.28), (A.29) and (A.30), we obtain

$$\begin{aligned} ([A,P](\chi u_n),\chi u_n)_{L^2(\Omega )}= & {} (\chi Pu_n,\overline{b}(x,D)(\chi u_n))_{L^2(\Omega )}\\&+(P(\chi u_n),r(x,D)(\chi u_n))_{L^2(\Omega )}\\&-(A(\chi u_n),\chi Pu_n)_{L^2(\Omega )}. \end{aligned}$$

Observing that

$$\begin{aligned} \sigma _0([A,P])=\frac{1}{ i}\{a,p\}, ~\hbox { where }a=\sigma _{1-m}(A)\hbox { and }p=\sigma _m(P), \end{aligned}$$

the second term of (A.27) is

$$\begin{aligned} J:=\lim _{n\rightarrow +\infty } i\left( [A,P](\chi u_n), \chi u_n \right) _{L^2(\Omega )}= & {} i\lim _{n\rightarrow +\infty }(\chi Pu_n,\overline{b}(x,D)(\chi u_n))_{L^2(\Omega )}\\&+i\lim _{n\rightarrow +\infty }(P(\chi u_n),r(x,D)(\chi u_n))_{L^2(\Omega )}\\&i i\lim _{n\rightarrow +\infty }(A(\chi u_n),\chi Pu_n)_{L^2(\Omega )}. \end{aligned}$$

Combining the fact that \(\chi P u_n \rightarrow 0\) in \(H^{1-m}_{comp}(\Omega )\) with \(\overline{b}(x,D)_{\chi _1}^{*}:H_{comp}^{1-m}(\Omega ) \longrightarrow L^2(\Omega )\), we obtain

$$\begin{aligned} |(\chi Pu_n,\overline{b}(x,D)(\chi u_n))_{L^2(\Omega )}|&=|(\overline{b}(x,D)_{\chi _1}^{*}(\chi Pu_n),\chi u_n)_{L^2(\Omega )}|\\&\le \Vert \overline{b}(x,D)_{\chi _1}^{*}(\chi Pu_n)\Vert _{L^2(\Omega )}\Vert \chi u_n\Vert _{L^2(\Omega )}\\&\rightarrow 0, \hbox { as } n \rightarrow \infty . \end{aligned}$$

Due to (A.32), it is necessary to prove that the term \(|(A(\chi u_n),\chi Pu_n)_{L^2(\Omega )}|\) converges to 0. In fact, \(A:L_{comp}^{2}(\Omega ) \longrightarrow H_{K}^{m-1}(\Omega )\) is a linear continuous operator. Therefore, A maps bounded sets to bounded sets. In particular, \(\{A(\chi u_n)\}\) is a bounded set of \(H_{K}^{m-1}(\Omega )\). Therefore,

$$\begin{aligned} |(A(\chi u_n),\chi Pu_n)_{L^2(\Omega )}|&\le \Vert A(\chi u_n)\Vert _{H^{m-1}_{K}(\Omega )}\Vert \chi Pu_n\Vert _{H^{1-m}_{K}(\Omega )}\\&\rightarrow 0, \hbox { as } n \rightarrow \infty . \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\{a,p\}(x,\xi )\,d\mu (x,\xi )=\lim _{n\rightarrow \infty }([A,P](\chi u_n),\chi u_n)_{L^2(\Omega )}=0, \end{aligned}$$

which concludes the proof. \(\square \)

Theorem A.7

Let X be a locally compact Hausdorff space and \(\mu \) be a positive Radon measure on X.

  1. (a)

    Let N be the union of all open \(U \subset X\) such that \(\mu (U)=0\). Then N is open and \(\mu (N)=0\). The complement of N is called the support of \(\mu \).

  2. (b)

    \(x \in \mathrm{supp}\mu \) if and only if \(\int _{X} f d\mu >0\) for every \(f \in C_{0}(X)\) with \(f(x)>0\).

Proof

a) clearly follows from \(\mu (N)=\sup \{\mu (K):K \subset N, K \hbox { compact}\}.\)

To prove b), suppose \(x \in \mathrm{supp}\mu =N^c\). Let \(f \in C_c(X,[0,1])\) such that \(f(x)>0\). By continuity, there is an open neighborhood U of x such that \(f(y)>\frac{1}{2}f(x)\) for all \(y \in U\). If \(\mu (U)=0\), then since U is open, \(U \subset N\), which contradicts \(x \in N^c\). Thus, \(\mu (U)>0\). Hence, \(\int _{X}fd\,\mu \ge \int _U f\, d\mu \ge \frac{1}{2}f(x)\mu (U)>0\).

Now, suppose \(x \notin \mathrm{supp}\mu \) and let \(F=\mathrm{supp}\mu \). Then, there exists an open neighbourhood U of x such that \(\mu (U)=0\) and \(U \cap F=\emptyset \). Let K be a compact set so that \(x \in K \subset U\). From Urysohn’s theorem, there exists a function \(f \in C_0(X,[0,1])\) such that \(f=1\) in K and \(f=0\) outside of a compact subset of U. In particular, \(f(x)=1>0\) and

$$\begin{aligned} \int _{X} f d\mu= & {} \int _{U} f d \,\mu +\int _{X \setminus U} f \, d\mu \\= & {} \int _{U}f d \, \mu +\int _{X \setminus U} 0 \, d\mu \\= & {} 0. \end{aligned}$$

\(\square \)

Theorem A.8

Let \(\{u_n\}_n\) be a bounded sequence in \(L_{loc}^2(\Omega )\) that weakly converges to zero and admits a microlocal defect measurement \(\mu \). Then, \((x_0,\xi _0) \notin \mathrm{supp}\mu \) if and only if there exists \(A \in \Psi _{c}^{0}(\Omega )\) essentially homogeneous such that \(\sigma _0(A)(x_0,\xi _0)\ne 0\) and \(A(\chi u_n) \rightarrow 0\) in \(L^2(\Omega )\) for all \(\chi \in C_{0}^{\infty }(\Omega )\).

Proof

If \((x_0,\xi _0) \notin \mathrm{supp}\mu \), then there exist open subsets \(U \subset \Omega \), \(V \subset S^{d-1}\) such that \((x_0,\xi _0) \in U \times V\) and \(U \times V \cap \mathrm{supp}\mu = \emptyset \). Let K be a compact neighborhood of \(\xi _0\), such that \(K \subset V\). Let \(\psi \in C^\infty (S^{d-1})\) such that \(\mathrm{supp}\psi \subset K\) and \(\psi =1\) in a neighborhood of \(\xi _0\) contained in K. Consider \(\phi \in C_{0}^{\infty }(\Omega )\) such that \(\phi =1\) in a neighborhood of \(x_0\) and \(\mathrm{supp}\phi \subset U\). Pick \(\eta \in C^\infty (\Omega )\) such that \(\eta =0\) near 0 and \(\eta =1\) at infinity. Define \(a(x,\xi )= \eta (\xi )\phi (x)\psi \left( \frac{\xi }{|\xi |}\right) \). Let A be the operator defined by \(a(\cdot ,\cdot )\). Note that \(\sigma _0(A)(x_0,\xi _0)=1\) and for all \(\chi \in C_{0}^{\infty }(\Omega )\)

$$\begin{aligned} \Vert A(\chi u_n)\Vert _{L^{2}(\Omega )}^{2}&= (A(\chi u_n), A(\chi u_n ))\nonumber \\&=(A(\chi u_n), A(\chi _1 \chi u_n ))\nonumber \\&=(A_{\chi _1}^{*}\circ A (\chi u_n),\chi u_n )\nonumber \\&\rightarrow \int _{\Omega \times S^{d-1} }|\sigma _0(A)(x,\xi )|^2 d \mu , \end{aligned}$$
(A.33)

where \(\chi _1 =1\) in a neighborhood of \(\mathrm{supp}\chi \). Since \(\mathrm{supp}\sigma _0(A) \subset \mathrm{supp}\phi \times \mathrm{supp}\psi \subset \Omega \times S^{d-1} \setminus \mathrm{supp}\mu \), we have

$$\begin{aligned} \int _{\Omega \times S^{d-1}}|\sigma _0(A)|^2 d \mu =0. \end{aligned}$$

Therefore, \(\Vert A(\chi u_n)\Vert _{L^{2}(\Omega )}\rightarrow 0\) in \(L^2(\Omega )\).

Conversely, if there exists \(A \in \Psi _{c}^{0}(\Omega )\) essentially homogeneous such that \(\sigma _0(A)(x_0,\xi _0)\ne 0\) and \(A(\chi u_n) \rightarrow 0\) in \(L^2(\Omega )\), from (A.33) and by the uniqueness of the limit, we obtain

$$\begin{aligned} \int _{\Omega \times S^{d-1}}|\sigma _0(A)|^2 d \mu =0. \end{aligned}$$

By using Theorem A.7, we deduce that \((x_0,\xi _0) \notin \mathrm{supp}\mu \). \(\square \)

We will call the vector field defined in \(\Omega \times {{\mathbb {R}}}^d\backslash \{0\}\) and given by

$$\begin{aligned} H_p(x,\xi )= \left( \frac{\partial p}{ \partial \xi _1} (x,\xi ), \cdots , \frac{\partial p}{\partial \xi _n} (x,\xi ); -\frac{\partial p}{\partial x_1} (x,\xi ), \cdots , -\frac{\partial p}{\partial x_n} (x,\xi )\right) \end{aligned}$$

a Hamiltonian field of p, denoted by \(H_p\).

The Lie derivative of a function f with respect to the Hamiltonian field \(H_p\) is given by \(H_p (f)=\{p,f\}\), where

$$\begin{aligned} \{p,f\}(x,\xi )=\sum _{j=1}^d \left( \frac{\partial p}{\partial \xi _j} \frac{\partial f}{\partial x_j}-\frac{\partial p}{\partial x_j} \frac{\partial f}{\partial \xi _j}\right) . \end{aligned}$$

A Hamiltonian curve of p is an integrable curve of the vector field \(H_p\), which is a maximal solution \(s\in I \mapsto (x(s),~\xi (s))\) to the Hamilton-Jacobi equations

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}=p_\xi (x,\xi )=\frac{\partial p}{\partial \xi }(x,\xi ), \quad \dot{\xi }=-p_x(x,\xi )=- \frac{\partial p}{\partial x},\\ \end{array}\right. \end{aligned}$$
(A.34)

where I is an open interval of \({\mathbb {R}}\). By null bicharacteristics we mean those integral curves of \(H_p\) along which \(p=0\).

Remark A.4

Since \(H_p p=0\), p is constant along the integral curves of \(H_p\). Indeed, let \(\alpha (t)=(H_p)_t(x_0,\xi _0)\) be an integral curve of \(H_p\) starting at \((x_0,\xi _0)\). Then,

$$\begin{aligned} 0= & {} H_p p (\alpha (t))\\= & {} d(p)_{\alpha (t)}H_p(\alpha (t))\\= & {} d(p)_{\alpha (t)}\alpha '(t)\\= & {} (p \circ \alpha )'(t). \end{aligned}$$

Now, from the connectedness of the domain of \(\alpha \), the remark follows.

Lemma A.2

Assume \(p(x_0,\xi _0)=0\) and \(\lambda \) is a \(C^\infty \) function on \(T^0\Omega \) with real values that never vanishes. Then, for all \(t \in (T_0,T_1)\) there exists \(s_t \in {\mathbb {R}}\) such that \((H_p)_t(x_0,\xi _0)=(H_{\lambda p})_{s_t}(x_0,\xi _0)\). Moreover, there exists \(S_0<0<S_1\) for which the mapping \(t \mapsto s_t\) is a \(C^\infty \) diffeomorphism from \((T_0,T_1)\) onto \((S_0,S_1)\).

Proof

Let us write \(b=\frac{1}{\lambda }\) and \(a=\lambda p\). Since \(p=ab\), using Remark A.4 we have \((ab)((H_p)_t(x_0,\xi _0)=0\). Since b never vanishes, we obtain \(a((H_p)_t(x_0,\xi _0))=0\) for all \(t \in (T_0,T_1)\). Thus,

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}(t)=\frac{\partial p}{\partial \xi }=\frac{\partial a}{\partial \xi }b +\frac{\partial b}{\partial \xi }a=b\frac{\partial a}{\partial \xi }(x(t),\xi (t)),\\ \dot{\xi }(t)=-\frac{\partial p}{\partial x}=-\frac{\partial a}{\partial x}b -\frac{\partial b}{\partial x}a=-b\frac{\partial a}{\partial x}(x(t),\xi (t)). \end{array}\right. \end{aligned}$$

Writing \(f(t)=b(x(t),\xi (t))\), we have

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}(t)=f(t)\frac{\partial a}{\partial \xi }(x(t),\xi (t)),\\ \dot{\xi }(t)=-f(t)\frac{\partial a}{\partial x}(x(t),\xi (t)), \end{array}\right. \end{aligned}$$

with \(x(0)=x_0\) and \(\xi (0)=\xi _0\). Now, let \((y(t),\eta (t))=(H_a)_t(x_0,\xi _0)=(H_{\lambda p})_t(x_0,\xi _0)\) and define

$$\begin{aligned} s(t)=\int _{0}^{t}f(s)ds. \end{aligned}$$

By virtue of the mean value theorem and the fact that f never vanishes, we have either \(f>0\) or \(f<0\). Let’s see that s is injective. Indeed, let \(t_1,t_2 \in (T_0,T_1)\) such that \(t_1 \ne t_2 \) and suppose by contradiction that \(s(t_1)=s(t_2)\). Without loss of generality, we can suppose \(t_1<t_2\). Then

$$\begin{aligned} \int _{0}^{t_1}f(s)ds=\int _{0}^{t_2}f(s)ds=\int _{0}^{t_1}f(s)ds+\int _{t_1}^{t_2}f(s)ds. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{t_1}^{t_2}f(s)ds=0. \end{aligned}$$

In the same fashion, if \(f>0\), then \(\int _{t_1}^{t_2}f(s)ds>0\) and if \(f<0\), then \(\int _{t_1}^{t_2}f(s)ds<0\), which yields a contradiction in both cases. Therefore, \(s(t_1)=s(t_2)\). Moreover, we have \(s'(t)=f(t)\ne 0\) for all \(t\in (T_0,T_1)\). Hence, s is a \(C^\infty \) injective local diffeomorphism. Since \(s(0)=0\), there exists an open neighbourhood of 0, \((S_0,S_1)\), such that s is a diffeomorphism from \((T_0,T_1)\) onto \((S_0,S_1)\). Let \((z(t),\gamma (t))=(y(s(t)),\eta (s(t)))\) and note that

$$\begin{aligned} \left\{ \begin{array}{l} \dot{z}(t)=\dot{y}(s(t))s'(t)=f(t)\frac{\partial a}{\partial \xi } (y(s(t)),\eta (s(t)))=f(t)\frac{\partial a}{\partial \xi }(z(t),\gamma (t))\\ \dot{\gamma }(t)=\dot{\eta }(s(t))s'(t)=-f(t)\frac{\partial a}{\partial x}(y(s(t)),\eta (s(t)))=-f(t)\frac{\partial a}{\partial x}(z(t),\gamma (t)), \end{array}\right. \end{aligned}$$

with \(z(0)=y(s(0))=y(0)=x_0\) and \(\gamma (0)=\eta (s(0))=\eta (0)=\xi _0\). By virtue of uniqueness of solutions, we obtain \((x(t),\xi (t))=(y(s(t)),\eta (s(t))\). Therefore, the bicharacteristics of \(\lambda p\) and p coincide modulo a reparametrization. \(\square \)

Theorem A.9

Let P be a differential operator of order m on \(\Omega \) and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0, associated with a microlocal defect measure \(\mu \). Then, the following statements are equivalent:

  1. (i)

    \(Pu_n \underset{n \rightarrow +\infty }{\longrightarrow }0 \hbox { strongly in } H_{loc}^{-m}(\Omega )~(m>0)\),

  2. (ii)

    \(\mathrm{supp}(\mu ) \subset \{(x,\xi )\in \Omega \times S^{d-1}: \sigma _m(P)(x,\xi )=0\}.\)

Proof

Observe that (i) is equivalent to \(\chi Pu_{n} \rightarrow 0\) in \(H^{-m}(\Omega )\) for every \(\chi \in C_{0}^{\infty }(\Omega )\). However, P is a properly supported operator, since it is a differential operator, so that there exists \(\psi \in C_{0}^{\infty }(\Omega )\) such that \(\chi Pu_{n} = \chi P(\psi u_{n})\) for every \(n \in {\mathbb {N}}\). Therefore, (i) is also equivalent to

$$\begin{aligned} (\Lambda ^{-m}(\chi P(\psi u_{n})), \Lambda ^{-m}(\chi P(\psi u_{n}) ))_{L^{2}}= & {} (\Lambda ^{-2m}\chi P(\psi u_{n}), \chi P(\psi u_{n}))_{L^{2}} \\= & {} (P^{*} \overline{\chi }\Lambda ^{-2m}\chi P( \psi u_{n}), \psi u_{n})_{L^{2}} \rightarrow 0. \end{aligned}$$

If we set \(B = P^{*}\overline{\chi }\Lambda ^{-m}\chi P \in \Psi _{c}^{0}(\Omega )\), then the principal symbol of B is

$$\begin{aligned} \sigma _{0}(B) = |\chi (x)|^{2}|\sigma _{m}(P) (x, \xi )|^{2}|\xi |^{-2m} \end{aligned}$$

and

$$\begin{aligned} (\Lambda ^{-m}(\chi P(\psi u_{n})), \Lambda ^{-m}(\chi P(\psi u_{n}) ))_{L^{2}} \rightarrow \int _{\Omega \times S^{d-1}}{|\chi (x)|^{2}|\sigma _{m}(P) (x, \xi )|^{2}}d\mu (x, \xi ). \end{aligned}$$

Employing Theorem A.7 we conclude that the condition

$$\begin{aligned} \int _{\Omega \times S^{d-1}}{|\chi (x)|^{2}|\sigma _{m}(P)(x, \xi )|^{2}}d\mu = 0 \end{aligned}$$

for every \(\chi \in C_{0}^{\infty }(\Omega )\) is equivalent to (ii), which completes the proof. \(\square \)

Theorem A.10

Let P be a self-adjoint differential operator of order m on \(\Omega \) and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) that weakly converges to 0, with a microlocal defect measure \(\mu \). Suppose that \(P u_n\) converges to 0 in \(H_{loc}^{-(m-1)}\). Then the support of \(\mu \), \(\mathrm{supp}(\mu )\), is a union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of p,  where p is the principal symbol of P.

Proof

We first consider the function \(q(x,\xi )=|\xi |^{1-m}p(x,\xi ),\) which is smooth on \(\Omega \times ({\mathbb {R}}^d\setminus \{0\})\) and homogeneous of degree 1 in the variable \(\xi .\)

We have already noticed that the null-bicharacteristics of q are reparametrizations of the null-bicharacteristics of p. Hence, it is enough to prove that \(\mathrm{supp}(\mu )\) is a union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of q.

Notice that \(\Omega \times S^{d-1}\) is covered by the bicharacteristics of q (that is, the integral curves of the Hamiltonian \(H_q\)). Since \(H^{1-m}_{loc}(\Omega )\) is continuously included in \(H^{-m}_{loc}(\Omega ),\) a previous result implies that \(\mathrm{supp}(\mu )\) is a subset of \(\{(x,\xi )\in \Omega \times S^{d-1}:p(x,\xi )=0\}=\{(x,\xi )\in \Omega \times S^{d-1}:q(x,\xi )=0\}.\)

If \(s\in I \mapsto (x(s),\xi (s))\) is a bicharacteristic in \(\Omega \times ({\mathbb {R}}^d\setminus \{0\}),\) in which q never vanishes, then the homogeneity of q in the second variable implies that \(q(x(s),\xi (s)/|\xi (s)|)\ne 0\) for all s. Hence, the curve \(s\in I \mapsto (x(s),\xi (s)/|\xi (s)|)\) never touches \(\mathrm{supp}(\mu ).\) It follows that \(\mathrm{supp}(\mu )\) is a subset of the union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of q.

To complete the proof, we must show that, for a null-bicharacteristic of q\((x(s),\xi (s)),\) defined on an interval I,  such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for some \(s_0\in I,\) we have \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for all \(s\in I.\)

We first notice that it is enough to consider a local version of the above assertion. Indeed, the set \(A=\left\{ t\in I: \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\right\} \) is closed in I. Moreover, A is open in I if for each \(s_0\in A,\) there exists \(\varepsilon >0\) such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu ),\) for all \(s\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I.\) In this case, we have \(A=I\) whenever \(A\ne \emptyset .\)

By the remarks in the above paragraphs, the proof reduces to show that for each \(s_0\in I\) such that \(\left( x(s_0), \frac{\xi (s_0)}{|\xi (s_0)|}\right) \in \mathrm{supp}(\mu ),\) there exists \(\varepsilon >0\) such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for all \(s\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I.\)

We will prove this by contradiction.

Assume that \(s_0\in I\) is such that \(\left( x(s_0), \frac{\xi (s_0)}{|\xi (s_0)|}\right) \in \mathrm{supp}(\mu )\) and for all \(\varepsilon >0\) we find \(s^{*}\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I\) such that \(\left( x(s^{*}), \frac{\xi (s^{*})}{|\xi (s^{*})|}\right) \not \in \mathrm{supp}(\mu ).\) Without loss of generality, we may assume \(s^*\in (s_0,s_0+\varepsilon ).\)

Let \((x_0,\xi _0)=(x(s_0),\xi (s_0))\) and \((x_0^{*},\xi _0^*)=(x(s^*),\xi (s^*)).\) By semigroup properties, we have \((x_0^{*},\xi _0^*)=(x(s^*-s_0,x_0,\xi _0),\xi (s^*-s_0,x_0,\xi _0)),\) where \((x(s,x,\xi ),\xi (s,x,\xi ))\) denote the bicharacteristics that satisfy \((x(0,x,\xi ),\xi (0,x,\xi ))=(x,\xi ).\) Notice that we also have \((x_0,\xi _0)=(x(s_0-s^*,x_0^*,\xi _0^*),\xi (s_0-s^*,x_0^*,\xi _0^*)).\)

Since we are working only locally, we can assume that we are working on a connected component of \(\Omega .\) Moreover, by choosing \(\varepsilon >0\) sufficiently small, we may assume that \(x_0^*\in B(x_0,r)\subset B[x_0,4r]\subset \Omega \) and that the bicharacteristics through the points on a compact set \(B[x_0,4r]\times S^{d-1}\subset \Omega \times ({\mathbb {R}}^d\setminus \{0\})\) are all defined on a same interval \((-\varepsilon ,\varepsilon ).\) Such interval contains \(s^*-s_0.\)

We claim that, by the continuous dependence on initial data, we may choose \(\varepsilon >0\) so that, for all \(s\in (-\varepsilon ,\varepsilon )\) with \(r\le 1/6\) sufficiently small and for all \((x,\xi )\in [B[x_0,4r]\setminus B(x_0,3r)]\times S^{d-1}\) we have \((x(s,x,\xi ),\xi (s,x,\xi ))\in (\Omega \setminus B[x_0,r])\times ({\mathbb {R}}^d\setminus \{0\}).\)

Indeed, for each \((\tilde{x},\tilde{\xi })\in [B[x_0,4r]\setminus B(x_0,3r)]\times S^{d-1},\) the solution \((x(s,\tilde{x},\tilde{\xi }),\xi (s,\tilde{x},\tilde{\xi }))\) is contained in \((\Omega \setminus B(x_0,11r/4))\times ({\mathbb {R}}^d\setminus B(0,3r))\) for s belonging to a sufficiently small interval \(I_{\tilde{x},\tilde{\xi }} =[-\delta _{\tilde{x},\tilde{\xi }},\delta _{\tilde{x},\tilde{\xi }}].\) Recall that the continuous dependence on initial data in the system \(\dot{x}=\partial q/\partial \xi \) and \(\dot{\xi }=-\partial q/\partial x\) implies that for each \(\varepsilon '>0,\) there exists \(\delta =\delta (\delta _{\tilde{x},\tilde{\xi }},\varepsilon ')>0\) such that for all \((x,\xi )\) satisfying \(\Vert (x,\xi )-(\tilde{x},\tilde{\xi })\Vert _{\max }<\delta ,\) we have \((x(s,x,\xi ),\xi (s,x,\xi ))\) defined on \(I_{\tilde{x},\tilde{\xi }}\) and

$$\begin{aligned} \Vert \left( x(s,x,\xi ),\xi (s,x,\xi )\right) -(x(s,\tilde{x},\tilde{\xi }), \xi (s,\tilde{x},\tilde{\xi }))\Vert _{\max }<\varepsilon ', \end{aligned}$$

for all \(s\in I_{\tilde{x},\tilde{\xi }}.\) Hereafter, we assume that \(\varepsilon _1<r/2\). For each \(s \in I_{\tilde{x},\tilde{\xi }}\) we define

$$\begin{aligned} A_{\tilde{x},\tilde{\xi }}^{s}=\{(x,\xi ) \in \Omega \times {\mathbb {R}}^{d}\setminus \{0\}: \Vert (x,\xi ) -(x(s,\tilde{x},\tilde{\xi }),\xi (s,\tilde{x},\tilde{\xi }))\Vert <\eta \}, \end{aligned}$$

where \(\eta <\min \{\delta ,r/4\}\). We also set

$$\begin{aligned} A_{\tilde{x},\tilde{\xi }}=\bigcup _{s \in I_{\tilde{x},\tilde{\xi }}} A_{\tilde{x},\tilde{\xi }}^{s} \end{aligned}$$

and

$$\begin{aligned} A=\bigcup _{(\tilde{x},\tilde{\xi }) \in B[x_0,4r] \setminus B(x_0,3r) \times S^{d-1}}A_{\tilde{x},\tilde{\xi }}. \end{aligned}$$

Note that the union of the sets \(A_{\tilde{x},\tilde{\xi }}\) with \((\tilde{x},\tilde{\xi }) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) does not intersect \(B(x_0,5r/2)\times B(0,11r/4)\). Hence by virtue of the compactness of \(B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) there exist \((\tilde{x}_1,\tilde{\xi }_1),\dots , (\tilde{x}_N,\tilde{\xi }_N) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) such that

$$\begin{aligned} B[x_0,3r]\setminus B(x_0,2r) \times S^{d-1} \subset \bigcup _{i=1}^{N} A_{\tilde{x}_i,\tilde{\xi }_i}. \end{aligned}$$

Therefore, given \((x_1,\xi _1) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\), there exists \(i_0 \in \{1,\dots , N\}\) such that

$$\begin{aligned} (x_1,\xi _1) \in A_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}} =\bigcup _{s \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}} A_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}^{s}. \end{aligned}$$

Thus, there exists \(s_0 \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\) such that

$$\begin{aligned} \Vert (x_1,\xi _1) -(x(s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}), \xi (s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}))\Vert <\eta \le \delta . \end{aligned}$$

Using the continuous dependence on data we deduce that

$$\begin{aligned} \Vert (x(s,x_1,\xi _1),\xi (s,x_1,\xi _1))-(x(s+s_0,\tilde{x}_{i_0}, \tilde{\xi }_{i_0}),\xi (s+s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}))\Vert<\varepsilon _1<\frac{r}{2} \end{aligned}$$

for all \(s \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\) such that \(s+s_0 \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\). Thus, for all \((a,b)\in B[x_0,r] \times B[0,r]\), we obtain

$$\begin{aligned}&\Vert (a,b) - (x(s,x_1,\xi _1),\xi (s,x_1,\xi _1))\Vert \\&\quad =\Vert (a,b) - (x(s+s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}), \xi (s+s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}))\Vert \\&\qquad - \Vert (x(s+s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0}), \xi (s+s_0,\tilde{x}_{i_0},\tilde{\xi }_{i_0})) -(x(s,x_1,\xi _1),\xi (s,x_1,\xi _1))\Vert \\&\quad \ge \frac{3r}{2}-\frac{r}{2}=r \hbox { for all } s \in I \hbox { such that } s+s_0 \in I, \end{aligned}$$

where I denotes the smallest of the intervals \(I_{\tilde{x}_i, \tilde{\xi }_i}\), for \(i \in \{1,\dots , N\}\). The situation described above is illustrated in Figure 1.

Fig. 1
figure 1

Proof diagram

Since \(\left( x_{0}^{*}, \frac{\xi _{0}^{*}}{|\xi _0^{*}|}\right) \not \in \mathrm{supp}(\mu ),\) it follows from Theorem A.7 that there exists \(g\in C^{\infty }_{0}(\Omega \times S^{d-1},[0,1])\) such that \(g\left( x_0^{*}, \frac{\xi _0^{*}}{|\xi _0^{*}|}\right) >0\) and \(\int _{\Omega \times S^{d-1}}gd\mu =0.\)

Let \(\phi \in C^{\infty }_0(\Omega )\) be such that \(\mathrm{supp}(\phi )\subset B[x_0^*,r_2]\subset B(x_0,r),\) \(\phi (x_0^*)=1\) and \(0\le \phi \le 1.\)

We then define \(\tilde{g}(x,\xi )=|\xi |^{1-m}\phi (x)g \left( x,\xi /|\xi |\right) ,\) \((x,\xi )\in \Omega \times ({\mathbb {R}}^{d}\setminus \{0\}).\) It follows that \(\tilde{g}\) is smooth on \(\Omega \times ({\mathbb {R}}^{d}\setminus \{0\}),\) homogeneous of degree \(1-m\) in the variable \(\xi ,\) and \(\tilde{g}\) has compact support in the variable x. Notice that

$$0\le \int _{\Omega \times S^{d-1}}\tilde{g}d\mu = \int _{\Omega \times S^{d-1}}\phi (x)g(x,\xi )d\mu \le \int _{\Omega \times S^{d-1}}gd\mu =0.$$

Hence, \(\int _{\Omega \times S^{d-1}}\tilde{g}d\mu =0.\)

Next, we use the bicharacteristics to bring the information given by g in a neighborhood of \(\left( x_{0}^{*}, \frac{\xi _{0}^{*}}{|\xi _0^{*}|}\right) \) to a neighborhood of \(\left( x_0, \frac{\xi _0}{|\xi _0|}\right) \).

For \((x,\xi )\in B(x_0,4r)\times S^{d-1},\) we define \(f(x,\xi )=\tilde{g}(x(s^*-s_0,x,\xi ),\xi (s^*-s_0,x,\xi )).\) It follows that f is well-defined since we previously established that the bicharacteristics \((x(s,x,\xi ),\xi (s,x,\xi ))\) are defined on a same interval which contains \(s^*-s_0,\) for all \((x,\xi )\in B(x_0,4r)\times S^{d-1}.\) Notice that f extends to a continuous function on \(\Omega \times S^{d-1}\) since \(f(x,\xi )=0\) for all \((x,\xi )\in (\Omega \setminus B[x_0,3r])\times S^{d-1}\) (recall that \((x(s^*-s_0,x,\xi ),\xi (s^*-s_0,x,\xi ))\) belongs to \((\Omega \setminus B[x_0,r])\times ({\mathbb {R}}^d\setminus \{0\}),\) for all \((x,\xi )\in (\Omega \setminus B[x_0,3r])\times S^{d-1}\)). It follows that \(f\in C_0(\Omega \times S^{d-1},[0,1]).\) In order to complete the proof, we will show that \(f(x_0,\xi _0/|\xi _0|)>0\) and \(\int _{\Omega \times S^{d-1}}fd\mu =0.\) Note that, by Theorem A.7, this implies a contradiction, since \((x_0,\xi _0/|\xi _0|)\in \mathrm{supp}(\mu )\).

Before we proceed, we note that for \(\lambda >0\) we have

$$\begin{aligned} (x(s,x,\lambda \xi ),\xi (s,x,\lambda \xi ))=(x(s,x,\xi ),\lambda \xi (s,x,\xi )) \end{aligned}$$
(A.35)

for all \((x,\xi )\in \Omega \times ({\mathbb {R}}^{d}\setminus \{0\}).\) Indeed, since q is homogeneous of degree one in the variable \(\xi ,\) it follows that \(\partial q/\partial \xi \) is homogeneous of degree zero and \(\partial q/\partial x\) is homogeneous of degree one. Hence,

$$\begin{aligned} \frac{d}{ds}x(s,x,\xi )=\frac{\partial q}{\partial \xi }(x(s,x,\xi ),\xi (s,x,\xi ))=\frac{\partial q}{\partial \xi }(x(s,x,\xi ),\lambda \xi (s,x,\xi )) \end{aligned}$$

and

$$\begin{aligned} \lambda \frac{d}{ds}\xi (s,x,\xi )=-\lambda \frac{\partial q}{\partial x}(x(s,x,\xi ),\xi (s,x,\xi ))=-\frac{\partial q}{\partial x}(x(s,x,\xi ),\lambda \xi (s,x,\xi )). \end{aligned}$$

It follows that both \((x(s,x,\xi ),\lambda \xi (s,x,\xi ))\) and \((x(s,x,\lambda \xi ),\xi (s,x,\lambda \xi ))\) are solutions through the point \((x,\lambda \xi ).\) By uniqueness we conclude that (A.35) holds.

It follows that

$$\begin{aligned} f(x_0,\xi _0/|\xi _0|)&=\tilde{g}(x(s^{*}-s_0,x_0,\xi _0/|\xi _0|),\xi (s^{*} -s_0,x_0,\xi _0/|\xi _0|))\\&=|\xi _0|^{m-1}\tilde{g}(x(s^{*}-s_0,x_0,\xi _0),\xi (s^{*}-s_0,x_0,\xi _0))\\&=|\xi _0|^{m-1}|\xi _0^{*}|^{1-m}\phi (x_0^*)g(x_0^{*}, \xi _0^{*}/|\xi _0^*|)>0. \end{aligned}$$

In order to compute \(\int _{\Omega \times S^{d-1}}fd\mu ,\) we use the notation

$$\begin{aligned} \phi _{s}(x,\xi )=(x(s,x,\xi ),\xi (s,x,\xi )) \end{aligned}$$

for \(s\in (-\varepsilon ,\varepsilon )\) and \((x,\xi )\in B(x_0,4r)\times S^{d-1}\). In particular,

$$\begin{aligned} \phi _{s^*-s_0}(x,\xi )=(x(s^*-s_0,x,\xi ),\xi (s^*-s_0,x,\xi )) \end{aligned}$$

and

$$\begin{aligned} 0\le \int _{\Omega \times S^{d-1}}fd\mu = \int _{\Omega \times S^{d-1}}\chi (\tilde{g}\circ \phi _{s^*-s_0})d\mu , \end{aligned}$$

where \(\chi \in C^\infty _0(B(x_0,4r))\) is such that \(0\le \chi \le 1\) and \(\chi \equiv 1\) on a neighborhood of \(B[x_0,3r].\)

We complete the proof by showing that

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\chi (\tilde{g}\circ \phi _{s^*-s_0})d\mu =\int _{\Omega \times S^{d-1}}\chi \tilde{g}d\mu , \end{aligned}$$

since

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\chi \tilde{g}d\mu \le \int _{\Omega \times S^{d-1}}\tilde{g}d\mu =0. \end{aligned}$$

For \(s\in (-\varepsilon ,\varepsilon ),\) we have

$$\begin{aligned} \frac{d}{ds}(\chi (\tilde{g}\circ \phi _{s}))(x,\xi )= & {} \chi (x)\frac{d}{d\tau } \left( \tilde{g}\circ \phi _{s+\tau }\right) (x,\xi )|_{\tau =0}\\= & {} \chi (x)\frac{d}{d\tau }\left( \tilde{g} \circ \phi _s \circ \phi _\tau \right) (x,\xi )|_{\tau =0}\\= & {} \chi (x)\{\tilde{g}\circ \phi _s,q\}(x,\xi ). \end{aligned}$$

Observing that \(\mathrm{supp}(H_q(\chi )(x,\xi ))\subset B(x_0,4r)\setminus B[x_0,3r]\), it follows that the function \((\tilde{g}\circ \phi _s)(x,\xi )H_q(\chi )(x,\xi )\) vanishes identically on \(\Omega \times S^{d-1}\). Thus, we also have

$$\begin{aligned} \{\chi (\tilde{g}\circ \phi _s),q\}(x,\xi )&=-H_q(\chi (\tilde{g}\circ \phi _s))(x,\xi )\\&=-\chi (x)H_q(\tilde{g}\circ \phi _s)-(\tilde{g}\circ \phi _s)(x,\xi )H_q(\chi )(x,\xi )\\&=-\chi (x)H_q(\tilde{g}\circ \phi _s)\\&=chi(x)\{\tilde{g}\circ \phi _s,q\}(x,\xi ). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{d}{ds}\int _{\Omega \times S^{d-1}}\chi (\tilde{g}\circ \phi _s)\,d\mu= & {} \int _{\Omega \times S^{d-1}}\{\chi (\tilde{g}\circ \phi _s),q\}\,d\mu \nonumber \\= & {} \int _{\Omega \times S^{d-1}}\{\chi (\tilde{g}\circ \phi _s),p\}\,d\mu , \end{aligned}$$
(A.36)

since \(H_q=|\xi |^{1-m}H_p\) on the support of \(\mu .\)

Applying Theorem A.6, it follows that

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\{\chi (\tilde{g}\circ \phi _s),p\}\,d\mu =0. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\Omega \times S^{d-1}}\chi (\tilde{g}\circ \phi _s)\,d\mu \end{aligned}$$

is constant on \((-\varepsilon ,\varepsilon ).\) Using \(s=0\) and \(s=s^{*}-s_0,\) we obtain

$$\begin{aligned} \int _{\Omega \times S^{d-1}} \chi (\tilde{g} \circ \phi _{s^*-s_0}) d \mu = \int _{\Omega \times S^{d-1}} \chi \tilde{g} d\mu , \end{aligned}$$

which completes the proof. \(\square \)

We finish this section by examining the case of the wave equation in an inhomogeneous medium:

$$\begin{aligned} P(x,D)u=-\rho (x) \partial _t^2u + \sum _{i,j=1}^d \partial _{x_i} (K(x) \partial _{x_j}u). \end{aligned}$$

whose principal symbol is given by

$$\begin{aligned} p(t,x,\tau ,\xi )=-\rho (x)\tau ^2 + \xi ^{\top } \cdot K(x) \cdot \xi , \hbox { where } \xi =(\xi _1,\cdots ,\xi _d), \end{aligned}$$
(A.37)

\(t \in {\mathbb {R}}\), \(x \in \Omega \subset {\mathbb {R}}^{d}\), \((\tau ,\xi ) \in {\mathbb {R}}^{d+1}\), \(\rho \in C^\infty (\Omega )\), \(0<\alpha \le \rho (x)\le \beta <\infty \), and \(K(x)=(k_{ij}(x))_{1 \le i,j \le d}\) is a positive-definite matrix satisfying

$$\begin{aligned} a|\xi |^2 \le \xi ^{\top } \cdot K(x) \xi \le b |\xi |^2, \end{aligned}$$

for \(0<a<b<\infty \).

Let’s describe the bicharacteristics of p. From Lemma A.2 the bicharacteristics do not change if we multiply p by a non-zero function. So we can study the Hamiltonian curves of

$$\begin{aligned} \tilde{p}(t,x,\tau ,\xi )=\frac{1}{2}\left( -\tau ^2 + \xi ^{\top } \cdot \frac{K(x)}{\rho (x)} \cdot \xi \right) . \end{aligned}$$
(A.38)

Proposition A.3

Up to a change of variables, the bicharacteristics of (A.38) are curves of the form

$$\begin{aligned} s \mapsto \left( s, y(s), \tau , -\tau \left( \frac{K(y(s))}{\rho (y(s))}\right) ^{-1} \dot{y}(s) \right) , \end{aligned}$$

where \(s \mapsto y(s)\) is a geodesic of the metric \(G=\left( \frac{K}{\rho }\right) ^{-1}\) on \(\Omega \), parameterized by the arc length.

Proof

Let’s define a curve

$$\begin{aligned} \begin{array}{cccl} \alpha :&{} I&{} \longrightarrow &{} \Omega \times {\mathbb {R}}^{d}\setminus \{0\}\\ &{} s &{} \mapsto &{} (t(s), x(s), \tau (s), \xi (s)) \end{array} \end{aligned}$$

such that

$$\begin{aligned} \left\{ \begin{aligned} \dot{t}(s)&= \frac{\partial \tilde{p}}{\partial \tau }(\alpha (s)),\\ \dot{x}_j(s)&= \frac{\partial \tilde{p}}{\partial \xi _j} (\alpha (s)), j = 1, \cdots ,d,\\ \dot{\tau }(s)&=-\frac{\partial \tilde{p}}{\partial t}(\alpha (s)),\\ \dot{\xi }_j(s)&=-\frac{\partial \tilde{p}}{\partial x_j}(\alpha (s)),~j= 1, \cdots ,d. \end{aligned}\right. \end{aligned}$$
(A.39)

From (A.38) and (A.39)\(_1\), we have

$$\begin{aligned} \dot{t}(s) = - \tau (s). \end{aligned}$$

Also, from (A.38) and (A.39)\(_2\) it follows that

$$\begin{aligned} \dot{x}(s) = \frac{ K(x(s))}{\rho (x(s))} \cdot \xi (s). \end{aligned}$$
(A.40)

Equation (A.38) and (A.39)\(_3\) ensure that

$$\begin{aligned} \dot{\tau }(s) = - \frac{\partial \tilde{p}}{\partial t} (\alpha (s)) = 0. \end{aligned}$$

Finally, from (A.39)\(_3\) we obtain

$$\begin{aligned} \dot{\xi }_j(s) =-\frac{\partial \tilde{p}}{\partial x_j} (\alpha (s)) =- \frac{1}{2} \sum _{ 1 \le i, j \le d} \frac{\partial }{\partial x_j} \left( \frac{k_{i j}(x(s))}{\rho (x(s))} \right) \xi _i(s)\cdot \xi _j(s). \end{aligned}$$

Thus, the sought after curve must satisfy the equations

$$\begin{aligned} \left\{ \begin{aligned}&\dot{t}(s) = -\tau (s),\\&\dot{x}(s) = \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s),\\&\dot{\tau }(s) = 0,\\&\dot{\xi }(s) = - \frac{1}{2} {(\xi (s))}^{\top } \cdot \nabla \left( \frac{K(x(s))}{\rho (x(s))}\right) \cdot \xi (s) \end{aligned}\right. \end{aligned}$$
(A.41)

Introducing the matrix \(G(x(s)) = \left( \frac{K(x(s))}{\rho (x(s))}\right) ^{-1},\) from the second equation of (A.41) we obtain

$$\begin{aligned} \xi (s) = G(x(s)) \dot{x}(s). \end{aligned}$$
(A.42)

Let \(f:\mathrm{GL}(n,{\mathbb {R}}) \longrightarrow \mathrm{GL} (n,{\mathbb {R}})\) be defined by \(f(X) = X^{-1}\) so that its derivative is given by \(d (f)_{A} (V) = - A^{-1} \cdot V \cdot A^{-1}.\) So,

$$\begin{aligned} \frac{\partial G}{\partial x_k} (x)&= d \left( f \circ \frac{K}{\rho }\right) _{x}(e_k)\\&= d(f)_{\left( \frac{K}{\rho }\right) (x)} \circ d \left( \frac{K}{\rho }\right) _{x} (e_k)\\&= d(f)_{\left( \frac{K}{\rho }\right) (x)} \left( \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) \right) \\&= - \left( \frac{K(x)}{\rho (x)}\right) ^{-1} \cdot \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) \cdot \left( \frac{K(x)}{\rho (x)}\right) ^{-1}. \end{aligned}$$

That is,

$$\begin{aligned} \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) =-\frac{K(x)}{\rho (x)} \frac{\partial G}{\partial x_k}(x) \frac{K(x)}{\rho (x)}. \end{aligned}$$
(A.43)

Therefore, from (A.42) and (A.43), we obtain

$$\begin{aligned} \dot{\xi }_k&= - \frac{1}{2} \sum _{1 \le i,j \le d} \frac{\partial }{\partial x_k} \left( \frac{k_{ij}(x(s)}{\rho (x)})\right) \xi _{i}(s) \cdot \xi _k(s)\nonumber \\&= -\frac{1}{2} \xi ^{\top } \cdot \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) \cdot \xi \nonumber \\&= - \frac{1}{2} (G(x)\dot{x} )^{\top } \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) \cdot G(x)\dot{x} \nonumber \\&= - \frac{1}{2} \dot{x}^{\top } G(x)^{\top } \cdot \frac{\partial }{\partial x_k} \left( \frac{K(x)}{\rho (x)}\right) \cdot G(x) \dot{x}\nonumber \\&= - \frac{1}{2} \dot{x}^{\top } G(x) \cdot \left( - \frac{K(x)}{\rho (x)} \cdot \frac{\partial G}{\partial x_k}(x) \cdot \frac{K(x)}{\rho (x)}\right) \cdot G(x) \dot{x}\nonumber \\&= \frac{1}{2} \dot{x}^{\top } G(x) \cdot \frac{K(x)}{\rho (x)} \cdot \frac{\partial G}{\partial x_k}(x) \cdot \frac{K(x)}{\rho (x)} \cdot G(x) \dot{x}\nonumber \\&= \frac{1}{2} \dot{x}^{\top } \cdot \frac{\partial G}{\partial x_k}(x) \cdot \dot{x}. \end{aligned}$$
(A.44)

Thus, from (A.42) and (A.44), we have

$$\begin{aligned} (G(x) \cdot \dot{x})^{\varvec{\cdot }} = \dot{\xi } =\frac{1}{2} \dot{x}^{\top } \cdot \nabla G(x) \cdot \dot{x}. \end{aligned}$$
(A.45)

Since \(\tilde{p}\) is 0 over each null bicharacteristic, it follows that

$$\begin{aligned} \xi (s)^{\top } \cdot \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s) = \tau (s)^{2}. \end{aligned}$$

And since \(\dot{\tau }(s) = 0,\) we have \(\tau (s) = \tau ,\) for some \(\tau \in {\mathbb {R}}.\) Thus,

$$\begin{aligned} \xi (s)^{\top } \cdot \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s) = \tau ^2. \end{aligned}$$

On the other hand, from (A.40)

$$\begin{aligned} \dot{x}(s)^{\top } \cdot G(x(s)) \cdot \dot{x}(s)&= \left( \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s)\right) ^{\top } \cdot G(x(s)) \cdot \left( \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s) \right) \nonumber \\&= \xi ^{\top } \cdot \left( \frac{K(x(s))}{\rho (x(s))}\right) ^{\top } \cdot G(x(s)) \cdot \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s)\nonumber \\&= \xi (s)^{\top }\cdot \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s)\nonumber \\&= \tau ^2, \end{aligned}$$
(A.46)

that is, \(\dot{x}(s)^{\top } \cdot G(x(s)) \cdot x(s)\) over each null bicharacteristic of \(\tilde{p}\). By (A.45) and (A.46) we can write

$$\begin{aligned} \frac{d}{ds} \frac{G(x(s)) \cdot \dot{x}(s)}{\sqrt{\dot{x}(s)^{\top } \cdot G(x(s)) \cdot \dot{x}(s)}} = \frac{1}{2} \frac{ \dot{x}(s)^{\top } \cdot \nabla G(x(s)) \cdot \dot{x}(s)}{\sqrt{\dot{x}(s)^{\top } \cdot G(x(s)) \cdot \dot{x}(s)}}. \end{aligned}$$
(A.47)

Defining the arc length functional L by

$$\begin{aligned} L(x, \dot{x}) = \sqrt{\dot{x}^{\top } \cdot G(x) \cdot {x}}, \end{aligned}$$

from (A.47) we obtain

$$\begin{aligned} \frac{d}{ds} \frac{\partial }{\partial \dot{x}} L(x, \dot{x}) = \frac{ \partial L}{\partial x}(x, \dot{x}). \end{aligned}$$

that is, L satisfies the Euler–Lagrange equations, so x is a geodesic.

Therefore, the bicharacteristics are curves of the form

$$\begin{aligned} \gamma (s) = (- \tau s, x(s), \tau , G(x(s)) \dot{x}(s)). \end{aligned}$$

Consider the function \(\alpha (s) = - \frac{s}{\tau },\) thus \(\alpha \) is the arc length parameter for x. In fact, it suffices to show that for \(y(s) = (x \circ \alpha )(s)\) we have

$$\begin{aligned} \Vert \dot{y}(s)\Vert _{y(s) = 1.} \end{aligned}$$

Indeed,

$$\begin{aligned} \Vert \dot{y}(s)\Vert _{y(s)}&= \Vert (x \circ \alpha )^{\varvec{\cdot }}(s) \Vert _{x(\alpha (s))}\\&= (x \circ \alpha )^{\varvec{\cdot }}(s)^{\top } \cdot G(x(\alpha (s))) \cdot (x \circ \alpha )^{\varvec{\cdot }}(s)\\&= (\dot{x}(\alpha (s))\dot{\alpha }(s))^{\top } \cdot G(x(\alpha (s))) \cdot \dot{x}(\alpha (s)) \dot{\alpha }(s)\\&= \left( \dot{x}(\alpha (s))\left( -\frac{1}{\tau }\right) \right) ^{\top } \cdot G(x(\alpha (s))) \cdot \dot{x}(\alpha (s)) \left( - \frac{1}{\tau }\right) \\&= - \frac{1}{\tau } \dot{x}(\alpha (s)) \cdot G(x(\alpha (s))) \cdot \dot{x}(\alpha (s)) \left( - \frac{1}{\tau }\right) \\&= \frac{1}{\tau ^{2}} \tau ^{2}\\&=1. \end{aligned}$$

Note that, \((x \circ \alpha )^{\varvec{\cdot }}(s) = \dot{x}(\alpha (s)) \dot{\alpha }(s) = - \frac{\dot{x}(\alpha (s))}{\tau }.\) So, \(\dot{x}(\alpha (s)) = - \tau (x \circ \alpha )^{\varvec{\cdot }}(s).\) Thus,

$$\begin{aligned} (\gamma \circ \alpha )(s)&= \gamma (\alpha (s))\\&= (- \tau \alpha (s), x(\alpha (s)), \tau , G(x(\alpha (s)))\dot{x}(\alpha (s))\\&= (- \tau \left( - \frac{s}{\tau }\right) , y(s), \tau , - \tau G(y(s)) \dot{y}(s))\\&= (s, y(s), \tau , - \tau G(y(s)) \dot{y}(s)). \end{aligned}$$

Therefore, less than one reparametrization, the bicharacteristics of p are curves of the form

$$\begin{aligned} s \mapsto (s, y(s), \tau , - \tau G(y(s)) \dot{y}(s)) \end{aligned}$$

where y(s) is a geodesic of the G metric parameterized by the arc length.

Conversely, consider \(s \mapsto y(s)\) a geodesic of the G metric parameterized by the arc length. Let us show that, less than one reparameterization of the curve \(\gamma \) given by

$$\begin{aligned} s \mapsto (s, y(s), \tau , - \tau G(y(s)) \dot{y}(s)) \end{aligned}$$

is a bicharacteristics of p. For this, we must prove that the component functions satisfy the Hamilton-Jacobi equations.

Let \( \beta (s) = - \tau s, \) then defining \(x(s) = y(\beta (s)),\) we have

$$\begin{aligned} \gamma (\beta (s))&= (\beta (s), y(\beta (s)), \tau , - \tau G(y(\beta (s))) \dot{y}(\beta (s))\\&= \left( - \tau s, x(s), \tau , - \tau G(x(s)) (y \circ \beta )^{\varvec{\cdot }} (s) \frac{1}{\dot{\beta }(s)}\right) \\&= (- \tau s, x(s), \tau , G(x(s)) \dot{x}(s)). \end{aligned}$$

Let \(t(s) = - \tau s\) and \(\xi (s) = G(x(s)) \dot{x}(s).\) Thus,

$$\begin{aligned} \dot{t}(s)&= - \tau \nonumber \\ \dot{x}(s)&= \frac{K(x(s))}{\rho (x(s))} \cdot \xi (s)\nonumber \\ \dot{\tau }(s)&= 0. \end{aligned}$$
(A.48)

It remains to show that

$$\begin{aligned} \dot{\xi }(s) = - \frac{1}{2} \xi (s)^{\top } \cdot \nabla \left( \frac{K(x(s))}{\rho (x(s))} \right) \cdot \xi (s), \end{aligned}$$

which is equivalent to

$$\begin{aligned} (G(x(s) \dot{x}(s))^{\varvec{\cdot }} = \frac{1}{2} \dot{x}(s) \cdot \nabla G(x(s)) \cdot \dot{x}(s). \end{aligned}$$

Since y(s) is parameterized by arc length, we have

$$\begin{aligned} \Vert \dot{y}(s)\Vert _{y(s)} = 1, \end{aligned}$$

that is,

$$\begin{aligned} 1 = \dot{y}(s)^{\top } \cdot G(y(s))\cdot \dot{y}(s). \end{aligned}$$

From the Euler–Lagrange equations,

$$\begin{aligned} (G(y(s)) \dot{y}(s))^{\varvec{\cdot }} = \frac{1}{2} (\dot{y}(s)^{\top } \cdot \nabla G(y(s)) \cdot \dot{y}(s)). \end{aligned}$$

Hence, taking \(f(s) = G(y(s)) \dot{y}(s)\), we have

$$\begin{aligned} (f \circ \beta )^{\varvec{\cdot }}(s) = \dot{f}(\beta (s)) \dot{\beta }(s) = \frac{1}{2} \left( \dot{y} (\beta (s))^{\top } \cdot \nabla G(y(\beta (s))) \cdot \dot{y}(\beta (s))\right) \dot{\beta }(s).\qquad \end{aligned}$$
(A.49)

Thus, from (A.49) and recalling that \(\dot{\beta }(s) = - \tau \), we obtain

$$\begin{aligned} (G(x(s)) \dot{x}(s))^{\varvec{\cdot }}&= (G(y(\beta (s)) (y \circ \beta )^{\varvec{\cdot }} (s))^{\varvec{\cdot }}\\&= (G(y(\beta (s))\dot{y}(\beta (s))\dot{\beta }(s))^{\varvec{\cdot }}\\&= - \tau (G(y (\beta (s)) \dot{y}( \beta (s)))^{\varvec{\cdot }}\\&= - \tau \frac{1}{2} \left( \dot{y} (\beta (s))^{\top } \cdot \nabla G (y(\beta (s))) \cdot \dot{y}(\beta (s))\right) \beta ^{\varvec{\cdot }}(s)\\&= \tau ^{2} \frac{1}{2} \left( \frac{\dot{x}(s)}{\tau }^{\top } \cdot \nabla G(x(s)) \cdot \frac{\dot{x}(s)}{\tau }\right) \\&= \frac{1}{2} (\dot{x}(s)^{\top } \cdot \nabla G(x(s)) \cdot \dot{x}(s)). \end{aligned}$$

Therefore,

$$\begin{aligned} \dot{\xi }(s) = \frac{1}{2} (\dot{x}(s))^{\top } \cdot \nabla G(x(s)) \cdot \dot{x}(s)), \end{aligned}$$

which ends the proof. \(\square \)

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Cavalcanti, M.M., Domingos Cavalcanti, V.N., Gonzalez Martinez, V.H. et al. Decay Rate Estimates for the Wave Equation with Subcritical Semilinearities and Locally Distributed Nonlinear Dissipation. Appl Math Optim 87, 2 (2023). https://doi.org/10.1007/s00245-022-09918-4

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