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Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.

Theorem 1:(Lagrangian Neighborhood Theorem) Let $(X,\omega)$ be a symplectic manifold and $L \subset X$ be a closed Lagrangian. Then there exists a neighborhood $U$ of $L$ in $X$ and a symplectomorphism $\varphi:U \simeq V \subset T^*L$ taking $L$ identically to the zero-section $L \subset T^*L$.

Now let $(W,\lambda)$ be a Liouville domain. That is, $W$ is a compact manifold with boundary, and $\lambda$ is a $1$-form on $W$ such that $d\lambda$ is symplectic and $\lambda|_{\partial W}$ is a contact form. Furthermore, let $L \subset W$ be a compact Lagrangian sub-manifold with Legendrian boundary $\partial L \subset \partial W$.

My question is whether the following version of the neighborhood theorem holds in this setting. It seems to me that if it is true, then it should be standard, but I can't find a reference.

Theorem 2 (Maybe?):There exists a neighborhood $U$ of $L$ in $W$ and a symplectomorphism of manifolds with boundary $\varphi:U \simeq V \subset T^*L$ taking $L$ identically to the zero-section $L \subset T^*L$.

**Remark On Proof Of Theorem 1:** The basic result that the usual Lagrangian neighborhood theorem depends on is the following lemma (see [1] or McDuff-Salamon).

Lemma:Let $X$ be a manifold with closed sub-manifold $S \subset X$, and let $\omega_0, \omega_1$ be two symplectic forms on $X$. Suppose that $\omega_0 = \omega_1$ on the fiber $T_sX$ for any $s \in S$.Then there exists neighborhoods $N_0$ and $N_1$ of $S$ and a symplectomorphism $\varphi:N_0 \to N_1$ with $\varphi|_S = \text{Id}$ and $\varphi^*\omega_1 = \omega_0$.

The proof is a version of the usual Moser trick. You find a $1$-form $\sigma$ in a neighborhood with $d\sigma = \omega_1 - \omega_0$ and then you integrate the vector-field $Z_t$ satisfying: $$\iota_{Z_t}\omega_t = -\sigma\quad\text{where}\quad\omega_t = (1-t)\omega_0 + t\omega_1$$ This gives you a family of diffeomorphisms with $\varphi^*_t\omega_t = \omega_0$ and you're done. If you try to run this proof on a sub-manifold $S \subset X$ with $\partial S \subset \partial X$, you run into the issue that $Z_t$ needs to be parallel to the boundary $\partial X$ in order for the flow to be well-defined. If I'm not mistaken, the criterion for this to be the case is: $$ T(\partial X)^{\omega_t} \subset \ker(\sigma) \text{ on }\partial X $$ Here $T(\partial X)^{\omega_t}$ is the characteristic foliation on $\partial X$ with respect to $\omega_t$, i.e. the symplectic perp to the tangent space to $\partial X$. It isn't clear to me that you can even accomplish the above inclusion for $\sigma$, or that you can upgrade $\sigma$ to a family $\sigma_t$ with this property.

**Speculation On Validity Of Theorem 2:** On a conceptual level, I can't decide whether or not Theorem 2 is too optimistic. Here is what makes me skeptical about it.

Theorem 2 would imply not only that the boundaries $\partial U \simeq \partial V$ of $U$ and $V$ were contactomorphic, but also that the characteristic foliations $T(\partial U)^{d\lambda}$ and $T(\partial V)^{d\lambda}$ on $U$ and $V$ near $\partial L$ were the same. The characteristic foliation of a contact hypersurface is generally very sensitive to the embedding of said hypersurface, and from that perspective a standard neighborhood in the vein of Theorem 2 would be a bit surprising.

I haven't pursued this idea enough to produce a counter-example unfortunately.