Displacement energy of compact Lagrangian submanifold from open subset

We prove that for any compact Lagrangian submanifold intersecting an open subset $U$ in tame symplectic manifold $(M,\omega)$, the Hofer displacement energy of $L$ from $U$ is positive, provided $L \cap U \neq \emptyset$. We also give an explicitlower bound in terms of an $\epsilon$-regularity type invariant for pseudo-holomorphic curves relative to $L$.