This is related to this post.
I think I am having problem with H2 meaning that it does not close properly.
I suspect this since I see myDB.lock.db
Not sure if this is relevant to your situation but have you tried adding a DBStarter listener?
http://www.h2database.com/html/tutorial.html, see the "Using a Servlet Listener to Start and Stop a Database" section.
The link suggests adding the following to web.xml:
org.h2.server.web.DbStarter
Please see discussion here (admittedly from 2008 so may be out of date) - apparently the fix applies to both embedded and non-embedded instances: http://groups.google.com/group/h2-database/browse_thread/thread/eb609d58c96f4317
Alternatively how are you using the connections? Are you sure you are cleaning up the connections properly?
I was having issues before, in my case I was using the connection with a JPA EntityManager and I forgot to close the EntityManager instance after use, which resulted in some issues:
@PersistenceUnit(unitName="myEm")
private EntityManagerFactory emf;
public void doStuff() {
EntityManager em = emf.createEntityManager();
...
em.close(); // forgot this line
}