Logical volume managers exist to ease disk space management. I will give you a brief overview of LVM so that you understand how it works and what is or isn't possible with it.

1. Basic Organisation
LVM consists of three main entities: disks (sometimes called physical volumes), disk groups (also called volume groups) and logical volumes.
Any number of disks (limited by implementation maximum values) can be collected to form volume groups. The volume groups can then be divided into logical volumes (again the minimum/maximum size and number of volumes is implementation dependent). A logical volume can exist on any physical volume(s) - this is hidden by the volume manager so that the filesystem sees and operates only on the logical volume (and applications in turn work only with the filesystem).

2. Advanced Organisation
A logical volume can be organised such that it occupies disks in one of several ways:
simple - a volume group consists of a single disk and multiple volumes exist on that disk. This is found on most home computers (akin to simple Windows partitions)
concatenation - a logical volume is larger than any single physical disk and is formed by concatenating multiple disks together. In this successive disks are used as the volume fills up.
striping - a volume may or may not be larger than a single disk, but is distributed over multiple disks to increase access speed. The distribution is in form of logical sectors being spread over multiple disks' phyiscal sectors.
mirroring - a logical volume is replicated within the volume group to prevent data loss due to loss of physical disks containing the logical volume. To achieve true mirroring, no sectors of the mirror volume should occupy any of the disks that are occupied by the volume being mirrored.

There is more, but probably would be better placed in the FAQ or Tutorials.

Now for your question:
Unix will not automatically allocate space on other physical volumes associated with the volume group. You can, however, extend the logical volume and the associated filesystem as the volume gets filled. About the last physical volume getting filled up, you can always extend the volume group itself by adding more physical volumes. This can be done to any extent (within the implementation limits).
Usually the LVM (I have used HP-LVM and Veritas VM), comes with the utilities that extend the filesystem without any impact to the applications running within the filesystem. In HP-LVM, you can use fsadm and in VxVM, you can use vxresize.