What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it allows for discontinuities in coefficients of PDEs where as the original PDE in differential form must have a "truly" smooth solution to be solved analytically. So my question is, firstly what is a weak formulation in the physical sense? and secondly, is it only used to handle PDEs with coefficient discontinuities in the sense that an analytical integration gives a "too smooth" result which isn't relevant in problems of nature?
Firstly, the important thing is not that the coefficients in the PDEs are allowed to be badly-behaved, but the solution itself
A weak formulation is often the natural way in which we get the PDE from the physics, perhaps most importantly from conservation laws. For example, we might get an equation for fluid flow from conservation of momentum, by considering an arbitrary region. Then (rate of change of momentum)=(nett flux of moment)+(force), written in an integral (weak!) form. By taking a small region, we get a differential formulation.
The weak solutions themselves are also physically relevant, because the evolution of the system leads to a discontinuity. For example, one-dimensional inviscid fluid flow is described by the Euler equations, but for certain boundary and initial conditions, these do not have a smooth solution: in particular, shock waves can form, where the pressure (say) is discontinuous. The PDEs on their own are ambiguous as to how to proceed, but a weak formulation will give a specific weak solution, using something like the Rankine-Hugoniot jump conditions. Similar phenomena happen for water waves breaking and models of traffic flow.
I'm not sure I've directly addressed your questions, but I hope that's enough to answer them!