Euler Lagrange equation - weak solutions?

[muzialis]

Hello there,

I was wondering if anybody could indicate me a reference with regards to the following problem.

In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer.
Can the Euler - Lagrange approach be enriched to cover piecewise smooth solutions, with a weak derivative?

Any reference or hint would be so much appreciated.

Thanks

Muzialis

 

[AlephZero]

This is one way to create finite element approximations where the approximating function is piecewise smooth over the region (line, surface, or volume) covered by each element.

Look for something on variational methods of formulating finite elements.

Note, there are other ways to create FE approximations, which may appear to be mathematically "simpler", and avoid the difficuilty that for some applications of FE it is hard to find a variational form to minimise, but an "advanced" text on the math of FE methods should cover variational methods.

I learned this stuff a long time ago, so I can't give you a personal recommendation for a good modern textbook or website - sorry about that.