I am currently reading "Introduction to the Calculus of Variations" by Bernard Dacorogna. In a certain chapter he proves higher regularity for a Minimizer of the following functional:
$I(u):= \int_\Omega f(x,u(x),\nabla u(x))dx$.
Here $\Omega\subset\mathbb{R}^n$ is open and bounded and $f(x,u,\xi) = g(x,\xi) - h(x)u$, $(x,u,\xi)\in \mathbb{\Omega}\times\mathbb{R}^n\times\mathbb{R}^n$, where $g$ is continuously differentiable and grows quadratically. Furthermore $h\in L^2(\mathbb{R}^n)$. I understand everything about the proof, it is perfectly fine. My question is just why we are interested in this particular $f$. I can imagine that it comes from some kind of problem from physics, but I don't know anything specific.
Can you help me out here? Moreover is there a source for a result with a more general $f$?
If we chose $g(x, \xi)=\frac{1}{2}|\xi|^2$ and $h \in L^2(\Omega)$ as an arbitrary function, we have $$ \frac{1}{2}\int_{\Omega} |\nabla u|^2-hu $$ Minimizing the above integral over all admissible function, say $u \in W^{1,2}(\Omega)$ with certain boundary conditions, we see that a minimizer satisfies the Euler-Lagrange equation (if $u$ is scalar valued): $$ -\Delta u=h $$ for all points in $\Omega$. You can also require certain boundary conditions.
A fair amount theories in theoretical physics involving variational formulations have the form $$ E=E_{kin}+E_{pot} $$ and one often choses $E_{kin}=\frac{1}{2}\int |\nabla u|^2$. Many problems fall under this category; see the Wikipedia page for "Poission's equation" for a few examples to check the relevance of the Euler-Lagrange equation of your problem.