Most techniques in the calculus of variations that I know of, deal with integrands of the form
$W(x, \phi(x), \nabla \phi(x)): \Omega \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \to \mathbb{R}$, $\Omega \subset \mathbb{R}^n$ that give rise to functionals of the form
$$J(\phi) = \int_{\Omega} W(x, \phi(x), \nabla \phi(x)) \ dx, \ \phi \in W^{k,p}(\Omega) \text{ for some suitable } k, p$$
Currently, however, I'm interested in functionals where the integrand cannot be expressed in the way above, because, for example, $\phi$ appears in a convolution $(\Psi \ast \phi)(x)$, $\Psi \in C_c^{\infty}(\mathbb{R}^n)$, or some type of integral transform acts on $\phi$.
Focusing on the case where the integrand is of the form $$ (\Psi \ast \phi)(x) + W(x, \phi(x), \nabla \phi(x))$$ for $\Psi \in L^q(\mathbb{R}^n)$, $\phi \in W^{k,p}(\mathbb{R}^n)$ and $k,p, q$, $W:\Omega \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \to \mathbb{R}$ so that this expression is well defined.
My question is: Is there an easy way to carry results (like existence of minimizers) from the classical theory over to this case?
Comments are greatly appreciated! Thank you.
Alejandro
The Key idea in Direct method is compactness, i.e., the coercivity, and the (weak) lower semi-continuous.
If in your Lagrange, (we usually call the integrand in energy function as Lagrange), especially the term $(\Phi\ast \phi)(x)$ is bounded below and process lower semi-continuous, then by direct method you will have a minimizers. (I assume the term $W$ has all properties that Direct method need, i.e., the coercivity and convexity)