Domain of definition of a Lagrangian in classical field theory In classical field theory one has the action:
$$S[\phi] = \int_{t_{0}}^{t_{1}}\int_{\Omega}\mathcal{L}(t,x,\phi(t,x),\dot{\phi}(t,x),\nabla\phi(t,x))dxdt$$
and we want to obtain the Euler-Lagrange equations for the Lagrangian density $\mathcal{L}$. In the physics literature, one usually takes the first variation with respect to a test field $\psi$ and considers the fields $\phi$ to be sufficiently differentiable and regular. To obtain the desired Euler-Lagrange equations, one assumes $\psi(t_{0})=\psi(t_{1}) = 0$ and that it has fast decay so the surface integrals obtained by using integration by parts vanish.
I want to understand better this argument under a more rigorous way. The way I see it, the set $\Omega$ is there to make things well-defined, but the final argument of the derivation of the Euler-Lagrange equation is that the equations follow because the analysis hold for every set $\Omega$. In my point of view, a better way of making sense of this is by using the following well-known result from analysis: if a function $f: \mathbb{R}^{n}\to \mathbb{R}$ is continuous (or smooth) and satisfies:
$$\int f(x)g(x)dx = 0$$
for every compactly supported $g: \mathbb{R}^{n}\to \mathbb{R}$, then $f = 0$.
It seems more reasonable for me to use this result instead of the usual "holds for every set $\Omega$" argument. If we fix $\Omega$, we can perform the calculations that lead to the Euler-Lagrange equation (for fields) by using compactly supported functions on $[t_{0},t_{1}]\times \Omega$. But this raises the question: what is a typical $\Omega$?
So my question is: is anything wrong with my reasoning? And if not, what is a typical $\Omega$ for a field theory? Open? Closed? Convex?
 A: PROPOSITION. Let us consider a region $\Omega \subset \mathbb{R}^4$ (viewed as classical or relativistic spacetime). We suppose that $\Omega$ is open and non-empty.
Let us consider a map ${\cal L}: \Omega \times \mathbb{R} \times \mathbb{R}^4 \to \mathbb{R}$  of class $C^2$ and the functional
$$I[\phi] := \int_{\Omega} {\cal L}(x,\phi(x), \partial \phi(x)) d^4x$$
defined for $\phi \in C^2(\Omega)$.
The following fact holds, for every $\phi_0 \in  C^2(\Omega)$ and every $\eta \in C^2(\Omega)$ whose support is included in $\Omega$.
$$\left.\frac{d}{d\epsilon}\right|_{\epsilon =0} I[\phi_0 + \epsilon\eta] :=\left\langle \left.\frac{\delta I}{\delta \phi}\right|_{\phi_0}, \eta \right\rangle$$
exists and defines a linear function in the variable $\eta$, where
$$\left\langle\left.\frac{\delta I}{\delta \phi}\right|_{\phi_0}, \eta \right\rangle = 
\int_{\Omega}\left[\frac{\partial {\cal L}}{\partial \phi} - \partial_\mu \left(\frac{\partial {\cal L}}{\partial_\mu \phi }\right) \right]|_{\phi_0} \eta\:  d^4x \:.$$
In particular $\phi_0$ satisfies the Euler-Lagrange equations on $\Omega$
$$\frac{\partial {\cal L}}{\partial \phi}|_{\phi_0} - \partial_\mu \left(\frac{\partial {\cal L}}{\partial_\mu \phi }|_{\phi_0}\right)=0$$
if and only if
$$\left.\frac{d}{d\epsilon}\right|_{\epsilon =0} I[\phi_0 + \epsilon\eta]=0\quad \mbox{for every $\eta \in C^2(\Omega)$ compactly supported in $\Omega$.}\tag{1}$$
Remark. The usual way to write down (1) is
$$\left.\frac{\delta I}{\delta \phi}\right|_{\phi_0} =0$$
and the notation $\delta \phi$ is very often used in place of $\eta$.
The proof of the above proposition is elementary: the first statement immediately arises by integrating by parts, the second one is a trivial consequence of the following result (you already stated): If $f \in C^0(\Omega)$, then
$$\int_\Omega f(x) \eta(x) d^nx =0 \quad \mbox{for all $\eta \in C^\infty(\Omega)$ compactly supported in $\Omega$}$$
is equivalent to $f(x)=0$ for all $x\in \Omega$.
If you want adding some further (actually here non-necessary mathematical machinery) you can interpret $\left.\frac{\delta I}{\delta \phi}\right|_{\phi_0}$ in terms of  Gateaux derivative of $I$. Furthermore

*

*$\Omega$ can be taken of the form $(t_1,t_2) \times \Omega_0$.


*One can also replace $(t_1,t_2)$ for $[t_1,t_2]$ with trivial changes and to assume that $\Omega_0 = \mathbb{R}^3$.


*The constraint that $\eta$ has compact support in $\Omega$ can be replaced by the request that $\eta=0$ on $\partial \Omega$ if $\Omega$ is the closure of an open set with regular orientable boundary. This framework can be adapted to the case $\Omega = [t_0,t_1]\times \Omega_0$ with trivial changes.
