The Complete integrals is used in the analysis of non linear first order PDE. I came across it in the books Partial Differential Equations by Evans Lawrence.
$$F(Du,u,x)=0$$
Where Du is the gradient of u. In very first page of chapter 3 it says u is function $u:U->R$ (U is subset of $R^n$ but while defining the complete integral it says $u=u(x;a)$ (where x and a belongs to X and A respectively) $C^2$ function in $U×A$, A being a open subset of $R^n$. Similar representation of u can also be seen in the definition of envelopes.
And considering the remark I found in consequent part of the book $D_au$, what does this symbol means is it gradient with respect to a vector ( if so how can we define it since it's only a parameter) or is it else?
So my question is:-
What is the domain of the function u here?
And
What is $D_au$ formally?
x is in the space $R^n$ and a is a (real) parameter so one could say "the domain is $R^{n+1}$" but a more hard-nosed person might say it is "$R^n$ with parameter a". "$D_au$" is the derivative of u with parameter a. For example "$D_a$ might be $\frac{\partial}{\partial x}+ a\frac{\partial}{\partial y}$ so that $D_1= \frac{\partial}{\partial x}+ \frac{\partial}{\partial y}$ and $D_2= \frac{\partial}{\partial x}+ 2\frac{\partial}{\partial y}$. $D_au= \frac{\partial u}{\partial x}+ a\frac{\partial u}{\partial y}$ for various values of a.