The question may be very basic, but I am a bit confused about the concepts, so it would be nice if you can clarify them for me and/or suggest some good references to fully understand them.
I am reading a proof for the existence of solutions to a variational problem. They consider a perturbation of this functional, obtaining a slightly different functional (this may be irrelevant for the question). The point is that at some point they claim that "computing the Hessian of this functional, you can see that this is an elliptic operator".
I am not sure what they mean with this:
- Is the Hessian the same as the second variation of the functional? I.e., if the functional is
$$
J[y]=\int_{x_{1}}^{x_{2}}L\left(x,y(x),y'(x)\right)\,dx
$$
then the Hessian is just
$$
\delta^2 J_y [V,V]=\frac{1}{2}\int_{x_{1}}^{x_{2}} (L_{yy}V^2 + 2F_{yy'}VV'+L_{y'y'}V'^2)dx?
$$
- If so, how to check it is an elliptic operator? Doing some computations, I can write the second variation as $$ \delta^2 J_y [V,V]=\frac{1}{2}\int_{x_{1}}^{x_{2}} (A(x)V''+B(x)V'+C(x)V)Vdx $$ if that helps.
It would be nice if for 2 you can recall the definition of an elliptic operator with a non-trivial example. An answer to any of both questions is appreciated!