2
$\begingroup$

It is known that the equation

$$ \Delta f = 0 $$

on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet Energy

$$ E(f) = \frac{1}{2}\int_\Omega \left\lVert \nabla f \right\rVert^2 dx $$

It happens however that again under certain conditions the operator $\Delta$ is selfadjoint, with the all the nice spectral properties coming from it.

I wonder now if there's a general formulation to find self adjoint operators as minimization of certain functionals which have certain properties. I would assume maybe convex and quadratic functionals might have such properties, but I don't know more in general.

Can anyone provide a reference?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Yes, one can obtain self-adjoint operators from quadratic forms, and it's a standard method. (Of course, there's no minimization to produce the operator, but that's just as well since $\Delta f=0$ is an equation, not an operator.) $\endgroup$
    – Christian Remling
    Commented Dec 4, 2023 at 15:38
  • $\begingroup$ I more meant "quadratic energies" not "quadratic forms". So something like $E(\alpha f) = \left| \alpha \right| ^2 E(f)$. Also in $\Delta f = 0$ I more meant the operator $\Delta$. I also think I meant if there's a general theorem between energies and selfadjoint operators (not necessarily quadratic). $\endgroup$
    – user8469759
    Commented Dec 5, 2023 at 3:01

0

Reset to default

You must log in to answer this question.