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The variational problem Poisson's equation reads "For given $f \in L^2(\Omega)$ find $u \in H_0^1(\Omega)$ such that $$\int_{\Omega} \nabla u \cdot \nabla v \, \mathrm dx= \int_{\Omega}fv \, \mathrm dx \quad \forall v\in H_0^1(\Omega).$$

Very often I read that it is equivalent to minimizing the energy functional $$J(v):=\frac{1}{2}a(v,v)-F(v)$$ where $a(u,v)=\int_{\Omega} \nabla u \cdot \nabla v \, \mathrm dx$ and $F(v)=\int_{\Omega}fv \, \mathrm dx.$ I get the proof, but what is the added value of introducing such equivalence, e.g. what relevance does the energy functional have? Sometimes I just see it randomly squeezed in and not being mentioned further in the literature.

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Depending on the context, there can be a few different answers.

  1. In cases outside of Poisson's equation, variational problems often emerge initially as finding the extrema of a functional which sometimes can be intepreted as "energy." For Poisson's equation this is not the case historically (the PDE was derived first, and the variational formulation later). But depending on what you study, it can be the case that more often than not, the PDE comes out of the optimization problem, and not vice versa.

  2. Knowing that a solution (should one exist) must be a minimizer of some energy functional can sometimes be useful for deducing that one must exist and/or be unique. For instance, given the form of the energy functional, one may be able to conclude that it is coercive and weakly lower semi-continuous. This is enough to conclude that a minimizer exists; moreover, such a minimizer can, in principle, be constructed by taking a weak infimizing sequence and upgrading the weak convergence to strong convergence (after possibly modding out the symmetries of the underlying problem: this is referred to sometimes as compensated/concentration compactness). In the case of Poisson's equation, the existence and uniqueness problems are relatively tame and may be possible to resolve with other methods, but this sort of thing becomes valuable for more difficult nonlinear PDEs. Therefore it's a good idea to learn how to solve Poisson's equation using energy methods, since this easy case generalizes to a large class of contexts.

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