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$E = \int_{\Omega}\left \| \nabla u(x)\right \|^2 dx$

So, Dirichlet's energy measures the integral of the squared norm of the gradient. Why squared norm? What would we get if we use just a norm? It's still going to be non-negative.

If I calculated a $u(x)$ that minimizes $E = \int_{\Omega}\left \| \nabla u(x)\right \| dx$ (not squared) would it be worse than Dirichlet? My goal is exactly what I specify: the gradient at any point should have minimum length.

Is the reason for the squared norm minimisation equivalent to the role of the square in the least squares fit? (here)

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Babis
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    extremizing the squared norm is equivalent to extremizing the norm. Also $$ |\nabla {u}|^2 = \nabla u \cdot \nabla u $$ to which the divergence theorem lends itself well. – user27182 Jan 29 '15 at 11:17

2 Answers2

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I think the square comes from physical modelling the bending energy.

But then, the square of a Hilbert space norm has several nice properties: it is uniformly convex and twice continuously differentiable with constant second derivative.

The Dirichlet problem $$ \min \int_\Omega \frac12|\nabla u|^2 - uf \ dx $$ is of course not equivalent to $$ \min \int_\Omega |\nabla u| - uf \ dx. $$ The latter problem is much harder to solve and analyze.

For instance, it is much more difficult to prove that the second problem has a solution with integrable gradient.

The minimizer of the Dirichlet energy satisfies $$ -\Delta u = f, $$ whereas the minimizer of the second problem satiesfies $$ -\nabla\cdot\left( \frac{\nabla u}{\|u\|} \right) =f. $$ To have a finite-dimensional analogue, consider the minimization of $$ \sum_{i=1}^n \frac12|a_i|^2 - a^Tc $$ and of $$ \sum_{i=1}^n |b_i| - b^Tc. $$ The first one has solution $a_i=c_i$, the second has a solution only if $|c_i|\le 1$ for all $i$ (which is then $b_i=c_i$).

daw
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  • I was talking about the Dirichlet energy, which is without -uf. I updated the question with more detail. – Babis Jan 29 '15 at 11:50
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I think in this form, the Dirichlet's energy is a strictly convex quadratic function and should have an UNIQUE minimum for reasonable boundary conditions . This also holds for the least squared norm minimisation.

arzack
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