Minimisation of Energy Functional

Question

Suppose I have the energy functional given by $$\Pi(u) : = \frac{1}{2} \int_{\Omega} k\nabla u \cdot \nabla u dx - \int_{\Omega} fu dx + \int_{\partial \Omega} \sigma_0 u^2 ds,$$ where $f \in L^2(\Omega)$ and $\sigma_0 \in C^{\infty}(\partial \Omega)$ and $\partial \Omega$ denotes the boundary of $\Omega$. Also note that $\sigma_0 > 0$.

How do I show this energy functional attains a minimum in $H^1(U)$?

According to Evan's, we need to show that $\Pi(u)$ is coercive and convex, but I haven't had any luck.

Answer 1

Note: This answers the first version of the question, in which $$\Pi(u) : = \frac{1}{2} \int_{\Omega} k\nabla u \cdot \nabla u dx - \int_{\Omega} fu dx - \int_{\Gamma_N} \sigma_0 u ds.$$

This functional is not coercive - try to plug in constant functions from $H^1(U)$. In fact, if a certain compatibility condition (which one?) on $(f,\sigma_0)$ is violated, $\Pi$ is not even bounded from below.

If this compatibility condition is satisfied, you have several possibilities to obtain the solvability:

• Discuss everything in the quotient space $H^1(U) / \mathbb R$ (where $\mathbb R$ denote the constant functions).
• Discuss everything in the subspace $\{ v \in H^1(U) \mid \int_U v \, \mathrm dx = 0\}$.
• Use a Fredholm argument: The necessary and sufficient optimality condition of your problem can be written as $A u = F$, with $A : H^1(U) \to H^1(U)^*$ and $F \in H^1(U)^*$. Denote by $E : H^1(U) \to H^1(U)^*$ the operator $(Eu)(v) = \int_U u \, v \, \mathrm dx$. Then, $A + E$ is invertible (by Lax-Milgram theorem). Due to Rellich-Kondrachov, $E$ is compact, hence $A$ is a Fredholm operator of index $0$. Now, it remains to study the kernel and cokernel of $A$.

Edit: Answer to the new question: Assuming some regularity of the boundary $\partial\Omega$ (Lipschitz is enough), one can show the existence of $c > 0$ such that $$\| u \|_{H^1(\Omega)}^2 \ge c \, \Big( \frac12 \int_\Omega |\nabla u|^2 \, \mathrm{d}x + \int_{\partial\Omega} \sigma_0 \, u^2 \, \mathrm{d}x\Big),$$ e.g., by a contradiction argument or, alternatively, by proving the inequality first for $u \in C^\infty(\overline\Omega)$. This yields the desired coercivity.

Answer 2

To show that the functional is coercive, perhaps try a weak convergence argument? See the proof of the Poincare inequality in Evan's Chapter 6.