# Construction of elliptic equation with Neumann boundary condition from a minimization problem

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $$B=B_1 \subset \mathbb{R}^3$$ and $$E : H^1(B) \to \mathbb{R}$$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $$u_0 \in W^{1,2}$$ to be the minimizer of the functional $$E$$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $$u_0$$ is the critical point of the functional, we let $$\xi \in K$$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0.$$ If we further require that $$\xi$$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0.$$ Suppose we also have that $$u \in H^2(B)$$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0.$$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $$u_0$$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $$H^2$$ bound of $$u$$? I think we can apply standard estimate to obtain $$H^2_{loc}$$. If we do not have the fact that $$u \in H^2(B)$$, we may hard to have the existence of $$\dfrac{\partial u}{\partial n}$$ on the boundary by trace theorem.

• From the context, it looks like your "$E:\mathbb{R}\to \mathbb{R}$" (third line) should be "$E:H^1(B)\to \mathbb{R}$". – DCM May 27 at 23:16
• Just to make sure I understand correctly: is the goal here to show that the minimising $u_0\in H^1(B)$ is a $H^2(B)$ function satisfying your Neumann problem? – DCM May 27 at 23:22
• @DCM Thanks for pointing out my mistake! – mnmn1993 May 28 at 5:12

Let $$u\in H^1(B)$$ be a minimising value of $$E:H^1(B)\to \mathbb{R}_+$$. Then $$E'(u)=0$$ in the sense that

$$E'(u)v = \int_B [\nabla u \cdot \nabla v + 4(u^2-1)uv ] dx - \int_{\partial B} Q'(u)v \hspace{.5pc}d\mathcal{H}^2 = 0$$

for all $$v\in H^1(B)$$. This implies, in particular, that

$$\int_B [-u \Delta v + 4(u^2-1)uv] \hspace{.5pc}dx = 0$$

for $$v\in \mathscr{D}(B)$$ (this uses the fact that $$v$$ and all its derivatives vanish on $$\partial B$$ if it has compact support in $$B$$). Having this last equation hold for all $$v\in \mathscr{D}(B)$$ is exactly the statement that

$$\Delta u = 4(u^2-1)u \hspace{1pc}\mbox{in B }$$

in the sense of distributions. The interior equation in the distributional sense is therefore just what you get from regarding $$E'(u)$$ itself as a distribution (i.e. by restricting it to $$\mathscr{D}(B))$$.

For each compact subset $$K$$ of $$B$$, take $$\psi_K\in \mathscr{D}(B)$$ with $$\psi_{|K}=1$$. Then considering the action of $$E'(u)$$ on $$\{v(1-\psi_K):v\in C^\infty(\bar B),\mbox{K\subset B compact}\}$$ should - I think - also give you a weak form of your boundary condition.

I might be wrong, but I think that it's only once you have your interior equation in a (weak or distributional) form which does a priori involve the assumption that $$u\in H^2(B)$$ that it's time time to start worrying about regularity. In particular, you may be able to establish that $$u\in H^2(B)$$ using the fact that it is a $$H^1$$ solution to your PDE.

• Yes, you are right. For a general $H^1(B)$ function, I think it may be have a well-defined derivative on the boundary and also the $\Delta u$ (ever in weak or distributional sense?). However my main goal is as the title, I do not know if I construct it in the correct way. I also look for another construction if mine is not good enough. – mnmn1993 May 28 at 5:16
• I think the general pattern you follow is sound. Assuming $E:H^1(B)\to \mathbb{R}$ is differentiable, a necessary condition for $u\in H^1(B)$ to be a minimiser is that $E'(u)v = 0$ for all $v\in H^1(B)$. Choosing $v\in H^1_0(B)$ gives you your "EL" equation, for example. The key thing is that $u$ has to satisfy the $E'(u)v = 0$ whatever $v\in H^1(B)$ you choose, which is the main mechanism through which you're able to deduce things about its behaviour. – DCM May 28 at 8:46
• In particular, I'd be inclined to see what you get by using Green's identity (thought of in a distributional sense, at least initially) in the equation in your third display, and seeing what drops out of that for different choices of $v$. – DCM May 28 at 8:47
• I think the slickest way to get the interior equation and the boundary equation alone is by choosing test functions $v$ which satisfy a suitable PDE of their own (a PDE designed to make one of the two integrals vanish). – DCM May 28 at 9:15
• I nice analogy here is 'plotting the graph of a function'. $E'(u)$ is a linear function(al) on $H^1(B)$; we learn about its `shape' behaviour by plugging in values and seeing what we get. The behaviour of $E'(u)$ then tells us things about $u$. – DCM May 28 at 9:18