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The Problem. Assume $\Omega \subset \mathbb{R}^2$ bounded and $u \in H^1(\Omega,\mathbb{C})$ is some fixed function. Now consider the variational problem $$ F_\lambda(v) = \frac{\lambda}{2} \int_{\Omega} \vert u-v \vert^2 + \frac{1}{2} \int_\Omega \vert Dv \vert^2+ \frac{1}{4} \int_\Omega (1-\vert v \vert^2)^2,$$ i.e. $$F_\lambda (v) = \int_\Omega L(Dv,v,x)$$ where $$ L(p,z,x)=\frac{\lambda}{2} \vert u(x)-z \vert^2+\frac{1}{2}\vert p \vert^2+\frac{1}{4}(1-\vert z \vert^2)^2. $$ Writing this as a system of real and imaginary part, I already showed that there is a solution and the associated Euler-Lagrange equation is $$ -\Delta v=\lambda(u-v)+v(1-\vert v \vert^2). \tag{1}$$

Therefore -my professor said- we infer that any solution $v$ to this variational problem is smooth.

The Question. How can we infer that?

What I tried so far. Every solution to the variational problem is a weak solution to (1), i.e. to a quasilinear elliptic equation. Now, regularity theory to quasilinear equations seems not to be very powerful. I read the relevant chapters of this and this book, but they only give me rather weak Hölder continuity. However, Evans states in chapter 8.3.2 that in some cases the solution is $C^\infty$ if only $L$ is $C^\infty$.

Is there a result that lets me infer smoothness? Is there some property of my equation that I have missed? Any hint to some literature would be much appreciated!

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You should note that you have a semilinear equation, not a quasilinear one.

Once you have $v \in H^1(\Omega)$, define $f = \lambda \, (u - v) + v \, (1-|v|^2)^2$ and observe $f \in W^{1,p}$, for some $p < 2$. This can be used to prove $v \in W^{3,p}$, if your domain is smooth enough.

Then, by bootstrapping, you get $f \in H^1$ (this is limited by the regularity of $u$!) and obtain $v \in H^3$. However, due to the regularity $u \in H^1$, you cannot do better. If $u$ would be more regular, than $v$ would also be more regular.

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  • $\begingroup$ Thank you! I'll try this and get back to you. $\endgroup$ – mjb Aug 10 '13 at 8:30
  • $\begingroup$ Thanks again, but I have trouble understanding the argument. Is there an easy way to show that $f \in W^{1,p}$ for some $p<2$? I can indeed show that $f \in W^{1,\frac{2}{5}}$, but only by long computations. Do you have any hint for me? Sorry to bother you with such beginner's questions, but I have very little experience with Sobolev embeddings. $\endgroup$ – mjb Aug 14 '13 at 8:20
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    $\begingroup$ The term which makes trouble is $v \, (1- |v|^2)^2$. Here, you can use that $H^1(\Omega)$ embeds into $L^p(\Omega)$ for all $p < \infty$ and a product rule for weak derivatives. $\endgroup$ – gerw Aug 14 '13 at 19:59

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