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These lecture notes by Piotr Hajłasz might have the introductory level you are looking for: The lectures will be divided into two almost independent streams. One of them is the theory of Sobolev spaces with numerous aspects which go far beyond the calculus of variations. The second stream is just calculus of variations. The theory of Sobolev spaces is a ...

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You are using the wrong space, that's all. The correct space is $$X=\{u \in H^1_{\textrm{loc}}(\mathbb R^2): u(\cdot+n)=u(\cdot) \quad \forall n=(n_1,n_2)\in \mathbb Z^2\}.$$ Then you apply Lax-Milgram and you are good to go. Look at Asymptotic Analysis for Periodic Structures, by Bensoussan, Lions, Papanicolaou (1979) chapter 1, for example. Answering ...

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My question has been answered in this MathStackexchange post: https://math.stackexchange.com/questions/4033589/sobolev-space-with-negative-index For every $k<0$, there exist a measure $\mu_k$ which is singular with respect to Lebesgue measure, and such that $\mu_k\in W^{k,2}(\mathbb R^n)$. So $W^{k,2}(\mathbb R^n)$ does not embed into the space of ...

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It is sufficient to focus on the unit ball $B$ in $\mathbb{R}^N$, $N\geq 2$. Consider the function $$u(x)=\sqrt{1-|x|}\cdot\left( \frac{1}{|\ln(1-|x|)|}\right)^\alpha$$ with $1/2<\alpha<N/2$, which is singular at the origin. PS. Note that the function is actually in $H^1_0(B)$.

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The result is mentioned in Triebel's Function Spaces and Wavelets on Domains Proposition 4.21. Also see Theorem 1.4 for my paper https://arxiv.org/abs/2110.14477

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