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Edit: This first paragraph is wrong: I think (a) is fine, because $1 \in L^p$ for $p \in [0,\infty]$ and therefore $f \in L^\infty$ implies $f \in L^p$. But I think, there is an issue with (b) as it …

This result seems to be a consequence of Dini's theorem (as noted by nonlinearism): If $\varphi^n$ converges to a continuous function $\varphi_\infty$, we have $\varphi_\infty = 0$. Hence, $\varphi < …

I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix …

You can manipulate the left-hand side to \begin{equation*} \lVert (Dv)^2 \rVert_{H^{3k-2}} = \lVert D v \rVert_{W^{3k-2,4}}^2 \le \lVert v \rVert_{W^{3k-1,4}}^2. \end{equation*} (Note that what w …

You have that $H_0^1(\Omega, D)$ is a subspace of $H^1(D)$ (equipped with the same norm). Due to the Lipschitz condition on $D$, $H^1(D)$ is compactly embedded in $L^2(D)$ (standard-Rellich-Kondrachov …

One positive answer is that this set is $p$-quasi-open, see some resource about capacity theory, e.g., here: https://math.stackexchange.com/questions/48776/capacity-theory-beginner-resources.

The set $K_A$ is essentially a polar of $A$. Indeed, we have $$ A = \{ x \in \mathbb R^n \mid l(x) \ge t \; \forall (l,t) \in K_A\} =: B.$$ The inclusion "$\subset$" is clear and in order to check "$\ …

If I understand your question correctly, we could have $g(x_j) = 0$ for all $j = 1, \ldots, n$. Thus, $Q_n[f g] = 0$. Further, if $f$ is a low degree polynomial, the right-hand side of (1) vanishes. H …