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Yes they are. Step 1 There exists measurable sections $e_1, e_2, \dotsc, e_m$, where $m = \dim M$, of $TM$ (measurable functions mapping a point $x$ to a vector of its tangent plane $T_xM$) such that …

Here is a variational argument to prove that the maximizers do not change sign. If $f \in H^1 (\mathbb{R}^N)$ be a maximizer, $u$ can be written as $$ f = f_+ - f_-, $$ with $f_+ \ne 0$ and $f_- \ …

You are minimizing under the constraint that $\rho \ge 0$. Hence your variation $\rho_m + t \eta$ might not be admissible. (Say if $\eta \ne 0$ and $\vert t\vert$ is large enough.) The trick is to co …

This seems to be a consequence of the Cauchy–Schwarz inequality: $$ \begin{split} \bigg\vert \sum_{\vert \alpha \vert = k} \sum_{\vert \beta \vert = k} \int_{\mathbb{R}^m} D^\alpha f D^\beta \phi \big …

One way to reformulate this is to consider a good extension $\bar{g}$ to the whole $\Omega$ of the function $g$, and then set $v = u - \bar{g}$. The function $v$ solves then the problem $$ \left\{ \b …

If $k \in (0, 2]$, we define the multiplier $$ m (\xi) = (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k. $$ We observe that if $\vert \xi \vert \ge 2$, then by differentiability $$ \big …

This follows from two facts: $(H^s \cap L^\infty) (\mathbb{T})$ is a Banach algebra (see for example in the framework of fractional spaces $W^{s, p}$ ($W^{s, 2} = H^s$) Bourgain, Brezis, Mironescu, …

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as $$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$ where $$ L^2_k (\mathbb{R}^2; \ …

The property is proved in the litterature (together with its $W^{1, p}$ counterpart): Augusto C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Different …

The map $(-\Delta + 1)^{-\frac{1}{2}}: L^p (\Omega) \to W^{1, p}_0 (\Omega)$ is a linear bijection when $\Omega$ is smooth and $1 < p < +\infty$, where $\Delta$ is the Laplacian with Dirichlet boundar …

The issue with $p = 1$ is that some definitions of fractional Sobolev spaces that were equivalent when $p > 1$ (by the Gagliardo seminorm that you gave, by interpolation between functional spaces, by …

Assume that $\Omega \subset \mathbb{R}^n$. I will prove that $C^1_c (\Omega)$ is dense in $H^1 (\Omega) \cap C^1 (\Omega) \cap C_0 (\Omega)$ endowed with the $H^1$ norm, where $$ C_0 (\Omega) = \{u …

The embedding is not compact this can be observed by considering the sequence $f_k (r) = f (r/k)/k^2$ for some given function $f \in C^\infty_c (0, +\infty) \setminus \{0\}$. One has for every $k \ge …

Here is a direct argument based on the definition by the Gagliardo norm $$ \Vert u \Vert_{H^{1/2}}^2 = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{\vert u (x) - u (y)\vert^2}{ \vert x - y \vert^{N …

If $\Omega \subseteq \mathbb{R}^N$ is an open set and $N \ge 2$, then any point $a \in \Omega$ is removable for weakly differentiable maps: for each function $u \in W^{1, 1} (\Omega \setminus \{a\})$, …