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If the magnetic field $\operatorname{curl} A$ is bounded, then the trace space is $$ \left\{ g \in L^2 (\mathbb{R}^{N - 1}, \mathbb{C}) ; \iint\limits_{\mathbb{R}^{N - 1} \times \mathbb{R}^{N - …

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 …

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 problem here is with the definition of $H^\ell (\partial D)$. Typically, $\partial D$ is a manifold and you can define this set by local charts. In order to define $H^\ell (\partial D)$ with $0 < …

Your question can be rephrased by asking whether one has a Hölder estimate $$ |u|_{W^{s, p}} \le C |u|_{W^{s, q}}, $$ when $p < q$ or whether $W^{s, q} \subset W^{s, p}$. There is no such embedding …

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 …

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 …

I think that the construction that you are proposing by harmonic extension does not work. Indeed consider on the unit disk $\mathbb{B}^2 \subset \mathbb{R}^2$ the function $u_n : \mathbb{B}^2 \to \mat …

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 …

The failure of the compactness of the embedding $W^{1, p} (\Omega) \subset L^{p^*} (\Omega)$ is local, that is, it can be exhibited in any ball. An abstract way of seeing the failure of the embeddin …

If $V_1 \ne \mathfrak{g}$, then $H W^{1, p} \ne W^{1, p}$. The construction is based on the notion of dilation on the Carnot group $\mathbb{G}$. The dilation $\delta_r$ is defined by for $X \in V_i$ …

This should follow from the nonhomogeneous trace inequalty $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2} + \lVert u \rVert_{L^2}, $$ and and from the classical Poincar …

This follows from its counterpart in the Euclidean space by local charts. If you want to have an estimate on the derivative $D^r f$, then you should impose some bounds on derivatives of the curvatur …

Both convergences imply the weak convergence in $L^2 (I)$, which has a unique limit.

As it is stated, this property does not hold: indeed consider the sequence of functinos $(u_n)_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by $$ u_n (t) = \sin (2n\pi t) v, $$ where $v \in V$ is …