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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 - …

I think that the trace is defined by the completion. Indeed for every $u \in H^1 (\Omega)$, you have $\Vert\operatorname{tr} u\Vert_{L^2 (\Omega)} \le \Vert u \Vert_{\varepsilon}$. Since $H^1 (\mathca …

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

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_- \ …

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 …

By Fubini's theorem and by taking functions that do not depend on the $\mathbb{S}^n$ variable, your estimate is equivalent with the inequality $$ \int_0^\infty \vert u \vert^2 \le C \int_0^\infty \v …

This is not the case. Functions can have a high level of integrability and still not have bounded mean oscillation. For example, let $u : B_1 \to \mathbb{R}$ be defined for $x \in B_1 \setminus \{0\} …

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 …

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 …

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 …

For $p > n$, assume that the Hölder continuous representative vanishes at some point $a \in Q^d$. Sinco $p > n$, by the Sobolev--Morrey embedding, for every $x \in Q^d$ $$ \vert u (x) \vert= \vert u …

Partial answer: according to Triebel (Theory of function spaces, 1983, Remark 2.7.5, p. 139), the trace of the Besov space $B^{1/p, p}_1 (\Omega)$ is $L^p (\partial \Omega)$, but the linear extension …

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 …

If you are interested, we have given an intrinsic definition of a weak derivative for maps between manifolds A. Convent et J. Van Schaftingen, emphasized Intrinsic colocal weak derivatives and Sobolev …

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 < …