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The answer by Bazin (https://mathoverflow.net/users/21907/bazin), Faa di Bruno's formula for vector valued functions, URL (version: 2012-09-04): https://mathoverflow.net/q/106339 is providing a formul …

Too long for comment. take for instance $f=0, g=0$. Then the mapping $h\mapsto u$ is a pseudo-differential operator which will have some Sobolev continuity properties for spaces $W^{s,p}$ with $p\in …

Up to some normalization, the harmonic oscillator $H$ is self-adjoint such that $$ \langle Hu, u\rangle=\sum_{k\ge 0}(\frac12+k) \vert u_k\vert^2, $$ and thus defining a self-adjoint $A$ by the equali …

Let me start by altering a bit your notations: we consider a differential operator $P$ defined by $$ P=\sum_{1\le j\le n}p_j(x) D^j, \quad D=-i\frac{d}{dx}. $$ The formal adjoint is (there is a typo i …

As a rather recent revisitation of Wigner's article, one may also quote James Glimm, who wrote in the article "Mathematical perspectives" (Bull. Amer. Math. Soc. (N.S.) 47 (2009), no. 1, 127–136), In …

I would write the Navier-Stokes equations on a Riemannian manifold $(\mathcal M,g)$ in a slightly different way. The unknown is still a time-dependent vector field $v$, to which you can associate a on …

Let us start with pointing out that $f\in H^\sigma$ is equivalent to $$ (\langle n\rangle^\sigma\hat f(n))_{n\in \mathbb Z}\in \ell^2(\mathbb Z), \quad \text{with $\langle n\rangle=\sqrt{1+n^2}$.} $$ …

With $q=1/p$, let me write your equation as $$ \vert D\vert u+ q u= h_0, \quad u(\pm 1)=0. $$ Multiplying the equation by $u$, we get $$\Vert{u}\Vert_{H^{1/2}_0}^2\le \Vert{u}\Vert_{H^{1/2}_0}^2+\unde …

From your assumptions, you have a $C^2$ function $\rho:\mathbb R^d\rightarrow \mathbb R$, such that $$ D=\{x\in \mathbb R^d, \rho(x)<0\}, \quad \partial D=\{x\in \mathbb R^d, \rho(x)=0\}, $$ and $ x\i …

Formally you get $ u(t)=e^{it p(D)} u_0 $ and this means that $$ u(t,x)=\int e^{i2π x\cdot \xi}e^{it p(\xi)}\widehat{u_0}(\xi) d\xi. $$ Obviously, you have to require something to give a meaning to …

You have the explicit Hadamard formula $$ \frac{1}{R(x)}=\limsup_n\vert a_n(x)\vert^{1/n}=\inf_n\bigl(\sup_{k\ge n}\vert a_k(x)\vert^{1/k}\bigr), $$ triggering semi-continuity properties for $1/R$: se …

Too long for a comment. Let us consider for $X\in \mathbb S^{n-1}$, $ \langle X,(e^{i \alpha k})_{1\le k\le n}\rangle_{\mathbb C^n}. $ The question at hand is $$ \max_{X\in \mathbb S^{n-1}}\vert\langl …

Let me assume that $a=-\infty, b=+\infty, x_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that $ I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt, $ …

Here is what I believe is a relevant example: consider the vector field $X$ in $\mathbb R^3$, $$ X=a_1(x_2, x_3)\frac{\partial}{\partial x_1}+a_2(x_1, x_3)\frac{\partial}{\partial x_2}+a_3(x_1,x_2)\fr …

Let $H$ be the Heaviside function (characteristic function of $(0,+\infty)$). The ODE $$ \dot x=H(x)\text{ on $t>0$}, \quad x(0)=0, $$ has solutions $ x_1(t) = 0 $ as well as $x_2(t)=t$. Thus non-uniq …