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Glaeser's Theorem says that a $C^\infty$ function $F$ on $\mathbb R^n$ which is invariant under permutation of the variables is a smooth function of the symmetric polynomials of $(x_1, \dots, x_n)$. …

(1) The most classical result, due to Aronszajn, and later to Calder\'{o}n and (independently) to H\"ormander. Take a second order elliptic operator $P$ with Lipschitz and real coefficients in the pri …

The Schwartz space itself, $\mathscr S(\mathbb R^n)$, can be viewed as a subspace of smooth functions defined on the sphere $\mathbb S^n$, the unit sphere of $\mathbb R^{n+1}.$ We recall that $$ \math …

With $H$ the Heaviside function (characteristic function of $\mathbb R_+$), you have $$ M(t)=\int_{\mathbb R^n} f(x) H(t-h(x)) dx $$ and thus, at least formally, $$ \dot M(t)=\int_{\mathbb R^n} f(x) \ …

The function $\widetilde W$ is a smooth function on the open set $\Phi(B)$ and is supported in the compact set$\Phi(I_1\times I_2)$. As a result it can be extended by 0 as you wish and this is simply …

I would like to describe the smooth global diffeomorphisms $\kappa:\mathbb R^n\rightarrow\mathbb R^n$ such that for all $x\in \mathbb R^n$, $$ \kappa'(x)\in O(n), \quad \text{i.e.}\quad ^t\!\kappa'(x) …

Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$ \int_{\mathbb R^N} f(x) dx =0. $$ Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector …

Let $\mathcal M$ be a compact $3D$ differentiable manifold and let $\alpha, \beta, \gamma$ be three one-forms on $\mathcal M$. I want to consider the scalar quantity $$ F(\alpha, \beta, \gamma)=\int_\ …

Let me quote the simplest and most classical result for a global inverse function theorem, due to Hadamard and Plastock (see L. Nirenberg, Topics in Nonlinear functional analysis, Courant LN,6, 2001). …

The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$. Question. Is it …

There is a famous open problem, whose solution is attributed to Paul Cohen, but no published paper seems to be available: There exists $f\in C^\infty(\mathbb R,\mathbb R_+)$ such that $f$ is not a f …

Let me answer to the local solvability question in the $C^\infty$ category. Take a complex-valued vector field $Z=X+iY, X,Y$ real-valued vector fields such that $Z$ is always non-zero (a "principal-ty …

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 …

Hörmander's definition of a pseudo-differential operator on an open subset $\Omega$ of $\mathbb R^n$ in the class $\text{Op}S^m$ ($m\in \mathbb R$) is the following: take a symbol $a$, that is a smoot …

Let me consider two autonomous vector fields $X_1, X_2$ on a compact smooth manifold $\mathcal M$ and assume that the Lipschitz condition is true for both of them. The flow $\psi_j$ of $X_j$ is define …