Search Results

No. Take for instance the following ODE where $0$ is a regular singular point, $$x y'=\lambda y.$$ The functions $cx_+^{\lambda}$ are solutions and are not analytic. If $\lambda$ is not a non-negative …

A most important result is missing in the previous answers, namely the characterization by Lars Hörmander of hypoellipticity in his seminal paper, On the theory of general partial differential operato …

Let $E$ be a fondamental solution of $\partial_{x_1}\square$. Then you have for $u$ compactly supported $$ u=u\ast \delta=u\ast (\partial_{x_1}\square E)= (\partial_{x_1}\square u)\ast E, $$ so that $ …

Let $Ai$ be the classical Airy function and let $(a_j)_{j\ge 1}$ be the strictly decreasing sequence of its zeroes: we have $a_{j+1}<a_j<\dots <a_2<a_1<0$, $\lim_{j\rightarrow +\infty}a_j=-\infty$. I …

Analytic regularity of a $C^\infty$ function can be characterized by using an infinite number of derivatives. A function $f\in C^\infty(\Omega)$ where $\Omega$ is an open subset of $\mathbb R^n$ is r …

I would like to give another example, a very classical one. Consider a smooth open subset $\Omega$ of $\mathbb R^n$ and let $T$ be a smooth vector field tangential to the boundary $\partial\Omega$. Th …

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 …

Your equation falls in the category of regular singular differential equation. Writing your equation as $$ x z'=a z+g(x,z), \tag{$\ast$}$$ the singularity is called regular because the exponent of th …

You will find a study of self-adjoint operators of type $$ \sum_{j=1}^r(X_j^*-iY_j^*)(X_j+iY_j), $$ where $X_j, Y_j$ are real-valued vector fields for instance in the Helffer-Nier Lecture Note (Lectur …

Writing $$ \langle-\Delta u + au, u\rangle_{L^2(\Omega)}=\langle f, u\rangle_{L^2(\Omega)}, $$ using the Dirichlet boundary condition, you get $$ \Vert \nabla u\Vert_{L^2(\Omega)}^2+\langle au, u\rang …

I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier transforma …

Consider the (stationary) Schrödinger equation $$ -\Delta u+ Vu=0, $$ or the differential inequality $\vert\Delta u \vert\le \vert V u\vert$, where $V$ is some "potential" function. The following uniq …

Solving the linear first-order $$ h'=h K^{1/2}e^{f/2}/\sqrt 2 \tag{$\ast$}$$ makes the rhs to vanish. We get then \begin{align} 2h''-f'h'-Ke^f h&=h' K^{1/2}e^{f/2}\sqrt 2+h K^{1/2}e^{f/2}f'(\sqrt 2)^{ …

Forget about finding a closed analytical expression for the fundamental matrix. Think about the simple case of a second order scalar equation such as the Airy equation $\ddot x=tx.$ There is neverthe …

I understand that your ODE is one-dimensional: in that case you can avoid Lipschitz continuity and replace it by transversality: The autonomous equation $$ \dot x =f(x),\quad x(0)=x_0, $$ has a uniqu …