fa.functional analysis - Taylor serie on a Riemannian manifold

ap.evaluation of pdes – Beltrami equation with harmonic coefficient Answer

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ap.evaluation of pdes – Beltrami equation with harmonic coefficient

Note that, in case you take $phi=0$, then the equation reduces to $w_y =0$, i.e., if $Dsubset C$ is the province of $w$ and $x:Dtomathbb{R}$ is the projection on the $x$-axis and has related fibers, then $w= h(x)$ for some $C^1$ duty $h:x(D)tomathbb{C}$, and that is the common resolution on such $D$.

Something related occurs in common: Write
$$
mathrm{d}w = w_z,mathrm{d}z + w_{barrier z},mathrm{d}barrier z
= w_z,(mathrm{d}z + mathrm{e}^{iphi(z)},mathrm{d}barrier z)
= mathrm{e}^{iphi(z)/2}w_zleft(mathrm{e}^{-iphi(z)/2}mathrm{d}z + mathrm{e}^{iphi(z)/2},mathrm{d}barrier zright).
$$

Then, setting $alpha = mathrm{e}^{-iphi(z)/2}mathrm{d}z + mathrm{e}^{iphi(z)/2},mathrm{d}barrier z$, we behold that $alpha$ is a real-valued $1$-form, and therefore all the time has a neighborhood integrating issue, i.e., it will probably breathe written domestically within the type $alpha = f,mathrm{d}u$ for some capabilities real-valued capabilities $u$ and $f>0$. Thus, if $Dsubsetmathbb{C}$ is a province such that $alpha$ can breathe written as $alpha = f,mathrm{d}u$ for some real-valued capabilities $u$ and $f>0$ on $D$ and the fibers of $u:Dto u(D)subset mathbb{R}$ are related, then any resolution of your equation on $D$ can breathe written within the type $w = h(u)$ for some $C^1$ duty $h:u(D)tomathbb{C}$, and each such $h$ that’s $C^1$ yields an answer. This is as a result of your equation for $w:Dtomathbb{C}$ reduces to $mathrm{d}w = p,mathrm{d}u$ for some duty $p:Dtomathbb{C}$.

The signficance of $phi$ being harmonic shouldn’t be actually limpid (aside from making certain that $alpha$ is real-analytic, in order that $u$ can breathe taken to breathe real-analytic too). Certainly, the conduct of $phi$ will learn which domains $Dsubsetmathbb{C}$ have the privilege configuration to uphold an integrating issue for $alpha$, however it isn’t limpid to me that simply requiring that $phi$ breathe harmonic provides you mighty simply accessible data alongside these strains.

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