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differential equations – actuality of options of a syestem of first bid PDEs retort

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differential equations – actuality of options of a syestem of first bid PDEs

Let $Omegasubset mathbb R^N$ breathe an launch, mild and bounded subset.

Given a $Ntimes N$, bounded and elliptic matrix of Hölder steady capabilities.
That is, $A(x)= {a_{ij}(x)}_{Ntimes N}$, $a_{ij}(x)in C^{0,alpha}(Omega)$ and for some fastened $C>0$,
$$
frac{1}{C} |xi|^2 leq langle A(x)xi,xi rangleleq C|xi|^2.
$$

$textbf{Question:}$

assassinate there animate a vector bailiwick $Phi$ (anticipated to breathe $C^{1,alpha}$ mild) such that the next system of PDEs is solved:
$$
|det(DPhi)(x)|^{1/2} DPhi (x)= A(x)
$$

We can too simplify the above drawback and contemplate into the next equal model
$$
D Phi(x) =frac{A(x)}{|det (A(x))|^{frac{1}{N+2}}}
$$

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