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## differential equations – actuality of options of a system 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 mounted $C>0$,

$$

frac{1}{C} |xi|^2 leq langle A(x)xi,xi rangleleq C|xi|^2.

$$

$textbf{Question:}$

Does 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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