ac.commutative algebra - Filtration over tensor product

differential equations – Solving a completely nonlinear first bid PDE retort

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differential equations – Solving a completely nonlinear first bid PDE

given a symmetric matrix of Holder steady features $A(x)$ such that
$$
frac{1}{C} |xi|^2 leq langle A(x)xi,xi rangle leq C |xi|^2
$$

discover a vector bailiwick $Phi$ such that
$$
D Phi(x)^t D Phi (x) = A(x)^tA(x)
$$

as a simplified model of the issue, if somebody may support me with the next totally nonlinear first bid PDE

given a Holder steady obligation $a(x)>lambda >0$, discover a obligation $Psi$ such that
$$
|nabla Psi(x)|^2= |a(x)|^2
$$

assassinate any of the above issues maintain an answer?

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