$$ \begin{cases} \frac{\partial S}{\partial t} + \frac{1}{2}\left((\nabla S)^2 + (x, \Omega^2 x) \right)= 0 \\ S|_{t=0} = (k,x) \end{cases}$$ Where $x \in \mathbf{R}^n,\ \Omega^2 $ - Positive-definite matrix, $k$ is constant vector
Solve Hamilton–Jacobi PDE
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multivariable-calculus
pde
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0Maybe you could add more context, for example tell us what $k$ is (constant or a function). – 2011-12-27
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0@Davide k is constant vector – 2011-12-27
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1How is $\partial S/\partial x$ a scalar? Is it supposed to be understood as $\nabla S\cdot x$? – 2011-12-27
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0@anon corrected – 2011-12-27