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I need to compute a Lipschitz constant for the function: $$ F(x,y)= \begin{pmatrix} \frac{1}{3}x^3+2xy\\ x^2+3xy \end{pmatrix}, $$ where $(x,y)\in[0,2]^2$. My first thought was to use the mean value theorem, and compute the operatornorm of the derivative. However this soon became very complicated.

So now I'm at a loss and can't figure out what to do. So how would one solve such a problem?

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You need a number $K$ with $$ \|F(x_1, y_1) - F(x_0, y_0)\| \le K D $$ where $D$ is the distance between the two points.

Letting $F_1(x, y) = x^3/3 + 2xy$, and similarly for $F_2$, the difference between $F_1(x_0, y_0)$ and $F_1(x_1, y_1)$ is no greater than the distance between the domain points, multiplied by the maximum magnitude of the gradient of $F_1$ on your rectangle. You can compute that max.

You can do the same for $F_2$. And then you can use these to estimate the max change in $(F_1, F_2)$ by the triangle inequality.

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