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I'm wondering if there are any special considerations/ideas/treatments of equations of the following sort, where if $\boldsymbol{x} = (x_1,x_2,x_3)$: $$\begin{bmatrix} \partial_{x_1} & \left(\partial_{x_3}(\cdot)\right)^2\\ a(\boldsymbol{x})\partial_{x_2}^2 & -4\\ \boldsymbol{b}^T\nabla(\cdot) & \partial_{x_1}^2 \end{bmatrix} \begin{bmatrix} u(\boldsymbol{x})\\ v(\boldsymbol{x}) \end{bmatrix} = \begin{bmatrix} \partial_{x_1}u + \left(\partial_{x_3}v\right)^2\\ a(\boldsymbol{x})\partial_{x_2}^2u-4v\\ \boldsymbol{b}^T\nabla u + \partial_{x_1}^2v \end{bmatrix}= \begin{bmatrix} f(\boldsymbol{x})\\ g(\boldsymbol{x})\\ h(\boldsymbol{x}) \end{bmatrix}$$

Obviously I just made this up, and I'm sure that monstrosity has no solution, but you get the picture: has anyone studied this extra layer of structure you can add in systems of PDEs where you apply an matrix to a vector-valued function whose entries are differential operators in their own right? Is there a particular calculus or set of computational rules that arise from these typesequations? Generally in indicial notation, $$L_{ij}u_j(\boldsymbol{x})=f_i(\boldsymbol{x})\quad\text{where $L_{ij}$ are (possibly nonlinear) differential operators}$$ If this is some obvious field/application of study, I clearly don't have the right key search terms to find this on my own. I'm sorry if this is an obvious question.

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