- #1

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- Homework Statement:
- Discover the classification of linear second order PDE yourself.

- Relevant Equations:
- Please see below

(1) ok.

(2) We start with ##\sigma(ξ) = a_{11} ξ_1^2 +2a_{12}ξ_1ξ_2 +a_{22}ξ_2^2ξ##

and we replace every ##ξ_iξ_j## with ##\partial_i\partial_ju##,

giving ##a_{11}\partial_x^2+2a_{12}\partial_x\partial_yu+1_{22}\partial_2^2##

(3) The given equation is the following.

##\sigma(ξ) = ξ^t A ξ ##

we take the product.

##\sigma(ξ) = \begin{pmatrix} ξ_1^t & ξ_2^t \end{pmatrix} \begin{pmatrix} a_{11} & 1_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} ξ_1 \\ ξ_2 \end{pmatrix} ##

giving

##\sigma(ξ) = a_{11}ξ_1^{t+1} +2a_{12}ξ_1ξ_2+a_{22}ξ_2^{t+1}##

which is different from the symbol given in part (2)

##\sigma(ξ) = a_{11} ξ_1^2 +2a_{12}ξ_1ξ_2 +a_{22}ξ_2^2ξ##

what is the ##t## used for, is this not a typo?

(4) The trace is the sum of the eigenvalues ##a_{11}+a_{22}##. knowing the the trace does not tell us anything about the eigenvalues. the determinant is given by ##det(A) = a_{11}a_{22}-(a_{12})^2 ##, which does not tell us ##a_{11}, a_{22}##. we have two equations and three unknowns, so we cannot use the trace and determinant for characterization.

(5) we could use matrix diagonalization (change of coordinates) to express ##A## as some matrix whose diagonal entries are ##a_{11}, a_{22}##. Doing this would cancel out the crossterm ##2a_{12}ξ_1ξ_2##. so

(a) ##\sigma(η) = \sigma(ξ) - 2a_{12}ξ_1ξ_2##

(b) ##\sigma(η) = \sigma(ξ) - a_{11} ξ_1^2-2a_{12}ξ_1ξ_2##

(c) ##\sigma(η) = \sigma(ξ) - 2a_{11} ξ_1^2-2a_{12}ξ_1ξ_2##

I definitely need to think more about this problem, but I value any suggestions.

(6) The highest order terms of the symbol controls the qualitative behavior of the solutions of PDEs, so we could use the symbols of every second order PDE by ignoring the lower order terms for a close approximation. And manipulate the symbols to fit the three cases by a change of coordinates.

(7) If we have non-constant coefficients, could we use the same procedure as before, but instead the eigenvalues will be functions, and not numbers?

(8) For higher dimensions, could we define elliptic, parabolic, and hyperbolic PDEs in just two variables and keep the other variables undisturbed?