3.1 The discriminant and its invariance
Consider the linear equation \begin {equation} \label {eq:secondorder} a u_{xx} + 2b u_{xy} + c u_{yy} + d u_x + e u_y + f u = g , \end {equation} with coefficients depending on \((x, y)\). Its principal part is \(a u_{xx} + 2b u_{xy} + c u_{yy}\), with symmetric matrix \(M = \begin {pmatrix} a & b \\ b & c \end {pmatrix}\).
Definition 3.1.1. At a point, equation (3.1) is hyperbolic if \(b^2 - ac > 0\), parabolic if \(b^2 - ac = 0\) (with \(M \neq 0\)), and elliptic if \(b^2 - ac < 0\).
The wave equation \(u_{tt} - u_{xx} = 0\) (\(a = -1\), \(c = 1\) in the variables \((x, t)\), \(b^2 - ac = 1\)) is hyperbolic; the heat equation \(u_t - u_{xx} = 0\) (\(a = -1\), \(b = c = 0\)) is parabolic; Laplace’s equation \(u_{xx} + u_{yy} = 0\) (\(b^2 - ac = -1\)) is elliptic. The names come from the conics \(ax^2 + 2bxy + cy^2 = 1\).
Theorem 3.1.2. Under a smooth change of variables \(\xi = \xi (x, y)\), \(\eta = \eta (x, y)\) with Jacobian \(J = \xi _x\eta _y - \xi _y\eta _x \neq 0\), equation (3.1) becomes \(Au_{\xi \xi } + 2Bu_{\xi \eta } + Cu_{\eta \eta } + (\text {lower order}) = g\), where \begin {align*} A &= a\xi _x^2 + 2b\xi _x\xi _y + c\xi _y^2, \\ B &= a\xi _x\eta _x + b(\xi _x\eta _y + \xi _y\eta _x) + c\xi _y\eta _y, \\ C &= a\eta _x^2 + 2b\eta _x\eta _y + c\eta _y^2 , \end {align*}
and \(B^2 - AC = J^2(b^2 - ac)\). In particular the type is unchanged by any change of variables.
Proof. By the chain rule \(u_x = u_\xi \xi _x + u_\eta \eta _x\), and differentiating again, the second derivatives of \(u\) in the new variables appear only through products of first derivatives of \(\xi \) and \(\eta \); the second derivatives of \(\xi \) and \(\eta \) multiply \(u_\xi \) and \(u_\eta \), and go into the lower-order terms. Collecting terms gives \(A\), \(B\), \(C\) above. In matrix form this says \[ \begin {pmatrix} A & B \\ B & C \end {pmatrix} = D M D^{\mathsf T}, \qquad D = \begin {pmatrix} \xi _x & \xi _y \\ \eta _x & \eta _y \end {pmatrix} , \] and taking determinants, \(AC - B^2 = (\det D)^2(ac - b^2) = J^2(ac - b^2)\). □
In \(n\) variables, the principal part \(\sum _{i,j}a_{ij}u_{x_ix_j}\) has a symmetric matrix \((a_{ij})\), and a linear change of variables replaces it by \(DAD^{\mathsf T}\); by Sylvester’s law of inertia the numbers of positive, negative and zero eigenvalues are invariant. The equation is elliptic if all eigenvalues have one sign, hyperbolic if all but one have one sign and the last the other, and parabolic if one is zero and the rest have one sign. The wave equation \(u_{tt} = c^2\Delta u\) has eigenvalues \(1, -c^2, \ldots , -c^2\), and is hyperbolic; the heat and Laplace equations in \(n\) dimensions are parabolic and elliptic. Other sign patterns (ultra-hyperbolic) can occur when \(n \geq 4\), but have no comparable role in physics.
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