Elliptic differential operator

Question

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are elliptic. Does anybody know how to do this, I am really puzzled by the wikipedia definition of elliptic differential operators.

Answer 1

The heat equation is parabolic: $$ \frac{\partial f}{\partial t} = \frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}} $$ Laplace's equation is elliptic: $$ \frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}} = 0 $$ The Wave equation is hyperbolic: $$ \frac{\partial^{2}f}{\partial t^{2}}=\frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}} $$ These designations are related to $y-x^{2}=C$, $x^{2}+y^{2}=C$ and $y^{2}-x^{2}=C$, whose level surfaces are generally parabolic, elliptic and hyperbolic, respectively.

Your equation is never classified as elliptic because such designations are applied only to Partial Differential Equations.

Answer 2

Your equation is only of first-order, only first derivatives with respect t only one variable appear there. There is no classification of such equations into elliptic, parabolic, hyperbolic equations.

If you talk about systems of first-order equations, they can be classified, if they can be written as second-order equations. E.g. the system $$ \frac{\partial f}{\partial t} = \frac{\partial g}{\partial x} , \ \frac{\partial g}{\partial t} = -\frac{\partial f}{\partial x} $$ is equivalent to $$ \frac{\partial^2 f}{\partial t^2} = \frac{\partial }{\partial t}\frac{\partial g}{\partial x} = \frac{\partial }{\partial x}\frac{\partial g}{\partial t} = - \frac{\partial^2 f}{\partial x^2} , $$ which is an elliptic equation.