Cauchy boundary condition

Partial differential equations

For partial differential equations, Cauchy boundary conditions specify both the function and the normal derivative on the boundary. To make things simple and concrete, consider a second-order differential equation in the plane

${\displaystyle A(x,y){\psi }_{xx}+B(x,y){\psi }_{xy}+C(x,y){\psi }_{yy}=F(x,y,\psi ,{\psi }_x,{\psi }_y),}$

where ${\displaystyle \psi (x,y)}$ is the unknown solution, ${\displaystyle {\psi }_x}$ denotes derivative of ${\displaystyle \psi }$ with respect to ${\displaystyle x}$ etc. The functions ${\displaystyle A,B,C,F}$ specify the problem.

We now seek a ${\displaystyle \psi }$ that satisfies the partial differential equation in a domain ${\displaystyle \mathrm{\Omega }}$, which is a subset of the ${\displaystyle xy}$ plane, and such that the Cauchy boundary conditions

${\displaystyle \psi (x,y)=\alpha (x,y),\mathbf{n}\cdot \mathrm{\nabla }\psi =\beta (x,y)}$

hold for all boundary points ${\displaystyle (x,y)\in \mathrm{\partial }\mathrm{\Omega }}$. Here ${\displaystyle \mathbf{n}\cdot \mathrm{\nabla }\psi }$ is the derivative in the direction of the normal to the boundary. The functions ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are the Cauchy data.

Notice the difference between a Cauchy boundary condition and a Robin boundary condition. In the former, we specify both the function and the normal derivative. In the latter, we specify a weighted average of the two.

We would like boundary conditions to ensure that exactly one (unique) solution exists, but for second-order partial differential equations, it is not as simple to guarantee existence and uniqueness as it is for ordinary differential equations. Cauchy data are most immediately relevant for hyperbolic problems (for example, the wave equation) on open domains (for example, the half plane).