User:DVD206/The Laplace-Beltrami operator and harmonic functions


 * a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation.
 * mean-value property
 * The value of a harmonic function is a weighted average of its values at the neighbor vertices.


 * maximum principle
 * Corollary: the maximum (and the minimum) of a harmonic functions occurs on the boundary of the graph or the manifold.


 * harmonic conjugate
 * One can use the system of Cauchy Riemann equations

$$ \begin{cases} \gamma u_x = v_y, \\ \gamma u_y = -v_x \end{cases} $$ to define the harmonic conjugate
 * analytic continuation
 * Analytic continuation is an extension of the domain of a given analytic function.