TITLE: A question on certain elliptic PDE
QUESTION [5 upvotes]: Consider the  elliptic  PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence   $$(LAP)\;\;\;\;\;\;U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations  are similar to the classical  Cauchi Riemann  and  Laplace equation.

1.Assume  that $U$ satisfies the "LAP" equation. Is there  a function $V$ such that the pair $(U,V)$ satisfies  the "CR" equations?


2.Is there  an infinite dimensional  algebra  of complex  functions $f(z)=U+iV$  which satisfies  the "CR" equations?If it is  the case, can one  find such algebra  which is  invariant under the  operator $D(U+iV)=U_{xx}+iV_{xx}$?(Motivated by classical holomorphic  differentiation $U_{x}+iV_{x}$)


3.Is there  a geometric  way (in term of Reimannian metric) to introduce the  above equations on an arbitrary Riemannian  manifold. (Similar to the  situation of conformal  maps  and Laplace  operator=the  composition of div and  gradient)?If the answer is  yes, what would  be the index of "LAP" operator? In the other  word, is  there  a  geometric  meaning  for these equations/

REPLY [2 votes]: All linear functions satisfy your CR equations, so it seems fairly natural that they should be included in any related function algebra $A$.
But there is no algebra of $C^1$ functions satisfying the CR equations and including a function apart from the algebra of constant functions.
(I assume that the algebra is supposed to be closed under pointwise multiplication.)
Let us write $z=x+iy$.
Suppose $F(z)=U(z)+iV(z)$ belongs to $A$.
Now also $z\mapsto x$ is in $A$, so $f(z)=xF(z)$ is in it.
Thus we have
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
and
$$\begin{cases} 2U_x+xU_{xx}=xV_{yy}\\xU_{yy}=-2V_x-xV_{xx} \end{cases}.$$
We thus have $U_x=V_x=0$ and we can get the same result for $x$ replaced by $y$, whence $F$ is constant.
Remark: Suppose $F=U+iV$ and $f=u+iv$ solve the CR equations.
If their product solves it too, we get
$$\begin{cases} U_xu_x-V_xv_x=U_yv_y+V_yu_y\\-U_yu_y+V_yv_y=U_xv_x+V_xu_x \end{cases}.$$
Therefore any pair of functions in your algebra should satisfy these equations.
I don't know if there is a nontrivial algebra if linear functions are excluded.