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.



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> 1.Assume that $U$ satisfies the "LAP" equation. Is there a function $V$ such that the pair $(U,V)$ satisfies the "CR" equations?
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> 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 $U\_{x}+iV\_{x}$)
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> 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/
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A question on certain elliptic PDE
1.If $U$ satisfies LAP then there exists a $V$ such that $(U,V)$ satisfies CR. In fact, $V$ is unique up to the addition of a term of the form $a + bx + cy + d xy$, where $a$, $b$, $c$, and $d$ are constants. This is an elementary consequence of the Frobenius Theorem. 


2.You need to specify what you mean by 'algebra'. The space of functions $F = U + i V$ that satisfy your CR is infinite dimensional. In fact, any real solution to your LAP can be written in the form
$$
U(x,y) = h\_1(x + \omega y) + h\_3(x + \omega^3 y) + h\_5(x + \omega^5 y) + h\_7(x + \omega^7 y)
$$
where $\omega = \mathrm{e}^{i\pi/4}$ and where each $h\_{2i+1}$ is holomorphic in one (complex) variable, satisfying $\overline{h\_1(z)}=h\_7(\bar z)$ and $\overline{h\_3(z)}=h\_5(\bar z)$. (I'll let you figure out the corresponding formula for $V$.) This is also provable by the Frobenius Theorem. **Added remark about algebras:** There are two infinite dimensional subspaces of the space of solutions of CR that are closed under multiplication: Just notice that the operator $C = {\partial\_x}^2+i\, {\partial\_y}^2$ factors as ${\partial\_x}^2+i\, {\partial\_y}^2 = \bigl({\partial\_x}+\omega^3\, {\partial\_y}\bigr)\bigl({\partial\_x}-\omega^3\, {\partial\_y}\bigr)$, and these two factors commute. Thus $C(F)=0$ is satisfied by any function $F = U+iV = h\_1(x + \omega y)$ where $h\_1$ is a holomorphic function of its argument (and these functions form a subalgebra) and also by any function $F = U+iV = h\_5(x + \omega^5 y)$ where $h\_5$ is a holomorphic function of its argument (and these functions also form a subalgebra). In fact, the space of solutions to $C(F)=0$ is almost the direct sum of these two subalgebras; they intersect in the space of constant functions and their sum is the whole space. Because these two factors also commute with $D={\partial\_x}^2$, the two subalgebras are each invariant under this operator.


3.Generally, no. Just see what you have to do for surfaces. The point is that the symbol of LAP on a surface will be a quartic form that is the sum of two real, linearly independent fourth powers, and such a quartic form on a surface will determine a splitting of the tangent bundle of the surface into a sum of two line bundles. Thus, there cannot be such an operator on a compact surface with nonvanishing Euler characteristic, e.g., the $2$-sphere.


