Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Question

Find an harmonic function in $R^n$ which

• It is a polynomial of degree 4 and is =1 at the origin.
• It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin.

Important definitions:

• An harmonic function is a solution of $\Delta u=0$.
• Theorem Let $f: C \to C$ be differentiable in C. Then $f(x+iy)$, $Re(f(x+iy))$, and $Im(f(x+iy))$ satisfy $\Delta u=0$.

Find an harmonic function in $R^n$ which it is a polynomial of degree 4 and is =1 at the origin.

Using the Theorem above, $$f(z)=f(x+iy)=(x+iy)^4=x^4+4ix^3y-6x^2y^2-4ixy^3+y^4$$

Is this correct?

Find an harmonic function in $R^n$ which it is a polynomial of degree 5 and its partial derivatives are both =0 at the origin.

Answer 1

This was sorted out in comments, but for completeness, a few additional remarks:

1. It suffices to construct such examples in two dimensions. A harmonic function of two variables can also be viewed as a harmonic function of $n$ variables that happens to be independent of $(n-2)$ of them.
2. The real part (or imaginary part) of $z^n$ is a harmonic polynomial of degree $n$. Since the $k$ order derivatives of $z^n$ vanish at the origin for $k\le n-1$, so do the partial derivatives of these polynomials.
3. Since any function of the form $ax+by+c$ is harmonic, one can add such a function to arrange any desired value and first-order partials at a particular point.