# Problem about solution of an PDE system

For the following system:

$$\begin{cases} |\nabla f|^2=1 \\\\ \Delta f=0 \end{cases}$$

where $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ and $$\nabla f$$, $$\Delta f$$ are Gradient and Laplacian of $$f$$, respectively.

Are there solutions for this system where the $$f$$ function is not an affine function?

No, there are no solutions $$f$$ where the function $$f$$ is not an affine function. To see this, note that differentiation of the identity $$|\nabla f|^2=1$$, once by $$x$$ and another time by $$y$$, $$|\nabla f|^2=f_x^2+f_y^2\equiv1$$ implies $$\langle\nabla f,(f_{xy},f_{yy})\rangle = 0$$ and also $$\langle\nabla f, (f_{xx},f_{xy})\rangle = 0$$ where $$\langle\cdot\rangle$$ is the Euclidean dot product. We have two vectors perpendicular to the (non-zero) gradient, so they must be proportional, i.e., there exists a number $$c(x,y)$$ such that $$(f_{xx},f_{xy})=c(x,y)(f_{xy},f_{yy})\quad\quad (1)$$ Note that if $$f_{yy}\neq 0, f_{xx}\neq 0$$ at a certain point, then $$c(x,y)\neq 0$$. So far we have only used the fact that the gradient has unit length. Now we use the second condition, that the Laplacian vanishes. This means that $$f$$ is harmonic, and so it is the real part of some analytic function $$F(x,y)=f(x,y)+ig(x,y)$$ Since $$F$$ is analytic, the Cauchy-Riemann equations are valid, so that $$f_x=g_y,\quad f_y=-g_x$$ Therefore $$f_{xy}=g_{yy}=-g_{xx}$$ Consequently, $$(f_{xy},f_{yy})=(g_{yy},f_{yy}),\quad (f_{xx},f_{xy})=(f_{xx},-g_{xx})$$ so that (1) implies $$(f_{xx},-g_{xx})=(f_{xx},f_{xy})=c(x,y)(f_{xy},f_{yy})=c(x,y)(g_{yy},f_{yy})=-c(x,y)(g_{xx},f_{xx})$$ but the vector $$(g_{xx},f_{xx})$$ is clearly perpendicular to the vector $$(-f_{xx},g_{xx})$$, so it cannot be proportional to it, unless $$c(x,y)=0$$, which implies that $$f_{xx}=f_{yy}=0$$, and subsequently also $$f_{xy}=0$$, which implies that $$f$$ is an affine function.
• Thank you very much, but if $f$ is not an harmonic function, (i.e. $\Delta f= a$ with $a$ constant not null), what's happen? – exxxit8 Aug 7 '19 at 12:14