I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the

• 1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value function $\{x_2 =\vert x_1 \vert \}.$

• 2.) In addition, I require that $\vert \nabla f_n \vert_{N_n}\vert=1$ and

• 3.) $\nabla^2 f_n \nabla f_n \vert_{N_n} = 0.$

I was almost able to construct such a sequence by choosing

$$f_n (x) = \Big(x_2 -\sqrt{x_1^2+ 1/n}\Big) \frac{\sqrt{2x_1^2 + 1/n}}{\sqrt{x_1^2+1/n}}.$$

This one satisfies 1 and 2 but not 3. Any thoughts?

I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the

• 1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value function $\{x_2 =\vert x_1 \vert \}.$

• 2.) In addition, I require that $\vert \nabla f_n \vert_{N_n}\vert=1$ and

• 3.) $\nabla^2 f_n \nabla f_n \vert_{N_n} = 0.$

I was almost able to construct such a sequence by choosing

$$f_n (x) = \Big(x_2 -\sqrt{x_1^2+ 1/n}\Big) \frac{\sqrt{x_1^2+1/n}}{\sqrt{2x_1^2 + 1/n}}.$$

This one satisfies 1 and 2 but not 3. Any thoughts?

I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the

• 1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value function $\{x_2 =\vert x_1 \vert \}.$

• 2.) In addition, I require that $\nabla f_n \vert_{N_n}=1$ and

• 3.) $\nabla^2 f_n \nabla f_n \vert_{N_n} = 0.$

I was almost able to construct such a sequence by choosing

$$f_n (x) = \Big(x_2 -\sqrt{x_1^2+ 1/n}\Big) \frac{\sqrt{2x_1^2 + 1/n}}{\sqrt{x_1^2+1/n}}.$$

This one satisfies 1 and 2 but not 3. Any thoughts?

I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the

• 1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value function $\{x_2 =\vert x_1 \vert \}.$

• 2.) In addition, I require that $\vert \nabla f_n \vert_{N_n}\vert=1$ and

• 3.) $\nabla^2 f_n \nabla f_n \vert_{N_n} = 0.$

I was almost able to construct such a sequence by choosing

$$f_n (x) = \Big(x_2 -\sqrt{x_1^2+ 1/n}\Big) \frac{\sqrt{2x_1^2 + 1/n}}{\sqrt{x_1^2+1/n}}.$$

This one satisfies 1 and 2 but not 3. Any thoughts?