PDE where the square of gradient of the unknown equals a given positive function

Question

Let $V(x)$ be a non-negative smooth function defined in a open domain $U\subset\mathbb{R}^n$. Suppose that $V(x)=0$ only at a given point $x_0\in U$. Consider the PDE $$|\nabla u|^2=V$$ with conditions that $u(x)>0$ for $x\neq x_0$ and $u(x_0)=0$. I expect that there is a unique solution for $u$. Does anyone can help me on this problem?

Answer 1

• As mentioned here When does the eikonal equation $\lvert Du \rvert^2 = f$ admit a local solution?, if we allow $V(x)=0$ at multiple locations, then we could get non-existence:

For $n=1$ or $2$, there is no $u\in C^1(D_\rho)$ for any $\rho>0$ that satisfies $|\nabla u|^2 = (xy)^{2n}$.

• For a strictly positive RHS, then the existence and uniqueness is more standard using viscosity solutions e.g. see the classical work of Ishii "A SIMPLE, DIRECT PROOF OF UNIQUENESS FOR SOLUTIONS OF THE HAMILTON-JACOBI EQUATIONS OF EIKONAL TYPE" and also "VISCOSITY SOLUTIONS OF THE EIKONAL EQUATIONS".
• In the article EIKONAL EQUATIONS IN METRIC SPACES, they go over the case of nonnegative $V(x)\geq 0$ using the notion of subsolutions. See also Solutions to the eikonal equation for an interesting discussion on local solution when allowing $V(x)\geq 0$.