# Why are viscosity solutions to Hamilton-Jacobi equations interesting?

This is a soft question. For my background, I am an undergraduate math major just finishing my first course in PDE. In my PDE class, we have been learning about viscosity solutions.

I am following the definitions in these notes

We say a continuous function $u : [0, T ] \times \Omega → \mathbb{R}$ is a subsolution ( alt supersolution) of the equation $u_t + H(x,t,u,\nabla u) =0$, if every time there exists a $C^1$ function $\varphi$ for which there is a point $(t_0, x_0) \in (0, T ] \times \Omega$ and $r>0$, such that $\varphi (t_0, x_0) = u(t_0, x_0)$, $\varphi (t,x)≥u(t,x)$ (alt $≤$) for all $(t,x) \in(t_0 −r,t_0]×B_r(x_0)$, then $\varphi_t +H(t,x,\varphi,\nabla \varphi)≤0$ (alt $≥0$).

A continuous function $u$ is a viscosity solution in $(0, T ] × Ω$ when it is both a subsolution and a supersolution.

Evans also discusses these solutions in Chapter 10 of his book on PDE. Evans motivates viscosity solutions by looking at a sequence of functions $u^\varepsilon$ that are solutions to $u_t + H(x,t,u,\nabla u) - \varepsilon \Delta u =0$. This equation has smooth solutions, because the Laplacian regularizes the Hamilton-Jacobi equation. Often we can assure that the sequence $u^\varepsilon$ is bounded and equicontinuous, giving some locally uniformly convergent subsequence, and these turn out to be viscosity solutions. (Though not all viscosity solutions arise in this way)

I am also aware that viscosity solutions can be defined for somewhat more general non-linear PDEs.

My questions are the following:

• Why are such generalized solutions interesting? Are they only interesting because we can prove the existence and uniqueness of such solutions? Or do viscosity solutions to Hamilton-Jacobi equations actually model anything?

• Are these solutions ever useful in applications? This may be just a more specific form of the above question.

• Why should the notion of viscosity solution for Hamilton-Jacobi equations be the "right" definition of generalized solution?

• I can see I've been down voted twice now. If anyone has suggestions on how I could improve the question or what's wrong with it, please let me know. I am quite interested in its answer. – msm Jun 7 '17 at 14:27