5.3.2 Viscosity solutions of PDEs

We are now ready to introduce the concept of a viscosity solution for PDEs. Consider a PDE of the form

$$F(x,v(x),\nabla v(x))=0$$ (5.30)

where $F:\mathbb{R}^n\times \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}$ is a continuous function. A viscosity subsolution of the PDE (5.34) is a continuous function $v:\mathbb{R}^n\to \mathbb{R}$ such that

$$F(x,v(x),\xi)\le 0\qquad \forall\,\xi\in D^+v(x),\ \forall\,x.$$ (5.31)

As we know, this is equivalent to saying that at every $x$ we must have $F(x,v(x),\nabla\varphi(x))\le 0$ for every $\mathcal C^1$ test function $\varphi$ such that $\varphi-v$ has a local minimum at $x$ . Similarly, $v$ is a viscosity supersolution of (5.34) if

$$F(x,v(x),\xi)\ge 0\qquad \forall\,\xi\in D^-v(x),\ \forall\,x$$ (5.32)

or, equivalently, at every $x$ we have $F(x,v(x),\nabla\varphi(x))\ge 0$ for every $\mathcal C^1$ function $\varphi$ such that $\varphi-v$ has a local maximum at $x$ . Finally, $v$ is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.

The above definitions of a viscosity subsolution and supersolution impose conditions on $v$ only at points where $D^+v$ , respectively $D^-v$ , is nonempty. We know that the set of these points is dense in the domain of $v$ . At all points where $v$ is differentiable, the PDE must hold in the classical sense. If $v$ is Lipschitz, then by Rademacher's theorem it is differentiable almost everywhere.

Note the lack of symmetry in the definition of a viscosity solution: the sign convention used when writing the PDE is important. In the above example, if we rewrite the PDE as $\vert\nabla v(x)\vert-1=0$ , then it is easy to see that $v(x)=\vert x\vert$ is no longer a viscosity solution.

The terminology ``viscosity solutions" is motivated by the fact that a viscosity solution $v$ of the PDE (5.34) can be obtained from smooth solutions $v_\varepsilon$ to the family of PDEs

$$F(x,v_\varepsilon (x),\nabla v_\varepsilon (x))=\varepsilon \Delta v_\varepsilon (x)$$ (5.33)

(parameterized by $\varepsilon >0$ ) in the limit as $\varepsilon \to 0$ . The operator $\Delta$ on the right-hand side of (5.37) denotes the Laplacian ( $\Delta v={v}_{{x_1}{x_1}}+\cdots+{v}_{{x_n}{x_n}}$ ); in fluid mechanics it appears in the PDE describing the motion of a viscous fluid. To understand the basic idea behind this convergence result, let $\xi\in D^-v(x_0)$ for some $x_0$ . Consider a test function $\varphi$ such that $\nabla \varphi(x_0)=\xi$ and $\varphi-v$ has a local maximum at $x_0$ . Assume that $\varphi\in\mathcal C^2$ (if not, approximate it by a $\mathcal C^2$ function). If $v_\varepsilon$ is close to $v$ for small $\varepsilon$ , then $\varphi-v_\varepsilon$ has a local maximum at some $x_\varepsilon$ near $x_0$ , implying that $\nabla \varphi(x_\varepsilon )=\nabla v_\varepsilon (x_\varepsilon )$ and $\Delta \varphi(x_\varepsilon )\le \Delta v_\varepsilon (x_\varepsilon )$ . Since $v_\varepsilon$ solves (5.37), this gives $F(x_\varepsilon ,v_\varepsilon (x_\varepsilon ),\nabla \varphi(x_\varepsilon ))\ge \varepsilon \Delta \varphi(x_\varepsilon ).$ Taking the limit as $\varepsilon \to 0$ , by continuity of $F$ we have $F(x_0,v(x_0),\nabla \varphi(x_0))\ge 0$ , which means that $v$ is a supersolution of (5.34). The argument showing that $v$ is a subsolution is similar.