Viscosity solution

In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems,[1] as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.

The classical concept was that a PDE

${\displaystyle F(x,u,Du,D^{2}u)=0}$

over a domain ${\displaystyle x\in \Omega }$ has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that ${\displaystyle x}$, ${\displaystyle u}$, ${\displaystyle Du}$, ${\displaystyle D^{2}u}$ satisfy the above equation at every point.

If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either ${\displaystyle Du}$ or ${\displaystyle D^{2}u}$ does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.

Definition

There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book[2] or the definition using semi-jets in the Users Guide.[3]

Degenerate elliptic

An equation ${\displaystyle F(x,u,Du,D^{2}u)=0}$ in a domain ${\displaystyle \Omega }$ is defined to be degenerate elliptic if for any two symmetric matrices ${\displaystyle X}$ and ${\displaystyle Y}$ such that ${\displaystyle Y-X}$ is positive definite, and any values of ${\displaystyle x\in \Omega }$, ${\displaystyle u\in \mathbb {R} }$ and ${\displaystyle p\in \mathbb {R} ^{n}}$, we have the inequality ${\displaystyle F(x,u,p,X)\geq F(x,u,p,Y)}$. For example, ${\displaystyle -\Delta u=0}$ is degenerate elliptic since in this case, ${\displaystyle F(x,u,p,X)=-{\text{trace}}(X)}$, and the trace of ${\displaystyle X}$ is the sum of its eigenvalues. Any real first order equation is degenerate elliptic.

Subsolution

An upper semicontinuous function ${\displaystyle u}$ in ${\displaystyle \Omega }$ is defined to be a subsolution of a degenerate elliptic equation in the viscosity sense if for any point ${\displaystyle x_{0}\in \Omega }$ and any ${\displaystyle C^{2}}$ function ${\displaystyle \phi }$ such that ${\displaystyle \phi (x_{0})=u(x_{0})}$ and ${\displaystyle \phi \geq u}$ in a neighborhood of ${\displaystyle x_{0}}$, we have ${\displaystyle F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\leq 0}$.

Supersolution

A lower semicontinuous function ${\displaystyle u}$ in ${\displaystyle \Omega }$ is defined to be a supersolution of a degenerate elliptic equation in the viscosity sense if for any point ${\displaystyle x_{0}\in \Omega }$ and any ${\displaystyle C^{2}}$ function ${\displaystyle \phi }$ such that ${\displaystyle \phi (x_{0})=u(x_{0})}$ and ${\displaystyle \phi \leq u}$ in a neighborhood of ${\displaystyle x_{0}}$, we have ${\displaystyle F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\geq 0}$.

Viscosity solution

A continuous function u is a viscosity solution of the PDE if it is both a supersolution and a subsolution.

Example

Consider the boundary value problem ${\displaystyle |u'(x)|=1}$, or ${\displaystyle F(u')=|u'|-1=0}$, on ${\displaystyle (-1,1)}$ with boundary conditions ${\displaystyle u(-1)=u(1)=0}$. The function ${\displaystyle u(x)=1-|x|}$ is the unique viscosity solution. To see this, note that the boundary conditions are satisfied, and ${\displaystyle |u'(x)|=1}$ is well-defined on the interior except at ${\displaystyle x=0}$. Thus, it remains to show that the conditions for subsolution and supersolution hold at ${\displaystyle x=0}$.

First, suppose that ${\displaystyle \phi (x)}$ is any function differentiable at ${\displaystyle x=0}$ with ${\displaystyle \phi (0)=u(0)=1}$ and ${\displaystyle \phi (x)\geq u(x)}$ near ${\displaystyle x=0}$. From these assumptions, it follows that ${\displaystyle \phi (x)-\phi (0)\geq -|x|}$. For positive ${\displaystyle x}$, this inequality implies ${\displaystyle \lim _{x\to 0^{+}}{\frac {\phi (x)-\phi (0)}{x}}\geq -1}$, using that ${\displaystyle |x|/x=sgn(x)=1}$ for ${\displaystyle x>0}$. On the other hand, for ${\displaystyle x<0}$, we have that ${\displaystyle \lim _{x\to 0^{-}}{\frac {\phi (x)-\phi (0)}{x}}\leq 1}$. Because ${\displaystyle \phi }$ is differentiable, the left and right limits agree and are equal to ${\displaystyle \phi '(0)}$, and we therefore conclude that ${\displaystyle |\phi '(0)|\leq 1}$, i.e., ${\displaystyle F(\phi ')\leq 0}$. Thus, ${\displaystyle u}$ is a subsolution. Moreover, the fact that ${\displaystyle u}$ is a supersolution holds vacuously, since there is no function ${\displaystyle \phi (x)}$ differentiable at ${\displaystyle x=0}$ with ${\displaystyle \phi (0)=u(0)=1}$ and ${\displaystyle \phi (x)\leq u(x)}$ near ${\displaystyle x=0}$. This implies that ${\displaystyle u}$ is a viscosity solution.

Discussion

Family of solutions ${\displaystyle u_{\epsilon }}$ converging toward ${\displaystyle u(x)=1-|x|}$.

The previous boundary value problem is an Eikonal equation in a single spatial dimension with ${\displaystyle f=1}$, where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of ${\displaystyle F}$. In particular, the viscosity solution to the PDE ${\displaystyle -F=0}$ with the same boundary conditions is ${\displaystyle u(x)=|x|-1}$. This can be explained by observing that the solution ${\displaystyle u(x)=1-|x|}$ is the limiting solution of the vanishing viscosity problem ${\displaystyle F(u')=[u']^{2}-1=\epsilon u''}$ as ${\displaystyle \epsilon }$ goes to zero, while ${\displaystyle u(x)=|x|-1}$ is the limit solution of the vanishing viscosity problem ${\displaystyle -F(u')=1-[u']^{2}=\epsilon u''}$.[4] One can readily confirm that ${\displaystyle u_{\epsilon }(x)=\epsilon [\ln(\cosh(1/\epsilon ))-\ln(\cosh(x/\epsilon ))}$ solves the PDE ${\displaystyle F(u')=[u']^{2}-1=\epsilon u''}$ for each epsilon. Further, the family of solutions ${\displaystyle u_{\epsilon }}$ converge toward the solution ${\displaystyle u=1-|x|}$ as ${\displaystyle \epsilon }$ vanishes (see Figure).

Basic properties

The three basic properties of viscosity solutions are existence, uniqueness and stability.

• The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations.[3] It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are

1. ${\displaystyle u+H(x,\nabla u)=0}$ with H uniformly continuous in x.
2. (Uniformly elliptic case) ${\displaystyle F(D^{2}u,Du,u)=0}$ so that ${\displaystyle F}$ is Lipschitz with respect to all variables and for every ${\displaystyle r\leq s}$ and ${\displaystyle X\geq Y}$, ${\displaystyle F(Y,p,s)\geq F(X,p,r)+\lambda ||X-Y||}$ for some ${\displaystyle \lambda >0}$.

• The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method [5] or for most equations using Perron's method.[6][7] There is a generalized notion of boundary condition, in the viscosity sense. The solution to a boundary problem with generalized boundary conditions is solvable whenever the comparison principle holds. [3]
• The stability of solutions in ${\displaystyle L^{\infty }}$ holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution). More generally, the notion of viscosity sub- and supersolution is also conserved by half-relaxed limits. [3]

History

The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation.[5] The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition of solution had actually been given earlier by Lawrence C. Evans in 1980.[8] Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.[9]

For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).[10]

In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes.[11] Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli.[12] Viscosity solutions have become a central concept in the study of elliptic PDE. In particular, Viscosity solutions are essential in the study of the infinity Laplacian.[13]

In the modern approach, the existence of solutions is obtained most often through the Perron method.[3] The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of viscosity solutions does not generally involve physical viscosity. Nevertheless, while the theory of viscosity solutions is sometimes considered unrelated to viscous fluids, irrotational fluids can indeed be described by a Hamilton-Jacobi equation.[14] In this case, viscosity corresponds to the bulk viscosity of an irrotational, incompressible fluid. Other names that were suggested were Crandall–Lions solutions, in honor to their pioneers, ${\displaystyle L^{\infty }}$-weak solutions, referring to their stability properties, or comparison solutions, referring to their most characteristic property.

References

1. Dolcetta, I.; Lions, P., eds. (1995). Viscosity Solutions and Applications. Berlin: Springer. ISBN 3-540-62910-6.
2. Wendell H. Fleming, H. M . Soner., eds., (2006), Controlled Markov Processes and Viscosity Solutions. Springer, ISBN 978-0-387-26045-7.
3. Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992), "User's guide to viscosity solutions of second order partial differential equations", American Mathematical Society. Bulletin. New Series, 27 (1): 1–67, arXiv:math/9207212, Bibcode:1992math......7212C, doi:10.1090/S0273-0979-1992-00266-5, ISSN 0002-9904
4. Barles, Guy (2013). "An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications". Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics. 2074. Berlin: Springer. pp. 49–109. doi:10.1007/978-3-642-36433-4_2. ISBN 978-3-642-36432-7.
5. Crandall, Michael G.; Lions, Pierre-Louis (1983), "Viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society, 277 (1): 1–42, doi:10.2307/1999343, ISSN 0002-9947, JSTOR 1999343
6. Ishii, Hitoshi (1987), "Perron's method for Hamilton-Jacobi equations", Duke Mathematical Journal, 55 (2): 369–384, doi:10.1215/S0012-7094-87-05521-9, ISSN 0012-7094
7. Ishii, Hitoshi (1989), "On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs", Communications on Pure and Applied Mathematics, 42 (1): 15–45, doi:10.1002/cpa.3160420103, ISSN 0010-3640
8. Evans, Lawrence C. (1980), "On solving certain nonlinear partial differential equations by accretive operator methods", Israel Journal of Mathematics, 36 (3): 225–247, doi:10.1007/BF02762047, ISSN 0021-2172
9. Crandall, Michael G.; Evans, Lawrence C.; Lions, Pierre-Louis (1984), "Some properties of viscosity solutions of Hamilton–Jacobi equations", Transactions of the American Mathematical Society, 282 (2): 487–502, doi:10.2307/1999247, ISSN 0002-9947, JSTOR 1999247
10. Jensen, Robert (1988), "The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations", Archive for Rational Mechanics and Analysis, 101 (1): 1–27, Bibcode:1988ArRMA.101....1J, doi:10.1007/BF00281780, ISSN 0003-9527
11. Barles, G.; Souganidis, P. E. (1991), "Convergence of approximation schemes for fully nonlinear second order equations", Asymptotic Analysis, 4 (3): 271–283, doi:10.3233/ASY-1991-4305, ISSN 0921-7134
12. Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7
13. Crandall, Michael G.; Evans, Lawrence C.; Gariepy, Ronald F. (2001), "Optimal Lipschitz extensions and the infinity Laplacian", Calculus of Variations and Partial Differential Equations, 13 (2): 123–129, doi:10.1007/s005260000065
14. Westernacher-Schneider, John Ryan; Markakis, Charalampos; Tsao, Bing Jyun (2019). "Hamilton-Jacobi hydrodynamics of pulsating relativistic stars". Classical and Quantum Gravity. arXiv:1912.03701.

External links

• N. Nadirashvili, Fully nonlinear elliptic equations