Skip to main content
SpringerLink
Log in

Non-linear Partial Differential Equations, Viscosity Solution Method in

  • Reference work entry
Encyclopedia of Complexity and Systems Science

  • 217 Accesses

Definition of the Subject

In order to investigate phenomena in both natural and social sciences, it is important to analyze solutions of PDEs derived from certainminimization principles such as calculus of variations.

Since it is hard to find classical solutions of PDEs in general, the first strategy is to look for weak solutions of those. For PDEs of divergencetype, the most celebrated notion of weak solutions is that in the distribution sense, which is formally derived through the integration by parts.

On the other hand, before viscosity solutions were introduced, there were several notions of weak solutions for PDEs of non-divergence typesuch as generalized solutions for first order PDEs by S.N. Kružkov.

For second order elliptic/parabolic PDEs of non-divergence type, it has turned out through much research that the notion of viscositysolutions is the most appropriate one by several reasons:

  1. (i)

    Viscosity solutions admit well-posedness (i.?e. existence, uniqueness, stability),

    ...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

Weak solutions:

In the study of mth order partial differential equations (abbreviated, PDEs), a function is informally called a classical solution of the PDE if (a) it is m-times differentiable, and (b) the PDE holds at each point of the domain by putting its derivatives there. However, it is not easy to find such classical solutions of PDEs in general except for some special cases.

The standard strategy to find a classical solution is first to look for a candidate of solutions, which becomes a classical solution if the property (a) holds for it. Here, such a candidate is called a weak solution of the PDE.

Viscosity solutions:

In 1981, for first order PDEs of non-divergence type, M.G. Crandall and P.-L. Lions in [15,16] (also [17]) introduced the notion of weak solutions, which are called viscosity solutions . The definition of those is the property which the limit of approximate solutions via the vanishing viscosity method admits.

Afterwards, the notion was extended to fully nonlinear second order elliptic/parabolic PDEs.

For general theory of viscosity solutions, [3,5,7,18,20] are recommended for the interested readers.

Throughout this article, to minimize the references, [18] will be often referred to instead of the original papers except for some pioneering works or those which appeared after [18].

Ellipticity /parabolicity :

General second order PDEs under consideration are

$$ F(x,u(x),Du(x),D^2u(x))=0\quad \text{in }\Omega \:. $$
(E)

Here, \( { u\colon\Omega\to\mathbf{R} } \) is the unknown function, \( { \Omega\subset \mathbf{R}^n } \) an open set, \( F\colon\Omega\times \mathbf{R}\times\mathbf{R}^n\times S^n \to\mathbf{R} \) a given (continuous) function, S n the set of \( { n\times n } \) symmetric matrices with the standard ordering, \( Du(x)=((\partial u)/(\partial x_1)(x),\ldots ,(\partial u)/(\partial x_n)(x)) \), and \( { D^2u(x)\in S^n } \) whose \( { (i,j) } \)th entry is \( (\partial^2 u)/(\partial x_i\partial x_j)(x) \).

According to the early literature in viscosity solution theory, (E) is called elliptic if

$$ X\leq Y \Longrightarrow F(x,r,p,X)\geq F(x,r,p,Y) $$

for \( { (x,r,p,X,Y)\in\Omega\times\mathbf{R}\times\mathbf{R}^n\times S^n \times S^n } \). It should be remarked that the opposite order has been also used.

When F does not depend on the last variables (i.?e. first order PDEs), it is automatically elliptic. Thus, the above notion has been called degenerate elliptic.

The evolution version of general PDEs is

$$ u_t(x,t)+F(x,t,u(x,t),Du(x,t),D^2u(x,t))=0\\ \text{in } Q_T := \Omega\times (0,T]\:. $$
(P)

Here, \( { u\colon Q_T\to\mathbf{R} } \) is the unknown function, \( { F\colon\Omega\times (0,T]\times\mathbf{R}\times\mathbf{R}^n\times S^n\to\mathbf{R} } \) a given function, \( { T > 0 } \), and \( { u_t(x,t)=(\partial u)/(\partial t)(x,t) } \). In (P), the notations \( { Du(x,t) } \) and \( { D^2u(x,t) } \) do not contain derivatives with respect to t.

Similarly, (P) is called parabolic if \( { F(\cdot ,t,\cdot ,\cdot ,\cdot) } \) is elliptic for each \( { t\in (0,T] } \).

Uniform ellipticity /uniform parabolicity :

Denoted by \( S^n_{\lambda,\Lambda}:=\{ A\in S^n | \lambda I\leq A\leq \Lambda I\} \) for fixed \( 0<\lambda\leq \Lambda \), the Pucci operators \( \mathcal{P}^\pm \colon S^n\to \mathbf{R} \) are defined by

$$ \begin{aligned} \mathcal{P}^+(X)& :=\max_{A\in S^n_{\lambda,\Lambda}}\{ -\mathrm{trace}(AX) \}\quad\text{and}\\ \mathcal{P}^-(X)& :=\min_{A\in S^n_{\lambda,\Lambda}}\{-\mathrm{trace}(AX)\}\:. \end{aligned} $$

Then, the PDE (E) is called uniformly elliptic if

$$ \mathcal{P}^-(X-Y)\leq F(x,r,p,X)-F(x,r,p,Y)\leq \mathcal{P}^+(X-Y) $$

for \( { (x,r,p,X,Y)\in \Omega\times\mathbf{R}\times\mathbf{R}^n\times S^n\times S^n } \). This is a fully nonlinear version of the standard uniform ellipticity. For the theory of second order uniformly elliptic PDEs, [24] is the standard text book.

Similarly, (P) is called uniformly parabolic if \( F(\cdot ,t,\cdot ,\cdot ,\cdot) \) is uniformly elliptic for each \( { t\in (0,T] } \).

Dynamic programming principle:

In stochastic control problems, the value function is determined by minimizing given cost functionals. The dynamic programming principle (abbreviated, DPP), which was established as the Bellman's principle of optimality, is a formula which the value function satisfies.

The DPP indicates that the value function isa viscosity solution of the associatedHamilton–Jacobi–Bellman (abbreviated, HJB) equation.

Bibliography

Primary Literature

  1. Alvarez O, Hoch P, Le Bouar Y, Monneau R (2006) Dislocation dynamics: short time existence and uniqueness of the solution. Arch Ration Mech Anal 85:371–414

    Google Scholar 

  2. Aronsson G, Crandall MG, Juutinen P (2004) A tour of the theory of absolutely minimizing functions. Bull Amer Math Soc 41:439–505

    MathSciNet  MATH  Google Scholar 

  3. Bardi M, Capuzzo Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston

    MATH  Google Scholar 

  4. Bardi M, Mannuci P (2006) On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Comm Pure Appl Anal 5:709–731

    MATH  Google Scholar 

  5. Bardi M, Crandall MG, Evans LC, Soner HM, Souganidis PE (1997) Viscosity Solutions and Applications. Springer, Berlin

    Google Scholar 

  6. Barles G (1993) Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations: A guided visit. Nonlinear Anal 20:1123–1134

    MathSciNet  MATH  Google Scholar 

  7. Barles G (1994) Solutions de Viscosité des Équations de Hamilton–Jacobi. Springer, Berlin

    MATH  Google Scholar 

  8. Barles G, Perthame B (1987) Discontinuous solutions of deterministic optimal stopping-time problems. Model Math Anal Num 21:557–579

    MathSciNet  MATH  Google Scholar 

  9. Barles G, Souganidis PE (1998) A new approach to front propagation problems: theory and applications. Arch Ration Mech Anal 141:237–296

    MathSciNet  MATH  Google Scholar 

  10. Cabré X, Caffarelli LA (2003) Interior C 2,a regularity theory for a class of nonconvex fully nonlinear elliptic equations. J Math Pures Appl 83:573–612

    Google Scholar 

  11. Caffarelli LA (1989) Interior a priori estimates for solutions of fully non-linear equations. Ann Math 130:180–213

    MathSciNet  Google Scholar 

  12. Caffarelli LP, Cabré X (1995) Fully Nonlinear Elliptic Equations. Amer Math Soc, Providence

    Google Scholar 

  13. Caffarelli LA, Crandall MG, Kocan M, Swi?ch A (1996) On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm Pure Appl Math 49:365–397

    Google Scholar 

  14. Caffarelli LA, Souganidis PE, Wang L (2005) Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm Pure Appl Math 58:319–361

    MathSciNet  MATH  Google Scholar 

  15. Crandall MG, Lions P-L (1981) Condition d'unicité pour les solutions generalisées des équations de Hamilton–Jacobi du premier ordre. CR Acad Sci Paris Sér I Math 292:183–186

    MathSciNet  MATH  Google Scholar 

  16. Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton–Jacobi equations. Tran Amer Math Soc 277:1–42

    MathSciNet  MATH  Google Scholar 

  17. Crandall MG, Evans LC, Lions P-L (1984) Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans Amer Math Soc 282:487–435

    MathSciNet  MATH  Google Scholar 

  18. Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc 27:1–67

    MathSciNet  MATH  Google Scholar 

  19. Evans LC (1992) Periodic homogenization of certain fully nonlinear partial differential equations. Proc Roy Soc Edinb 120:245–265

    MATH  Google Scholar 

  20. Fleming WH, Soner HM (1993) Controlled Markov Processes and Viscosity Solutions. Springer, Berlin

    MATH  Google Scholar 

  21. Friedman A, Souganidis PE (1986) Blow-up of solutions of Hamilton–Jacobi equations. Comm Partial Differ Equ 11:397–443

    MathSciNet  MATH  Google Scholar 

  22. Giga Y (2002) Viscosity solutions with shocks. Comm Pure Appl Math 55:431–480

    MathSciNet  MATH  Google Scholar 

  23. Giga Y (2006) Surface Evolutions Equations: A Level Set Approach. Birkäuser, Basel

    Google Scholar 

  24. Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer, New York

    MATH  Google Scholar 

  25. Gozzi F, Sritharan SS, Swi?ch A (2005) Bellman equations associated to optimal control of stochastic Navier–Stokes equations. Comm Pure Appl Math 58:671–700

    Google Scholar 

  26. Gutiérrez CE (2001) The Monge–Ampère Equation. Birkhäuser, Boston

    Google Scholar 

  27. Ishii H (1987) Perron's method for Hamilton–Jacobi equations. Duke Math J 55:369–384

    MathSciNet  MATH  Google Scholar 

  28. Ishii H, Koike S (1993/94) Viscosity solutions of functional-differential equations. Adv Math Sci Appl 3:191–218

    Google Scholar 

  29. Jensen R (1988) The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch Rat Mech Anal 101:1–27

    MATH  Google Scholar 

  30. Jensen R (1993) Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch Ration Mech Anal 123:51–74

    MATH  Google Scholar 

  31. Karatzas I, Shreve SE (1998) Methods of Mathematical Finance. Springer, New York

    MATH  Google Scholar 

  32. Koike S (1995) On the state constraint problem for differential games. Indiana Univ Math J 44:467–487

    MathSciNet  MATH  Google Scholar 

  33. Koike S, Swiech A (2004) Maximum principle for fully nonlinear equations via the iterated comparison function method. Math Ann 339:461–484

    Google Scholar 

  34. Lions P-L (1983) Bifurcation and optimal stochastic control. Nonlinear Anal 7:177–207

    MathSciNet  Google Scholar 

  35. Lions PL, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 326:1085–1092; 326:753–741

    MathSciNet  ADS  MATH  Google Scholar 

  36. Lions PL, Souganidis PE (2000) Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 331:783–790

    MathSciNet  ADS  MATH  Google Scholar 

  37. Lions PL, Souganidis PE (2003) Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Comm Pur Appl Math 56:1501–1524

    MathSciNet  MATH  Google Scholar 

  38. Manfredi JJ (2002) A version of the Hopf–Lax formula in the Heisenberg group. Comm Partial Differ Equ 27:1139–1159

    MathSciNet  MATH  Google Scholar 

  39. Nadirashvili N, Vladut S (2007) Nonclassical solutions of fullynonlinear elliptic equations. Geom Funct Anal 17:1283–1296

    Google Scholar 

  40. Nadirashvili N, Vladut S (2008) Singular viscosity solutions to fullynonlinear elliptic equations. J Math Pures Appl 89(9):107–113

    Google Scholar 

  41. Savin O (2005) C 1 regularity for infinity harmonic functions in two dimensions. Arch Ration Mech Anal 176:351–361

    MathSciNet  MATH  Google Scholar 

  42. Soner MH (1986) Optimal control with state-space constraint I. SIAM J Control Optim 24:552–562

    MathSciNet  ADS  MATH  Google Scholar 

Books and Reviews

  1. Bellman R (1957) Dynamic Programming. Princeton University Press, Princeton

    MATH  Google Scholar 

  2. Evans LC, Gangbo W (1999) Differential Equations Methods for the Monge–Kantrovich Mass Transfer Problem. Amer Math Soc, Providence

    Google Scholar 

  3. Fleming WH, Rishel R (1975) Deterministic and Stochastic Optimal Control. Springer, New York

    MATH  Google Scholar 

  4. Koike S (2004) A Beginner's Guide to the Theory of Viscosity Solutions. Math Soc Japan, Tokyo. Corrections: http://133.38.11.17/lab.jp/skoike/correction.pdf

  5. Lions P-L (1982) Generalized Solutions of Hamilton–Jacobi Equations. Pitman, Boston

    MATH  Google Scholar 

  6. Maugeri A, Palagachev DK, Softova LG (2000) Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley, Berlin

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saitama University, Saitama, Japan

    Shigeaki Koike

Authors
  1. Shigeaki Koike
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Koike, S. (2009). Non-linear Partial Differential Equations, Viscosity Solution Method in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_366

Download citation

Publish with us

Policies and ethics

Search

Navigation