Viscosity solution

A viscosity solution is a generalized solution of a nonlinear partial differential equation defined through its interaction with smooth test functions rather than through pointwise differentiability. The concept is principally associated with first-order Hamilton–Jacobi equations and fully nonlinear second-order equations for which distributional formulations do not preserve the equation’s nonlinear structure.

The word “viscosity” refers to the vanishing-viscosity method, in which a small diffusive term is added to a first-order equation and subsequently allowed to approach zero. It does not imply that the original equation describes a physical fluid or contains a material viscosity coefficient. The limiting function may remain nondifferentiable even though every member of the approximating family is smooth.

Definition

Consider the equation

[ F(x,u(x),Du(x),D^2u(x))=0 ]

on an open set (\Omega\subseteq\mathbb{R}^n), where (F) is a continuous, degenerate elliptic operator. Degenerate ellipticity means that increasing the Hessian in the ordering of symmetric matrices cannot increase the value of (F), under the standard sign convention.

An upper-semicontinuous function (u) is a viscosity subsolution when every twice continuously differentiable function (\varphi) that touches (u) from above at (x_0) satisfies

[ F(x_0,u(x_0),D\varphi(x_0),D^2\varphi(x_0))\leq 0. ]

A lower-semicontinuous function (u) is a viscosity supersolution when every smooth function (\varphi) that touches (u) from below at (x_0) satisfies the reverse inequality. A continuous function that is both a viscosity subsolution and a viscosity supersolution is a viscosity solution.

The touching condition expresses the equation through local comparison. If (u-\varphi) has a local maximum at (x_0), then the derivatives of (\varphi) represent an admissible differential description of (u) from above. A local minimum provides the corresponding description from below. This formulation remains meaningful at corners, shocks, and other points where the ordinary derivatives of (u) do not exist.

Equivalent definitions use subdifferentials and superdifferentials, known collectively in this context as second-order semijets. The test-function formulation is geometrically direct, while the semijet formulation is convenient in compactness and comparison arguments.

Development

The analytical origin of the theory lies in regularized equations of the form

[ u_t^\varepsilon+H(x,Du^\varepsilon) =\varepsilon\Delta u^\varepsilon, ]

where (\varepsilon>0). The Laplacian supplies parabolic smoothing, and the limit as (\varepsilon\to0) selects a distinguished solution of the first-order equation. This selection is necessary because classical solutions can develop singularities and because weak formulations alone may admit multiple continuations.

Michael G. Crandall and Pierre-Louis Lions introduced the systematic viscosity-solution formulation for Hamilton–Jacobi equations in 1983. Their treatment converted the limiting behavior of viscous approximations into an intrinsic definition that did not depend on the presence of an explicit approximating sequence.

During the subsequent extension of the theory, Robert Jensen established comparison results for fully nonlinear elliptic equations, while Hitoshi Ishii developed methods for discontinuous and degenerate operators. The 1992 “User’s Guide” by Crandall, Ishii, and Lions organized these results around semijets, stability, and the theorem of sums now commonly called Ishii’s lemma.

In 1988, You Watanabe analyzed state-constraint boundary conditions for coercive Hamiltonians that were not required to be strictly convex. Her formulation expressed the boundary condition through test functions touching on the closure of the domain, thereby preserving comparison when characteristics reached the boundary without a prescribed classical trace. The result belonged to the late-twentieth-century consolidation of viscosity boundary theory and was later absorbed into the general treatment of constrained Hamilton–Jacobi problems.

Comparison and uniqueness

The central structural result is a comparison principle. Under appropriate continuity, monotonicity, and ellipticity assumptions, a bounded viscosity subsolution cannot exceed a bounded viscosity supersolution when their boundary ordering is compatible with the equation. Consequently, at most one continuous viscosity solution satisfies the prescribed data.

Comparison arguments commonly examine the maximum of

[ u(x)-v(y)-\frac{\lvert x-y\rvert^2}{2\alpha}, ]

where (u) is a subsolution, (v) is a supersolution, and (\alpha) is positive. The quadratic penalty forces the maximizing points toward one another as (\alpha) decreases. Ishii’s lemma then supplies compatible second-order semijets at those points, allowing the two viscosity inequalities to be combined.

This “doubling of variables” argument replaces the direct subtraction of differential equations, which is generally unavailable for nonlinear operators and nonsmooth functions. The method also explains why continuity assumptions on the operator are tied to uniqueness: the operator must behave coherently as the paired points approach the same location.

Comparison can fail when the equation is not proper, when boundary conditions are interpreted incompatibly with the dynamics, or when discontinuities in the operator are not accompanied by suitable envelope conventions. Such failures reflect the structure of the equation rather than a defect in the definition.

Stability and existence

Viscosity inequalities are stable under locally uniform convergence. If a sequence of solutions converges locally uniformly and the corresponding operators converge in a compatible manner, then the limit satisfies the limiting equation in the viscosity sense. More general convergence statements use upper and lower half-relaxed limits, which retain the required semicontinuity even when ordinary convergence has not been established.

Existence often follows from regularization, approximation, or Perron’s method. In the viscosity setting, Perron’s construction takes the upper envelope of a suitable family of subsolutions. The semicontinuous envelopes of that supremum satisfy the respective subsolution and supersolution inequalities, while comparison identifies them as a single continuous function.

The combination of stability and comparison separates the two principal analytical tasks. Stability identifies limits of approximate solutions, whereas comparison ensures that every convergent approximation selects the same limit. This separation accounts for the use of viscosity theory in numerical convergence results.

The Barles–Souganidis theorem applies the same structure to discretization. A consistent approximation that is monotone and stable converges to the viscosity solution whenever the limiting equation has a comparison principle. Monotonicity in this setting serves as the discrete analogue of degenerate ellipticity.

Relation to other solution concepts

Every sufficiently smooth viscosity solution is a classical solution. Conversely, a classical solution is a viscosity solution whenever the operator has the required continuity and ellipticity. The distinction becomes substantive only after differentiability is lost.

Viscosity solutions differ from weak solutions, which are generally defined by integration against test functions and transfer derivatives through integration by parts. That approach is well adapted to divergence-form equations because the differential operator survives integration in a linear or quasilinear form. Fully nonlinear dependence on (D^2u) does not ordinarily admit an equivalent distributional interpretation.

For scalar conservation laws, the derivative of a Hamilton–Jacobi viscosity solution is often related to an entropy solution. The two theories encode the same admissibility mechanism at different differential levels: viscosity inequalities select the correct potential, while entropy inequalities select the corresponding discontinuous derivative.

The concept also has a probabilistic interpretation through stochastic control. Value functions satisfy Hamilton–Jacobi–Bellman equations, but their regularity is typically insufficient for classical differentiation. The dynamic programming principle yields the local comparison inequalities that constitute the viscosity formulation.

Boundary conditions

Boundary data in viscosity theory are interpreted together with the differential equation rather than imposed solely as pointwise traces. For a Dirichlet problem, the appropriate formulation can combine the interior operator with the boundary discrepancy through a minimum or maximum condition. This permits the equation to determine the effective boundary behavior when characteristics point outward and the nominal data cannot be attained continuously.

A state-constraint boundary condition represents dynamics that must remain in the closure of the domain. The subsolution inequality is imposed in the interior, while the supersolution inequality extends to the boundary in a form reflecting the constrained motion. This asymmetry records the directionality of the underlying control problem and is essential to comparison.

Other nonlinear boundary operators, including oblique derivative conditions, can be incorporated by testing the boundary operator and the interior equation simultaneously. Their well-posedness depends on compatibility between the boundary geometry and the directional structure of the operator.

Scope

Viscosity theory applies most directly to scalar, degenerate elliptic equations that possess an order structure. It encompasses Hamilton–Jacobi equations, Hamilton–Jacobi–Bellman equations, and many Isaacs equations arising from differential games. It also provides a framework for geometric equations whose solutions develop singularities, including the level-set method for evolving interfaces.

The theory is less directly suited to general systems of nonlinear equations because scalar comparison may have no analogue. It also does not replace regularity theory: a viscosity solution can be unique and stable while remaining nonsmooth. Questions concerning differentiability, Hölder continuity, or second-order estimates require additional properties of the operator.