02.18.05 · analysis / parabolic-hyperbolic

Viscosity Solutions of Hamilton-Jacobi Equations

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Anchor (Master): Evans §10.1-§10.2; Crandall-Evans-Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations (Trans. AMS 282, 1984); Crandall-Ishii-Lions, User's guide to viscosity solutions (Bull. AMS 27, 1992); Bardi-Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhäuser 1997); Barles, Solutions de viscosité des équations de Hamilton-Jacobi (Springer 1994)

Intuition Beginner

Some equations describe a moving front: the edge of a wildfire, the boundary of the region a robot can reach in a fixed time, the level sets of the distance to a wall. Each such front obeys a rule connecting how fast it moves to which way it is pointing. Written as an equation for the arrival-time surface, this rule is a Hamilton-Jacobi equation. The trouble is that fronts develop corners. Two parts of a fire meet and form a crease; the distance-to-a-square has ridges. At a corner the surface has no well-defined slope, so the equation, which is a statement about slopes, has nothing to say there. The classical notion of a solution simply breaks.

You might hope that any reasonable patched-together surface, smooth except along a few creases, would count as a solution. It does not work, and the reason is the heart of the subject. There are usually many ways to glue smooth pieces along corners, and most of them are physically wrong. A surface can have a downward crease, like a valley, or an upward crease, like a roof ridge. For the arrival-time problem only one kind of crease actually occurs, and a bare equation about slopes cannot tell the two apart, because at a crease there is no slope to plug in. We need an extra principle that selects the correct gluing.

The principle comes from a thought experiment. Add a tiny amount of smoothing to the equation, the same diffusion that rounds off a sharp temperature spike in a heated bar. Smoothing rounds every corner, so the smoothed problem has an honest smooth solution with a genuine slope everywhere. Now shrink the amount of smoothing toward zero. The smooth solutions settle down onto one particular cornered surface, never the wrong gluings. That limiting surface is the one nature picks, and it is called the viscosity solution, because the smoothing term plays the role of a small viscosity in a fluid.

The limit can be detected without ever doing the smoothing. The trick is to test a candidate with smooth probe surfaces. Slide a probe down until it just touches the candidate from above at one point; the probe has a slope there even if the candidate does not. Demand that the equation, fed the probe's slope, point one way. Then slide a probe up until it touches from below, and demand the equation point the other way. A surface passing both touching tests everywhere is exactly the vanishing-smoothing limit. This two-sided condition is the definition of a viscosity solution, and it turns a question about nonexistent slopes into one about the slopes of probes.

Why bother with all this care? Because the selected surface is the one that carries the real information: the true arrival time, the true reachable set, the true cost of an optimal plan. Get the gluing wrong and the answer is a fiction. Viscosity solutions are the rigorous bookkeeping that keeps the answer honest at every corner.

Visual Beginner

The picture to hold is a cornered surface being probed from above and from below by smooth surfaces that just touch it.

Read the left panel first. The black V is a surface with a downward corner at its tip. A smooth probe lowered from above can rest on that corner, touching at one point, and at that touch point the probe has a perfectly good slope even though the V does not. The supersolution test asks the equation, fed the probe's slope, to satisfy one inequality at every such touch-from-above point.

Now the right panel. A smooth probe raised from below this same V cannot reach the corner, because the corner pokes downward into the way; the probe can only kiss the surface along its straight arms, where the surface was already smooth. The subsolution test applies at those touch-from-below points. A downward corner has many places to touch from above and few from below; an upward corner is the reverse. This asymmetry is how the two tests tell a correct crease from a wrong one. The small inset shows the rounded surface a little smoothing would give; as the smoothing vanishes, those surfaces converge to the single cornered surface passing both tests.

Worked example Beginner

We watch the touching tests pick out the correct cornered solution for the simplest interesting equation. On the line, ask for a function whose slope, in absolute value, equals one everywhere: the size of the slope is one. With the value pinned to zero at both ends of the interval from minus one to one, this is the arrival-time problem for a front moving inward at unit speed from both walls. The honest answer is the distance to the nearer wall.

Step 1. Write the candidate. The distance-to-the-nearer-wall function is $u (x) = 1 - ∣ x ∣$ : it climbs from zero at the left wall up to one at the center, then descends back to zero at the right wall. Its slope is plus one on the left half and minus one on the right half, so the size of the slope is one wherever the slope exists. There is a single downward... no, an upward corner at the center, a roof ridge.

Step 2. Test from below at the ridge. Lower a smooth probe so it touches $u$ from below at the center peak. Such a probe sits under the roof and can just touch the tip; at the tip its slope is some number $p$ between minus one and plus one (it cannot be steeper than the arms it is squeezed between). The subsolution test asks that the size of $p$ be at most one. Every such $p$ has size at most one, so the test passes at the ridge.

Step 3. Test from above at the ridge. Raise a smooth probe to touch $u$ from above at the center. A probe resting on top of a roof ridge would have to bend sharply downward on both sides to stay above the descending arms, which a smooth probe cannot do at a single point while staying above. So there is no touch-from-above point at the ridge, and the supersolution test is vacuous there. Off the ridge $u$ is smooth with slope size exactly one, so both tests hold by plugging in the real slope.

Step 4. Rule out the impostor. Consider instead $v (x) = ∣ x ∣ - 1$ , the upside-down tent with a downward valley at the center. It also has slope size one off the corner and the same boundary values. But raise a smooth probe to touch $v$ from above at the valley: now the probe can rest in the valley with any slope $p$ of size up to one, and the supersolution test, which for this equation demands the size of $p$ be at least one there, fails for every $p$ of size below one. So $v$ flunks. The two-sided touching test keeps the roof and discards the valley.

What this tells us: both tent functions solve "slope size equals one" away from the corner, yet only the roof $u = 1 - ∣ x ∣$ is the viscosity solution, and it is the physically correct arrival time. The touching tests did exactly what a bare slope equation could not: they read the corner and chose the right one.

Check your understanding Beginner

Exercise (easy, multiple choice).

In the viscosity definition, why do we test a cornered candidate surface with smooth probe surfaces instead of using the candidate's own slope?

A. Because smooth probes are easier to draw. B. Because at a corner the candidate has no well-defined slope, but a probe that just touches it does. C. Because the candidate is never continuous. D. Because the equation only makes sense for straight lines.

Hint

The whole problem was that the equation talks about slopes, and corners have none.

Answer

B. Because at a corner the candidate has no well-defined slope, but a probe that just touches it does. The touching probe carries a genuine slope at the contact point, so the equation can be evaluated there even though the cornered candidate cannot supply a slope of its own. Feedback-correct: the probe lends its slope to a point where the candidate has none. Feedback-wrong: A is irrelevant; C is false, viscosity solutions are continuous; D is false, the equation handles curved smooth surfaces fine.

Formal definition Intermediate+

Throughout, $Ω \subseteq R^{n}$ is open, $T > 0$ , and the Hamiltonian $H : R^{n} \times Ω \to R$ , written $H = H (p, x)$ with $p \in R^{n}$ the gradient slot, is continuous. The unknown $u : Ω \times (0, T] \to R$ is required only to be continuous, not differentiable. The evolutionary Hamilton-Jacobi equation is $u_{t} + H (D u, x) = 0 on Ω \times (0, T],$ with $D u = D_{x} u$ the spatial gradient. The sign convention is fixed once here: $H$ enters with a plus sign, matching the optimal-control form $u_{t} + H (D u, x) = 0$ with $H (p, x) = sup_{a} {- f (a, x) \cdot p - ℓ (a, x)}$ used below. The stationary equation $H (D u, x) = 0$ is the special case with no $t$ -dependence; the eikonal equation $∣ D u ∣ = 1$ is the model instance.

Definition (test function touching). A function $φ \in C^{1} (Ω \times (0, T))$ touches $u$ from above at $(x_{0}, t_{0})$ if $u - φ$ has a local maximum at $(x_{0}, t_{0})$ with $u (x_{0}, t_{0}) = φ (x_{0}, t_{0})$ ; it touches from below if $u - φ$ has a local minimum there. At a touching point $φ$ supplies the derivatives $(φ_{t}, D φ) (x_{0}, t_{0})$ that $u$ may lack.

Definition (viscosity subsolution). A continuous $u$ is a viscosity subsolution of $u_{t} + H (D u, x) = 0$ if for every $φ \in C^{1}$ touching $u$ from above at an interior point $(x_{0}, t_{0})$ , $φ_{t} (x_{0}, t_{0}) + H (D φ (x_{0}, t_{0}), x_{0}) \leq 0.$

Definition (viscosity supersolution). A continuous $u$ is a viscosity supersolution if for every $φ \in C^{1}$ touching $u$ from below at an interior point $(x_{0}, t_{0})$ , $φ_{t} (x_{0}, t_{0}) + H (D φ (x_{0}, t_{0}), x_{0}) \geq 0.$

A viscosity solution is a continuous function that is simultaneously a sub- and a supersolution. The asymmetry — subsolutions tested from above with $\leq$ , supersolutions from below with $\geq$ — is the whole content: it is the sign-bookkeeping that survives the vanishing-viscosity limit.

Equivalent semijet formulation. The superdifferential and subdifferential of $u$ at $(x_{0}, t_{0})$ are $D^{+} u (x_{0}, t_{0}) = {(a, p) : u (x, t) \leq u (x_{0}, t_{0}) + a (t - t_{0}) + p \cdot (x - x_{0}) + o (∣ x - x_{0} ∣ + ∣ t - t_{0} ∣)},$ and $D^{-} u$ with the reversed inequality. A pair $(a, p) \in D^{+} u (x_{0}, t_{0})$ is exactly $(φ_{t}, D φ) (x_{0}, t_{0})$ for some $φ \in C^{1}$ touching from above. Thus $u$ is a subsolution iff $a + H (p, x_{0}) \leq 0$ for all $(a, p) \in D^{+} u$ , and a supersolution iff $a + H (p, x_{0}) \geq 0$ for all $(a, p) \in D^{-} u$ . The two formulations are interchangeable; the test-function version is more convenient for stability arguments, the semijet version for comparison proofs.

Definition (Legendre transform and the Lagrangian). For convex $H (\cdot, x)$ the Lagrangian is the Legendre-Fenchel conjugate $L (q, x) = H^{*} (q, x) = p \in R^{n} sup (p \cdot q - H (p, x)),$ and convex duality gives $H = L^{*}$ in return. For the spatially homogeneous $H = H (p)$ convex and coercive ( $H (p) /∣ p ∣ \to \infty$ as $∣ p ∣ \to \infty$ ), $L = H^{*}$ is finite, convex, and superlinear; this is the data of the Hopf-Lax formula.

Counterexamples to common slips Intermediate+

The sign convention is load-bearing. Replacing $H$ by $- H$ swaps the roles of sub- and supersolution and converts the roof solution of $∣ D u ∣ = 1$ into the (wrong) valley. A viscosity solution of $u_{t} + H = 0$ is not a viscosity solution of $- u_{t} - H = 0$ read with the same inequalities; one must track which side the test function touches. The convention here, $u_{t} + H (D u, x) = 0$ with subsolutions touched from above, is the one consistent with vanishing viscosity $u_{t} + H (D u, x) = ε Δ u$ , $ε ↓ 0$ .
A.e.-solutions are not viscosity solutions. Both $1 - ∣ x ∣$ and $∣ x ∣ - 1$ solve $∣ u^{'} ∣ = 1$ at every point off the corner, hence almost everywhere, yet only the first is the viscosity solution. Demanding the PDE merely a.e. (or even in the distributional sense for first-order equations) fails to select; the touching condition at the measure-zero corner set is exactly the extra information.
Test functions must be allowed to touch only at the marked point. The definition uses a local max/min of $u - φ$ , not a global one, and the value-matching $u (x_{0}, t_{0}) = φ (x_{0}, t_{0})$ can always be arranged by adding a constant. If no $C^{1}$ function touches from a given side at a point (as from below at a downward corner), the corresponding inequality is vacuous there — vacuity is permission, not failure.
Continuity is essential; differentiability is not assumed and not concluded. A viscosity solution can have a dense corner set (the distance function to a Cantor set). At any point where $u$ happens to be differentiable, both inequalities collapse to the classical equation $u_{t} + H (D u, x) = 0$ holding pointwise, which is the consistency statement.

Key theorem with proof Intermediate+

Theorem (consistency with classical solutions). Let $u \in C^{1} (Ω \times (0, T))$ . Then $u$ is a viscosity solution of $u_{t} + H (D u, x) = 0$ if and only if $u$ is a classical solution, i.e. $u_{t} (x, t) + H (D u (x, t), x) = 0$ at every point. Moreover, if $u$ is any viscosity solution and $u$ is differentiable at $(x_{0}, t_{0})$ , then the equation holds classically at $(x_{0}, t_{0})$ ^{[Crandall-Lions 1983]} ^{[Evans 2010 §10.1]}.

Proof. ( $\Rightarrow$ ) Viscosity $\Rightarrow$ classical at points of differentiability. Suppose $u$ is a viscosity solution and differentiable at $(x_{0}, t_{0})$ , with spacetime gradient $(u_{t}, D u) (x_{0}, t_{0}) =: (a, p)$ . Differentiability means $(a, p) \in D^{+} u (x_{0}, t_{0})$ and $(a, p) \in D^{-} u (x_{0}, t_{0})$ simultaneously: the first-order Taylor expansion gives both the $\leq$ and the $\geq$ envelope. The subsolution property applied to $(a, p) \in D^{+} u$ gives $a + H (p, x_{0}) \leq 0$ ; the supersolution property applied to $(a, p) \in D^{-} u$ gives $a + H (p, x_{0}) \geq 0$ . Hence $a + H (p, x_{0}) = 0$ : the equation holds classically. If $u \in C^{1}$ everywhere, this holds at every point, so $u$ is a classical solution.

( $\Leftarrow$ ) Classical $\Rightarrow$ viscosity. Let $u \in C^{1}$ satisfy $u_{t} + H (D u, x) = 0$ pointwise, and let $φ \in C^{1}$ touch $u$ from above at $(x_{0}, t_{0})$ , so $u - φ$ has a local maximum there. Both $u$ and $φ$ are $C^{1}$ , so the interior maximum of the differentiable function $u - φ$ forces its gradient to vanish: $(u_{t} - φ_{t}) (x_{0}, t_{0}) = 0$ and $(D u - D φ) (x_{0}, t_{0}) = 0$ . Therefore $φ_{t} (x_{0}, t_{0}) = u_{t} (x_{0}, t_{0})$ and $D φ (x_{0}, t_{0}) = D u (x_{0}, t_{0})$ , and $φ_{t} (x_{0}, t_{0}) + H (D φ (x_{0}, t_{0}), x_{0}) = u_{t} (x_{0}, t_{0}) + H (D u (x_{0}, t_{0}), x_{0}) = 0 \leq 0.$ So the subsolution inequality holds. The identical computation with $φ$ touching from below (an interior minimum of $u - φ$ , again forcing equal gradients) gives the supersolution inequality with the value $0 \geq 0$ . Hence $u$ is a viscosity solution. $□$

Bridge. Consistency is the foundational reason the viscosity notion deserves to be called a solution concept at all: where the candidate is smooth it reproduces the classical equation exactly, and the touching tests add content only at the corners the classical theory could not reach. This is exactly the relationship the vanishing-viscosity limit forces — a uniform limit of smooth solutions of $u_{t} + H = ε Δ u$ is smooth-where-it-can-be and cornered-where-it-must-be — and it is dual to the conservation-law picture, where Kruzhkov entropy solutions play the same selecting role for $D u$ . The central insight is that the asymmetry between the two inequalities is what a one-sided limit of diffusion leaves behind: diffusion rounds maxima down and minima up, and in the limit that bias becomes the rule that a subsolution may only be touched from above. This builds toward the comparison principle, where the doubling-of-variables argument turns the two one-sided inequalities into a global ordering and hence uniqueness, and it appears again in the Hopf-Lax formula, where the explicit minimizing-path representation is shown to satisfy precisely these touching tests, and once more in dynamic programming, where the value function of an optimal-control problem is identified as the unique viscosity solution by exactly this characterization.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Show that $u (x) = 1 - ∣ x ∣$ is a viscosity solution of $∣ u^{'} ∣ - 1 = 0$ on $(- 1, 1)$ by checking both touching tests at the corner $x = 0$ . Use the convention that a subsolution of $H (D u) = 0$ is tested from above with $H (D φ) \leq 0$ and a supersolution from below with $H (D φ) \geq 0$ .

Hint

A $C^{1}$ function $φ$ touching $u = 1 - ∣ x ∣$ from above at $0$ would need $u - φ$ to have a local max at $0$ ; check whether any such $φ$ exists. From below, $φ^{'} (0) = p$ must satisfy $∣ p ∣ \leq 1$ .

Answer

Off $0$ , $u$ is smooth with $∣ u^{'} ∣ = 1$ , so both tests reduce to the classical identity. At $x = 0$ , take the from-above test: a $C^{1}$ graph staying above the upward corner $1 - ∣ x ∣$ and equal at $0$ would itself need a downward corner at $0$ , which a $C^{1}$ function cannot have. So no $C^{1}$ $φ$ touches from above at $0$ , the subsolution inequality is vacuous, and it holds.

For the from-below test, $φ \leq 1 - ∣ x ∣$ with equality at $0$ squeezes the graph under both arms, so any touching $φ$ has $∣ φ^{'} (0) ∣ = 1$ (a slope of size below $1$ would let the graph poke above an arm). Then $H (φ^{'} (0)) = ∣ φ^{'} (0) ∣ - 1 = 0 \geq 0$ , satisfying the supersolution test. Both tests pass, so $u$ is a viscosity solution. The valley $∣ x ∣ - 1$ fails: it admits from-above probes with $∣ p ∣ < 1$ , giving $H (p) = ∣ p ∣ - 1 < 0$ where $\geq 0$ is required.

Exercise 5 (medium, symbolic).

Compute the Legendre transform $L (q) = H^{*} (q) = sup_{p} (p \cdot q - H (p))$ of the quadratic Hamiltonian $H (p) = \frac{1}{2} ∣ p ∣^{2}$ on $R^{n}$ , and write the resulting Hopf-Lax formula $u (x, t) = min_{y} {t L ((x - y) / t) + g (y)}$ explicitly.

Hint

Maximize $p \cdot q - \frac{1}{2} ∣ p ∣^{2}$ over $p$ by setting the gradient to zero: $q - p = 0$ .

Answer

The supremum of $p \cdot q - \frac{1}{2} ∣ p ∣^{2}$ is attained at $p = q$ (gradient $q - p = 0$ ), giving $L (q) = q \cdot q - \frac{1}{2} ∣ q ∣^{2} = \frac{1}{2} ∣ q ∣^{2}$ . So the kinetic Hamiltonian is self-dual, $H = L$ . The Hopf-Lax formula reads $u (x, t) = y \in R^{n} min {\frac{∣ x - y ∣ ^{2}}{2 t} + g (y)},$ the inf-convolution of $g$ with the parabola $∣ z ∣^{2} / (2 t)$ . This is the unique viscosity solution of $u_{t} + \frac{1}{2} ∣ D u ∣^{2} = 0$ , $u (\cdot, 0) = g$ — the free-particle Hamilton-Jacobi equation, whose characteristics are straight lines and whose value is the least action $\frac{1}{2} ∣ x - y ∣^{2} / t$ over straight paths from $y$ to $x$ in time $t$ , plus the initial cost $g (y)$ .

Exercise 6 (medium, symbolic).

Show that the Hopf-Lax function $u (x, t) = min_{y} {t L ((x - y) / t) + g (y)}$ satisfies the dynamic-programming identity $u (x, t) = z min {(t - s) L (\frac{x - z}{t - s}) + u (z, s)}, 0 < s < t .$

Hint

A straight path from $y$ to $x$ in time $t$ passes through the point $z = y + \frac{s}{t} (x - y)$ at time $s$ ; split the action across $[0, s]$ and $[s, t]$ and minimize in stages.

Answer

For convex $L$ , the minimizing path in the definition of $u (x, t)$ is the straight segment from $y$ to $x$ traversed at constant velocity $(x - y) / t$ ; the same velocity $L ((x - y) / t)$ is spent on each subinterval. Let $z = y + \frac{s}{t} (x - y)$ be the location at time $s$ . The action splits as $t L ((x - y) / t) = s L ((z - y) / s) + (t - s) L ((x - z) / (t - s))$ because the velocity is constant, $(z - y) / s = (x - z) / (t - s) = (x - y) / t$ . Minimizing first over $y$ (equivalently over the intermediate $z$ and then the tail), $u (x, t) = z min {(t - s) L (\frac{x - z}{t - s}) + y min [s L (\frac{z - y}{s}) + g (y)]} = z min {(t - s) L (\frac{x - z}{t - s}) + u (z, s)} .$ The inner minimization is precisely $u (z, s)$ . This semigroup (Bellman) identity is the engine behind the Hopf-Lax solution: it is the dynamic-programming principle, and differentiating it formally produces the Hamilton-Jacobi equation $u_{t} + H (D u) = 0$ .

Exercise 7 (hard, symbolic).

Prove stability under uniform convergence: if $u_{k}$ are viscosity subsolutions of $u_{t} + H (D u, x) = 0$ with $H$ continuous, and $u_{k} \to u$ locally uniformly, then $u$ is a viscosity subsolution.

Hint

Let $φ$ touch $u$ from above at $(x_{0}, t_{0})$ with a strict local max of $u - φ$ . Use uniform convergence to produce nearby local maxima $(x_{k}, t_{k})$ of $u_{k} - φ$ with $(x_{k}, t_{k}) \to (x_{0}, t_{0})$ ; apply the subsolution inequality for $u_{k}$ and pass to the limit using continuity of $H$ and $D φ$ .

Answer

Let $φ \in C^{1}$ touch $u$ from above at $(x_{0}, t_{0})$ ; by subtracting $∣ x - x_{0} ∣^{2} + ∣ t - t_{0} ∣^{2}$ from $φ$ we may assume $u - φ$ has a strict local max at $(x_{0}, t_{0})$ on a closed ball $\overset{ˉ}{B}$ , with the strict-max-preserving modification leaving $D φ (x_{0}, t_{0})$ and $φ_{t} (x_{0}, t_{0})$ unchanged. On $\overset{ˉ}{B}$ , $u_{k} - φ \to u - φ$ uniformly, so the maximizers $(x_{k}, t_{k})$ of $u_{k} - φ$ over $\overset{ˉ}{B}$ converge to the unique strict maximizer $(x_{0}, t_{0})$ of $u - φ$ (a standard consequence of uniform convergence and strictness), and for large $k$ they lie in the interior. Each $(x_{k}, t_{k})$ is an interior local max of $u_{k} - φ$ , so $φ$ touches $u_{k}$ from above there, and the subsolution property of $u_{k}$ gives $φ_{t} (x_{k}, t_{k}) + H (D φ (x_{k}, t_{k}), x_{k}) \leq 0$ . Let $k \to \infty$ : $φ \in C^{1}$ makes $φ_{t}, D φ$ continuous, $H$ is continuous, and $(x_{k}, t_{k}) \to (x_{0}, t_{0})$ , so $φ_{t} (x_{0}, t_{0}) + H (D φ (x_{0}, t_{0}), x_{0}) \leq 0$ . Thus $u$ is a subsolution. (The supersolution case is identical with min replacing max and the reversed inequality, so locally uniform limits of viscosity solutions are viscosity solutions — the property that makes vanishing viscosity work.)

Exercise 8 (hard, symbolic).

Run the doubling-of-variables estimate for the stationary comparison principle in one step. Let $u$ (subsolution) and $v$ (supersolution) of $u + H (D u, x) = 0$ on a bounded $Ω$ , $H$ uniformly continuous and satisfying $∣ H (p, x) - H (p, y) ∣ \leq ω (∣ x - y ∣ (1 + ∣ p ∣))$ , with $u \leq v$ on $\partial Ω$ . For $ε > 0$ define $Φ_{ε} (x, y) = u (x) - v (y) - \frac{1}{2 ε} ∣ x - y ∣^{2}$ and let $(x_{ε}, y_{ε})$ maximize it. Show $\frac{1}{ε} ∣ x_{ε} - y_{ε} ∣^{2} \to 0$ and deduce $sup_{Ω} (u - v) \leq 0$ .

Hint

Compare $Φ_{ε} (x_{ε}, y_{ε})$ with $Φ_{ε} (x_{ε}, x_{ε})$ to bound the penalty. At the max, $p_{ε} = (x_{ε} - y_{ε}) / ε$ lies in $D^{+} u (x_{ε})$ and in $D^{-} v (y_{ε})$ . Subtract the sub- and supersolution inequalities.

Answer

Since $(x_{ε}, y_{ε})$ maximizes $Φ_{ε}$ , $Φ_{ε} (x_{ε}, y_{ε}) \geq Φ_{ε} (x_{ε}, x_{ε}) = u (x_{ε}) - v (x_{ε})$ , giving $\frac{1}{2 ε} ∣ x_{ε} - y_{ε} ∣^{2} \leq v (x_{ε}) - v (y_{ε}) \leq ω_{v} (∣ x_{ε} - y_{ε} ∣)$ by uniform continuity of $v$ ; hence $∣ x_{ε} - y_{ε} ∣ \to 0$ and, feeding this back, $\frac{1}{ε} ∣ x_{ε} - y_{ε} ∣^{2} \to 0$ . The map $x \mapsto \frac{1}{2 ε} ∣ x - y_{ε} ∣^{2}$ is the $C^{1}$ test function touching $u$ from above at $x_{ε}$ , so $p_{ε} := (x_{ε} - y_{ε}) / ε \in D^{+} u (x_{ε})$ and the subsolution gives $u (x_{ε}) + H (p_{ε}, x_{ε}) \leq 0$ . The map $y \mapsto - \frac{1}{2 ε} ∣ x_{ε} - y ∣^{2}$ touches $v$ from below at $y_{ε}$ with the same gradient $p_{ε} \in D^{-} v (y_{ε})$ , so the supersolution gives $v (y_{ε}) + H (p_{ε}, y_{ε}) \geq 0$ . Subtract: $u (x_{ε}) - v (y_{ε}) \leq H (p_{ε}, y_{ε}) - H (p_{ε}, x_{ε}) \leq ω (∣ x_{ε} - y_{ε} ∣ (1 + ∣ p_{ε} ∣)) .$ Now $∣ p_{ε} ∣ = ∣ x_{ε} - y_{ε} ∣/ ε$ , so $∣ x_{ε} - y_{ε} ∣ (1 + ∣ p_{ε} ∣) = ∣ x_{ε} - y_{ε} ∣ + ∣ x_{ε} - y_{ε} ∣^{2} / ε \to 0$ , whence the right side $\to 0$ . Finally, for any $x \in Ω$ , $u (x) - v (x) = Φ_{ε} (x, x) \leq Φ_{ε} (x_{ε}, y_{ε}) \leq u (x_{ε}) - v (y_{ε})$ (dropping the nonnegative penalty), and the last quantity $\to\leq 0$ (its limsup is $\leq 0$ ; the boundary condition handles the case where maximizers approach $\partial Ω$ , where $u - v \leq 0$ ). Hence $sup_{Ω} (u - v) \leq 0$ : the subsolution lies below the supersolution, which is comparison, and uniqueness follows by symmetry.

Advanced results Master

The definition organizes a complete well-posedness theory: comparison gives uniqueness, Hopf-Lax and Perron give existence, vanishing viscosity gives the selection principle its name, and optimal control gives the meaning. Each is a refinement of the touching condition.

Theorem 1 (comparison principle). Let $H = H (p, x)$ be continuous with the structure condition $∣ H (p, x) - H (p, y) ∣ \leq ω (∣ x - y ∣ (1 + ∣ p ∣))$ for a modulus $ω$ , and consider $u_{t} + H (D u, x) = 0$ on $Ω \times (0, T]$ . If $u$ is a bounded upper-semicontinuous subsolution, $v$ a bounded lower-semicontinuous supersolution, and $u \leq v$ on the parabolic boundary $(Ω \times {0}) \cup (\partial Ω \times [0, T])$ , then $u \leq v$ throughout $Ω \times (0, T]$ ^{[Crandall-Lions 1983]} ^{[Crandall-Ishii-Lions 1992]}. The proof doubles the variables, maximizing $u (x, t) - v (y, s) - \frac{1}{2 ε} (∣ x - y ∣^{2} + ∣ t - s ∣^{2}) - \frac{η}{T - t}$ over $(Ω \times (0, T))^{2}$ ; the penalty $\frac{1}{2 ε} ∣ x - y ∣^{2}$ forces the spatial maximizers together at rate $∣ x_{ε} - y_{ε} ∣^{2} / ε \to 0$ (Exercise 8), the term $\frac{η}{T - t}$ keeps the maximum off the final time, and feeding the common penalty-gradient $p_{ε}$ into the sub- and supersolution inequalities and subtracting yields a contradiction unless $u \leq v$ . Uniqueness of viscosity solutions with given initial-boundary data is the immediate corollary: two solutions are each sub- and supersolution of the other, so each lies below the other.

Theorem 2 (Hopf-Lax formula). Let $H = H (p)$ be convex and superlinear ( $H (p) /∣ p ∣ \to \infty$ ), $L = H^{*}$ its Legendre transform, and $g$ Lipschitz. Then $u (x, t) = y \in R^{n} min {t L (\frac{x - y}{t}) + g (y)}$ is Lipschitz, satisfies $u (\cdot, 0) = g$ , is differentiable a.e. with $u_{t} + H (D u) = 0$ at every point of differentiability, and is the unique viscosity solution of $u_{t} + H (D u) = 0$ , $u (\cdot, 0) = g$ ^{[Hopf 1965]} ^{[Evans 2010 §10.1]}. The formula is the dynamic-programming value of the calculus-of-variations problem $in f {\int_{0}^{t} L (\dot{ξ}) d s + g (ξ (0)) : ξ (t) = x}$ , whose minimizers over the convex superlinear $L$ are straight lines (Exercise 6 gives the Bellman semigroup); the verification that $u$ passes both touching tests is where convexity of $H$ (equivalently of $L$ ) is used, exactly as convexity drove lower semicontinuity in the direct method 02.18.04.

Theorem 3 (existence by vanishing viscosity). For continuous coercive $H$ and bounded uniformly continuous $g$ , the regularized problems $u_{t}^{ε} + H (D u^{ε}, x) = ε Δ u^{ε}, u^{ε} (\cdot, 0) = g,$ are uniformly parabolic 02.13.03 and have smooth solutions $u^{ε}$ , bounded with equicontinuity estimates independent of $ε$ . By Arzelà-Ascoli a subsequence converges locally uniformly, $u^{ε} \to u$ , and the stability theorem (Exercise 7) shows the limit $u$ is a viscosity solution of $u_{t} + H (D u, x) = 0$ ^{[Crandall-Lions 1983]}. This is the construction that names the theory: the small Laplacian is the artificial viscosity, the touching-test inequalities are precisely the sign conditions that survive the limit, and comparison (Theorem 1) makes the limit independent of the subsequence. The heat-kernel smoothing of 02.13.03 is the exact mechanism: $ε Δ$ rounds corners by Gaussian convolution, and the inherited one-sided maximum-principle bias is the asymmetry built into the definition.

Theorem 4 (existence by Perron's method). Assume comparison (Theorem 1) holds and there exist a subsolution $\underline{u}$ and a supersolution $\overline{u}$ with $\underline{u} \leq \overline{u}$ and matching boundary data. Then $u (x, t) = sup {w (x, t) : w a subsolution, \underline{u} \leq w \leq \overline{u}}$ is a viscosity solution ^{[Crandall-Ishii-Lions 1992]}. The proof uses two structural facts: the pointwise supremum of subsolutions (more precisely its upper-semicontinuous envelope) is a subsolution, and a subsolution that fails to be a supersolution at some point can be bumped up there, contradicting maximality. Perron's method decouples existence from any explicit formula or regularization, requiring only comparison plus barriers; it is the standard existence route once $H$ is too general for Hopf-Lax and too irregular for clean vanishing-viscosity estimates.

Theorem 5 (optimal control and dynamic programming). Consider the controlled dynamics $\dot{ξ} = f (ξ, a)$ , $a \in A$ , with running cost $ℓ$ and terminal cost $g$ , and value function $u (x, t) = in f_{a} {\int_{0}^{t} ℓ (ξ, a) d s + g (ξ (0)) : ξ (t) = x}$ . Then $u$ is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation $u_{t} + H (D u, x) = 0$ with $H (p, x) = sup_{a \in A} {- f (x, a) \cdot p - ℓ (x, a)}$ ^{[Bardi-Capuzzo-Dolcetta 1997]} ^{[Lions 1982]}. The dynamic-programming principle, $u (x, t) = in f_{a} {\int_{s}^{t} ℓ d r + u (ξ (s), s)}$ , is the exact analogue of the Hopf-Lax semigroup (Exercise 6); it holds for merely continuous $u$ , and its infinitesimal form is the touching condition, not the classical PDE — which is why value functions, generically nondifferentiable at shocks of the optimal feedback, are viscosity but not classical solutions. This is the structural payoff: viscosity solutions are the unique solution concept under which the value function of a control problem is the solution of its HJB equation.

Synthesis. The viscosity theory is the foundational reason first-order Hamilton-Jacobi equations have a well-posed weak solution theory at all, and the entire edifice is generated by one asymmetric device: a subsolution may be touched only from above and a supersolution only from below, and the sign distinguishing the two is the trace of a vanishing diffusion. Putting these together, comparison (Theorem 1) converts the two one-sided inequalities into a global ordering by doubling the variables, and this is exactly the mechanism that delivers uniqueness; existence is then supplied three ways — explicitly by Hopf-Lax for convex $H$ (Theorem 2), constructively by vanishing viscosity for coercive $H$ (Theorem 3), and abstractly by Perron given comparison (Theorem 4) — each of which generalises a classical idea past the corner where classical solutions die.

The central insight is that the convex case is self-dual through the Legendre transform: $H$ convex makes $L = H^{*}$ a Lagrangian, the Hopf-Lax minimum is the least action over straight characteristics, and the dynamic-programming semigroup is dual to the PDE in the same way the calculus-of-variations existence theorem 02.18.04 is dual to its Euler-Lagrange equation. This builds toward the bridge from the classical Hamilton-Jacobi equation of mechanics 09.05.02, where the complete integral is built from smooth characteristics that cease to exist once they cross, to the modern theory, where the value function continues uniquely past the caustic; the characteristics of mechanics are the optimal paths of control, and where they collide the viscosity solution records the surviving minimum. This selection principle appears again in the level-set front propagation and stochastic-control equations that share one analytic backbone.

Full proof set Master

Proposition 1 (a viscosity solution differentiable at a point solves classically there). If $u$ is a viscosity solution of $u_{t} + H (D u, x) = 0$ and $u$ is differentiable at $(x_{0}, t_{0})$ , then $u_{t} (x_{0}, t_{0}) + H (D u (x_{0}, t_{0}), x_{0}) = 0$ .

Proof. Differentiability gives a first-order expansion $u (x, t) = u (x_{0}, t_{0}) + a (t - t_{0}) + p \cdot (x - x_{0}) + o (∣ x - x_{0} ∣ + ∣ t - t_{0} ∣)$ with $(a, p) = (u_{t}, D u) (x_{0}, t_{0})$ . The " $\leq$ " reading of this expansion is the statement $(a, p) \in D^{+} u (x_{0}, t_{0})$ , and the " $\geq$ " reading is $(a, p) \in D^{-} u (x_{0}, t_{0})$ ; both hold because the remainder is $o (\cdot)$ , hence both above and below the affine part to first order. The subsolution property applied to $(a, p) \in D^{+} u$ yields $a + H (p, x_{0}) \leq 0$ , and the supersolution property applied to $(a, p) \in D^{-} u$ yields $a + H (p, x_{0}) \geq 0$ . Together $a + H (p, x_{0}) = 0$ . $□$

Proposition 2 (uniqueness from comparison). If $u_{1}, u_{2}$ are viscosity solutions of $u_{t} + H (D u, x) = 0$ on $Ω \times (0, T]$ with the same parabolic-boundary data and $H$ satisfies the comparison hypotheses (Theorem 1), then $u_{1} = u_{2}$ .

Proof. $u_{1}$ is a subsolution and $u_{2}$ a supersolution with $u_{1} \leq u_{2}$ on the parabolic boundary (equal data), so comparison gives $u_{1} \leq u_{2}$ on $Ω \times (0, T]$ . Exchanging the roles, $u_{2}$ is a subsolution and $u_{1}$ a supersolution with the same boundary ordering $u_{2} \leq u_{1}$ , so comparison gives $u_{2} \leq u_{1}$ . Hence $u_{1} = u_{2}$ . $□$

Proposition 3 (Hopf-Lax is a subsolution, $H$ convex). Let $H = H (p)$ be convex superlinear, $L = H^{*}$ , $g$ Lipschitz, and $u (x, t) = min_{y} {t L ((x - y) / t) + g (y)}$ . Then at every point $(x_{0}, t_{0})$ where $u$ is differentiable, $u_{t} (x_{0}, t_{0}) + H (D u (x_{0}, t_{0})) \leq 0$ , and the differentiable-a.e. Lipschitz function $u$ is a viscosity subsolution.

Proof. The dynamic-programming identity (Exercise 6) gives, for $h > 0$ small, $u (x_{0}, t_{0}) \leq h L (q) + u (x_{0} - h q, t_{0} - h)$ for every fixed $q \in R^{n}$ (take the straight path of velocity $q$ arriving at $x_{0}$ ). Rearranging, $\frac{u ( x _{0} , t _{0} ) - u ( x _{0} - h q , t _{0} - h )}{h} \leq L (q) .$ At a point of differentiability the left side tends, as $h ↓ 0$ , to $u_{t} (x_{0}, t_{0}) + q \cdot D u (x_{0}, t_{0})$ . Hence $u_{t} (x_{0}, t_{0}) + q \cdot D u (x_{0}, t_{0}) \leq L (q)$ for every $q$ , so $u_{t} (x_{0}, t_{0}) + q sup (q \cdot D u (x_{0}, t_{0}) - L (q)) \leq 0.$ The supremum is $L^{*} (D u (x_{0}, t_{0})) = H (D u (x_{0}, t_{0}))$ by convex duality ( $H = L^{*}$ since $H$ is convex and lower-semicontinuous). Thus $u_{t} + H (D u) \leq 0$ at points of differentiability. The touching-from-above test then holds: where $φ$ touches $u$ from above at a differentiability point the gradients agree (Proposition 1 mechanism); a standard argument upgrades the a.e. inequality to all touching points because at a from-above contact $D φ$ is a supergradient of $u$ , and the inf-convolution structure makes every superdifferential element a limit of gradients at differentiability points. $□$

Proposition 4 (Hopf-Lax is a supersolution, $H$ convex). Under the hypotheses of Proposition 3, $u$ is also a viscosity supersolution, hence the viscosity solution.

Proof. The dynamic-programming identity gives the reverse selection: $u (x_{0}, t_{0}) = min_{z} {h L ((x_{0} - z) / h) + u (z, t_{0} - h)}$ , and the minimizing $z = z_{h}$ satisfies $(x_{0} - z_{h}) / h \to q_{*}$ where $q_{*}$ achieves $H (D u) = q_{*} \cdot D u - L (q_{*})$ at a differentiability point. Let $φ$ touch $u$ from below at $(x_{0}, t_{0})$ , so $φ \leq u$ near $(x_{0}, t_{0})$ with equality there. Using the minimizing path, $φ (x_{0}, t_{0}) = u (x_{0}, t_{0}) = h L (\frac{x _{0} - z _{h}}{h}) + u (z_{h}, t_{0} - h) \geq h L (\frac{x _{0} - z _{h}}{h}) + φ (z_{h}, t_{0} - h) .$ Rearrange and divide by $h$ : $\frac{φ ( x _{0} , t _{0} ) - φ ( z _{h} , t _{0} - h )}{h} \geq L (\frac{x _{0} - z _{h}}{h})$ . As $h ↓ 0$ with $(x_{0} - z_{h}) / h \to q_{*}$ , the left side tends to $φ_{t} (x_{0}, t_{0}) + q_{*} \cdot D φ (x_{0}, t_{0})$ and the right to $L (q_{*})$ , so $φ_{t} + q_{*} \cdot D φ \geq L (q_{*})$ , giving $φ_{t} + (q_{*} \cdot D φ - L (q_{*})) \geq 0$ . Since $q_{*} \cdot D φ - L (q_{*}) \leq sup_{q} (q \cdot D φ - L (q)) = H (D φ)$ would be the wrong direction, one uses instead that $q_{*}$ is the maximizer for $D u = D φ$ at the contact (the from-below touching forces $D φ (x_{0}, t_{0}) \in D^{-} u$ , and for the inf-convolution $D^{-} u$ is a single gradient where the minimizer is unique), giving $q_{*} \cdot D φ - L (q_{*}) = H (D φ)$ . Hence $φ_{t} + H (D φ) \geq 0$ . With Proposition 3, $u$ is a viscosity solution; uniqueness (Proposition 2) identifies it as the solution. $□$

Proposition 5 (vanishing-viscosity limits are viscosity subsolutions). Let $u^{ε}$ solve $u_{t}^{ε} + H (D u^{ε}, x) = ε Δ u^{ε}$ classically and $u^{ε} \to u$ locally uniformly as $ε ↓ 0$ . Then $u$ is a viscosity subsolution of $u_{t} + H (D u, x) = 0$ .

Proof. Let $φ \in C^{1}$ touch $u$ from above at $(x_{0}, t_{0})$ , with (after the strict-max modification of Exercise 7, taking $φ \in C^{2}$ which is available by mollifying without changing first derivatives at $x_{0}$ ) a strict local max of $u - φ$ . Local uniform convergence produces interior maximizers $(x_{ε}, t_{ε})$ of $u^{ε} - φ$ with $(x_{ε}, t_{ε}) \to (x_{0}, t_{0})$ . At such an interior max of the smooth $u^{ε} - φ$ : $\partial_{t} (u^{ε} - φ) = 0$ , $D (u^{ε} - φ) = 0$ , and the second-order condition $Δ (u^{ε} - φ) \leq 0$ , i.e. $Δ u^{ε} (x_{ε}, t_{ε}) \leq Δ φ (x_{ε}, t_{ε})$ . Therefore $u_{t}^{ε} = φ_{t}$ and $D u^{ε} = D φ$ at $(x_{ε}, t_{ε})$ , and the equation gives $φ_{t} (x_{ε}, t_{ε}) + H (D φ (x_{ε}, t_{ε}), x_{ε}) = ε Δ u^{ε} (x_{ε}, t_{ε}) \leq ε Δ φ (x_{ε}, t_{ε}) .$ Let $ε ↓ 0$ : the right side $\to 0$ since $Δ φ$ is bounded near $(x_{0}, t_{0})$ , and continuity of $φ_{t}, D φ, H$ gives $φ_{t} (x_{0}, t_{0}) + H (D φ (x_{0}, t_{0}), x_{0}) \leq 0$ . So $u$ is a subsolution. (The supersolution direction uses interior minima of $u^{ε} - φ$ , where $Δ u^{ε} \geq Δ φ$ , reversing the inequality.) $□$

Connections Master

The vanishing-viscosity construction of existence (Theorem 3) runs on the parabolic-smoothing machinery of 02.13.03: the regularized equation $u_{t}^{ε} + H (D u^{ε}) = ε Δ u^{ε}$ is a perturbation of the heat equation, its solvability and the gradient bounds come from the heat-kernel and maximum-principle theory there, and the one-sided maximum-principle bias inherited in the limit is precisely the asymmetry of the sub/supersolution definitions. That unit's smoothing-operator viewpoint is the analytic origin of the word "viscosity."
The Hopf-Lax formula (Theorem 2) is the dynamic-programming twin of the direct method of 02.18.04: there a variational integral is minimized over a Sobolev class by coercivity plus weak lower semicontinuity, here the action $\int_{0}^{t} L (\dot{ξ}) d s + g (ξ (0))$ is minimized over paths, with the same Legendre-duality between the convex Lagrangian $L$ and the Hamiltonian $H = L^{*}$ converting the minimization into the PDE. Convexity plays the identical structural role — it is what makes the minimizer (a straight characteristic) the solution, just as it made the weak limit a minimizer there.
The viscosity solution is the modern continuation past the caustic of the classical Hamilton-Jacobi equation of mechanics 09.05.02: there Hamilton's principal function is built from a complete integral along non-crossing characteristics and ceases to be single-valued once characteristics focus, whereas the viscosity solution continues uniquely as the value function, selecting at each point the surviving least-action minimum. The characteristics of 09.05.02 are exactly the optimal trajectories of the control interpretation (Theorem 5), and the viscosity solution is what mechanics' multivalued $S$ collapses to under the minimum.
The comparison principle's doubling-of-variables and the resulting uniqueness feed forward into the level-set method for front propagation and into second-order theory: the same touching-test definition extends verbatim to fully nonlinear degenerate-elliptic equations $F (D^{2} u, D u, u, x) = 0$ , where the Crandall-Ishii-Lions theory governs Hamilton-Jacobi-Bellman-Isaacs equations of stochastic control and differential games, the analytic backbone shared with the elliptic regularity discussion of 02.13.03.

Historical & philosophical context Master

The need for a generalized solution of $u_{t} + H (D u) = 0$ was recognized through the theory of conservation laws, to which Hamilton-Jacobi equations are linked by differentiation: in one space dimension, if $u$ solves the Hamilton-Jacobi equation then $v = u_{x}$ solves the scalar conservation law $v_{t} + H (v)_{x} = 0$ . Peter Lax, in his 1957 work on hyperbolic systems ^{[Lax 1957]}, and Eberhard Hopf, in his 1965 paper on generalized solutions of first-order equations ^{[Hopf 1965]}, gave explicit formulas — the Lax-Oleinik and Hopf representations — for the convex case, isolating the inf/sup-convolution structure now called Hopf-Lax. Stanislav Kruzhkov's 1975 eikonal-type results ^{[Kruzhkov 1975]} supplied a definition through doubling techniques for the convex Hamilton-Jacobi case, and approaches via semiconcavity captured the convex theory, but a definition valid for non-convex $H$ , stable under limits, and yielding uniqueness in full generality remained open.

That definition arrived in the 1983 Transactions of the American Mathematical Society paper of Michael Crandall and Pierre-Louis Lions ^{[Crandall-Lions 1983]}, which introduced viscosity solutions through the test-function inequalities, proved consistency and the vanishing-viscosity convergence that named the concept, and established uniqueness by comparison. The companion 1984 paper of Crandall, Lawrence Evans, and Lions ^{[Crandall-Evans-Lions 1984]} gave the equivalent sub/superdifferential formulation and streamlined the proofs. Pierre-Louis Lions's 1982 monograph ^{[Lions 1982]} connected the theory to optimal control and the Hamilton-Jacobi-Bellman equation, identifying value functions as viscosity solutions, work cited in his 1994 Fields Medal. The extension to second-order fully nonlinear equations, with the doubling-of-variables comparison technology in its mature form, was codified in the 1992 Crandall-Ishii-Lions "User's Guide" ^{[Crandall-Ishii-Lions 1992]}, the standard reference. The optimal-control synthesis was developed in the monographs of Bardi and Capuzzo-Dolcetta ^{[Bardi-Capuzzo-Dolcetta 1997]} and Barles ^{[Barles 1994]}.

Bibliography Master

@article{CrandallLions1983,
  author  = {Crandall, Michael G. and Lions, Pierre-Louis},
  title   = {Viscosity solutions of Hamilton-Jacobi equations},
  journal = {Transactions of the American Mathematical Society},
  volume  = {277},
  number  = {1},
  year    = {1983},
  pages   = {1--42}
}

@article{CrandallEvansLions1984,
  author  = {Crandall, Michael G. and Evans, Lawrence C. and Lions, Pierre-Louis},
  title   = {Some properties of viscosity solutions of Hamilton-Jacobi equations},
  journal = {Transactions of the American Mathematical Society},
  volume  = {282},
  number  = {2},
  year    = {1984},
  pages   = {487--502}
}

@article{CrandallIshiiLions1992,
  author  = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis},
  title   = {User's guide to viscosity solutions of second order partial differential equations},
  journal = {Bulletin of the American Mathematical Society (New Series)},
  volume  = {27},
  number  = {1},
  year    = {1992},
  pages   = {1--67}
}

@article{Hopf1965,
  author  = {Hopf, Eberhard},
  title   = {Generalized solutions of non-linear equations of first order},
  journal = {Journal of Mathematics and Mechanics},
  volume  = {14},
  year    = {1965},
  pages   = {951--973}
}

@article{Lax1957,
  author  = {Lax, Peter D.},
  title   = {Hyperbolic systems of conservation laws II},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {10},
  year    = {1957},
  pages   = {537--566}
}

@article{Kruzhkov1975,
  author  = {Kru{\v{z}}kov, Stanislav N.},
  title   = {Generalized solutions of the Hamilton-Jacobi equations of eikonal type},
  journal = {Matematicheskii Sbornik (N.S.)},
  volume  = {98(140)},
  year    = {1975},
  pages   = {450--493}
}

@book{Lions1982,
  author    = {Lions, Pierre-Louis},
  title     = {Generalized Solutions of Hamilton-Jacobi Equations},
  series    = {Research Notes in Mathematics},
  volume    = {69},
  publisher = {Pitman},
  year      = {1982}
}

@book{BardiCapuzzoDolcetta1997,
  author    = {Bardi, Martino and Capuzzo-Dolcetta, Italo},
  title     = {Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations},
  series    = {Systems \& Control: Foundations \& Applications},
  publisher = {Birkh\"auser},
  year      = {1997}
}

@book{Barles1994,
  author    = {Barles, Guy},
  title     = {Solutions de viscosit\'e des \'equations de Hamilton-Jacobi},
  series    = {Math\'ematiques \& Applications},
  volume    = {17},
  publisher = {Springer},
  year      = {1994}
}

@book{Evans2010,
  author    = {Evans, Lawrence C.},
  title     = {Partial Differential Equations},
  edition   = {2},
  series    = {Graduate Studies in Mathematics},
  volume    = {19},
  publisher = {American Mathematical Society},
  year      = {2010}
}

Prerequisites

02.18.04
02.13.03
09.05.02

Tier anchors

beginner: Strogatz-style picture of a wavefront that keeps moving even after it develops a corner; the modern slogan 'the right weak solution is the one a tiny bit of diffusion would have chosen', recast as touching a candidate from above and below with smooth test functions
intermediate: Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §10.1 (definitions, consistency, Hopf-Lax) and §10.2 (uniqueness via doubling of variables); Crandall-Lions, Viscosity solutions of Hamilton-Jacobi equations (Trans. AMS 277, 1983)
master: Evans §10.1-§10.2; Crandall-Evans-Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations (Trans. AMS 282, 1984); Crandall-Ishii-Lions, User's guide to viscosity solutions (Bull. AMS 27, 1992); Bardi-Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhäuser 1997); Barles, Solutions de viscosité des équations de Hamilton-Jacobi (Springer 1994)

References

Crandall-Lions — Viscosity solutions of Hamilton-Jacobi equations · Transactions of the American Mathematical Society 277 (1983), 1-42
Crandall-Evans-Lions — Some properties of viscosity solutions of Hamilton-Jacobi equations · Transactions of the American Mathematical Society 282 (1984), 487-502
Crandall-Ishii-Lions — User's guide to viscosity solutions of second order partial differential equations · Bulletin of the American Mathematical Society (N.S.) 27 (1992), 1-67
Hopf — Generalized solutions of non-linear equations of first order · Journal of Mathematics and Mechanics 14 (1965), 951-973
Lax — Hyperbolic systems of conservation laws II · Communications on Pure and Applied Mathematics 10 (1957), 537-566
Kruzhkov — Generalized solutions of the Hamilton-Jacobi equations of eikonal type · Matematicheskii Sbornik (N.S.) 98(140) (1975), 450-493
Lions — Generalized Solutions of Hamilton-Jacobi Equations · Pitman Research Notes in Mathematics 69 (1982)
Bardi-Capuzzo-Dolcetta — Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations · Birkhäuser, Systems & Control (1997)
Barles — Solutions de viscosité des équations de Hamilton-Jacobi · Springer, Mathématiques & Applications 17 (1994)
Evans — Partial Differential Equations, 2e · AMS Graduate Studies in Mathematics 19 (2010), §10.1-§10.2

Estimated time

beginner: 25m
intermediate: 65m
master: 105m