2.1 An introduction to the viscosity theory
2.1.1 The continuous case
In this subsection we introduce the classic theory of viscosity solutions for Hamilton-Jacobi first-order equations; so we look at continuous functions as candidate solutions.
This theory has been introduced by M.G. Crandall and P.L. Lions in [46, 47] and then developed by H. Ishii, G. Barles, L.C. Evans and many others. In particular, we suggest to see [10, 14, 54, 84] for a very useful and clear treatment of the viscosity theory for the first-order equations.
We start giving the definition of viscosity subsolutions and viscosity supersolutions.
Definition 2.1. Let Ω ⊂ Rn an open set, u ∈ C(Ω) is a viscosity subsolution [ v. supersolution] of a first-order equation
H(x, u, Du) = 0, x ∈ Ω, (2.3)
at a point x0 ∈ Ω, if and only if, for any test-function ϕ ∈ C1(Ω) such that u − ϕ has a local maximum [minimum] at x0 ∈ Ω, then
H(x0, ϕ(x0), Dϕ(x0)) ≤ 0 [≥ 0]. (2.4) A continuous function u (defined on Ω) is a viscosity solution at a point x0, if and only if, it is as a v. subsolution as a v. supersolution, at x0.
Moreover, it is a viscosity solution in an open set Ω, if and only if, it is a viscosity solution at any x ∈ Ω.
Remark 2.2. If u(x) − ϕ(x) has a local maximum [minimum] at x0, we can assume that the maximum [minimum] is zero and moreover, adding some quadratic perturbation, we can suppose also that it is strict, i.e. locally unique (see [43], Lemma 1.1).
Example 2.3. Set I = (−1, 1), we want to explicitly show that u(x) = |x| is a v. solution of the eikonal equation
−|Du(x)| + 1 = 0, x ∈ I. (2.5)
Moreover, it is unique from general uniqueness results (see [10]).
Remark that, whenever x0 6= 0, x0 is an extremal-point for u − ϕ where the function is differentiable, then D(ϕ − u)(x0) = 0, that is
Dϕ(x0) = Du(x0) = 1.
The problem is at the point x0 = 0, since there Du doesn’t exist.
So let ϕ ∈ C1(I) such that u − ϕ attends a local minimum at 0.
Using Remark 2.2, we can assume u(0) = ϕ(0). So there exists r > 0 such that, for any x ∈ Br(0), it holds
0 ≤ ϕ(x) − ϕ(0) ≤ u(x) − u(0) = |x|, then
0 ≤ ϕ(x) − ϕ(0)
|x| ≤ 1.
Passing to the limit, as x → 0+, we can conclude that |Dϕ(0)| ≤ 1 and so the supersolution-condition is satisfied. While, if we assume that u − ϕ attends a local maximum (equal to 0) at the point 0, then we find 1 ≤ Dϕ(0) ≤ 0, that is impossible. So there is not any function ϕ where we have to verify the subsolution-condition. Therefore u(x) = |x| is a v. solution of (2.5).
Since the equation (2.5) doesn’t depend explicitly on u, any translated function of u is still a v. solution of a such equation.
In the same way, one can verify that v(x) = −u(x) = −|x| is a v.
solution of
|Du(x)| − 1 = 0, x ∈ I, (2.6)
while u(x) = |x| doesn’t satisfy, in the viscosity sense, equation (2.6). In fact, if we look at ϕ(x) = −x2, then 0 is a local minimum-point for |x| − (−x2).
Nevertheless |Dϕ(0)| = 0, so the subsolution-condition is not satisfied.
In the next propositions we show that the definition of viscosity solu-tions is consistent. In fact, C1-viscosity solutions are also classic solutions and locally Lipschitz continuous viscosity solutions are almost everywhere solutions, too.
Proposition 2.4. Let Ω an open set and u ∈ C1(Ω). Then u is a viscosity solution, if and only if, it is a classical solution.
Proof. To prove that C1-viscosity solutions are also classical solutions is triv-ial. In fact, we can choose, as test-function, u itself and, therefore, it satisfies the equation in the classical sense.
In order to get the inverse implication, we can observe that, for any test-function ϕ, if x0 is an extremal point of u − ϕ where both of the functions are differentiable, then Du(x0) = Dϕ(x0). So if u satisfies the equation, any test-function does so and, therefore, u is a viscosity solution.
Previous proposition can be written punctually, as follows.
Remark 2.5. If u is a viscosity solution at a point where it is differentiable, then it satisfies the equation in the classical sense at this point.
Proposition 2.6. Let Ω an open set and u a viscosity solution in Ω.
If u ∈ Lip(Ω), then u is also an almost everywhere solution. The reverse implication is not true.
Proof. The result follows immediately by Remark 2.5 and the Rademacher’s Theorem.
To show that the reverse implication does not hold, it is sufficient to look at the eikonal equation |Du| = 1 in (−1, 1). We know that all the functions (2.2) are almost everywhere solutions of the vanishing Dirichlet problem (2.1), while only u(x) = −|x| + 1 solves this, in the viscosity sense.
The previous proposition means that viscosity solutions select, in some optimal sense, the best solution among all the almost everywhere solutions.
In particular, we can note that, in the eikonal example (2.1), the v. solution is the biggest among all the everywhere solutions.
The next result generalizes another fact, seen in Example 2.3.
Proposition 2.7. If u is a v. solution of the Hamilton-Jacobi equation (2.3), then v(x) = −u(x) is a v. solution of −H(x, −v, −Dv) = 0.
The proof of this result is trivial, so we don’t recall it.
Note that, in general, v(x) = −u(x) is not a v. solution of H(x, v(x), Dv(x)) = 0, since the equation is nonlinear.
Remark 2.8. In [47] (Corollary 1.8) one can find a result concerning non-linear changes of the solutions. More precisely, there is showed that, for any Φ ∈ C1(R), with Φ0 > 0 and Φ(R) = R, if u is a v. subsolution [v.
supersolution] of the equation (2.3), then v = Φ(u) is a v. subsolution [v.
supersolution] of H(x, Φ−1(v), (Φ−1)0(v)Dv) = 0.
Now we want to investigate the stability-properties of the v. solutions.
The key is the stability of the maximum-points and the minimum-points.
Proposition 2.9. Let Ω an open set, u ∈ C(Ω) and ϕ ∈ C1(Ω). Assume that x0 is a maximum-point [minimum-point] of u − ϕ in Br(x0). If un ∈ C(Ω) are such that
n→∞lim un(x) = u(x),
for any x ∈ Ω, let xn a sequence of minimum-points for un− ϕ in Br(x0), then xn→ x, as n → ∞, and moreover
n→∞lim un(xn) = u(x0).
The proof of this result and the next one are easy and so we will show these directly in the semicontinuous case (Proposition 2.19 and Proposition 2.20).
Proposition 2.10. Let Hn and H continuous in all the variables and such that Hn → H, as n → ∞. Let un v. solutions of Hn(x, un, Dun) = 0 and
u(x) = lim
n→+∞un(x).
If u is a continuous function, then u is a v. solution of H(x, u, Du) = 0.
Other key-properties of v. subsolutions and v. supersolutions are their behavior w.r.t. the operations of minimum and maximum, respectively.
Proposition 2.11. Let u, v ∈ C(Ω) v. subsolutions [v. supersolutions], then u ∨ v [u ∧ v] is a v. subsolution [v. supersolution] of the same equation.
The previous proof is also trivial. One can find it and the non-trivial proof of the next optimal property in [10] (Proposition 2.1).
Proposition 2.12. Let u ∈ C(Ω) a v. subsolution such that u ≥ v for any v. subsolutions v ∈ C(Ω), then u is a v. solution.
Generalizing Proposition 2.11 to a generic family of functions, it is very useful in order to study marginal-functions and to prove existence-results, using some Dynamical Programming Principle.
Proposition 2.13. Let v ∈ F a family of v. subsolutions [v. supersolutions]
of H(x, v, Dv) = 0 and
u(x) = sup
v∈F
v(x) [u(x) = inf
v∈Fv(x)].
If u is a continuous function, then it is a v. subsolution [v. supersolutions]
of the same Hamilton-Jacobi equation.
here we omit the proof but we will show this directly for lower semicon-tinuous viscosity solutions (see Proposition 2.21).
Remark 2.14 (stationary and evolutive cases). Writting H(x, u, Du) = 0, we can express, at the same time, the stationary case and the evolutive one.
Nevertheless, sometimes, can be useful to use two different notation: the previous one, in particular, for the stationary case, while we express the evolutive case, separating the time-variable and the space-variable and, most of all, the corresponding derivatives, i.e.
ut+ H(t, x, u, Du) = 0, (t, x) ∈ Ω ⊂ [0, +∞) × Rn, (2.7) where Du means Dxu.
Now we quote some existence- and uniqueness-results for continuous v.
solutions of Hamilton-Jacobi equations.
We start by recalling some different methods to get the existence of viscosity solutions.
One of the most used methods consists in giving representative for-mulas and explicitly proving that they are v. solutions. This method is characteristic in the optimal control theory (and in the calculus of variations).
The solution can be defined as the value function of a suitable associated functional to minimize. Now we don’t investigate this method because we will see it, in detail, studying the Hopf-Lax function (Sec. 2.3).
Another very used method is the method of the characteristics. This consists in solving a nonlinear PDE using a suitable associated system of ODE. More precisely, let us consider the Dirichlet problem
(H(x, u, Du) = 0, in Ω,
u = g, in Γ ⊂ ∂Ω. (2.8)
Note that also a Cauchy problem can be written as (2.8), setting Ω = (0, +∞) × Rn and so ∂Ω = {0} × Rn.
Given a Hamiltonian H(x, z, p), differentiable in all its variables, it is possi-ble to associate to (2.8) a system of ODE, called equations of characteristics, (see [54], Sec. I.3.2, to find an explicit proof of this claim), that is
˙x(s) = DpH(x(s), z(s), p(s)),
˙z(s) = DpH(x(s), z(s), p(s)) · p(s),
˙
p(s) = −DxH(x(s), z(s), p(s)) − DzH(x(s), z(s), p(s))p(s),
(2.9)
Setting z(·) = u(x(·)), we are able to solve the system of ODE (2.9), using the solution of PDE (2.8) and vice-versa.
Another general method in order to prove existence-results, is to ap-proximate the original equation by a family of equations, easier than
the original one. Then, by suitable estimates, one can get (extracting a subsequence) the convergence of the approximating solutions. So, by their limit-function (in some suitable sense), it is possible to find a solution of the original equation. In particular, for nonlinear first-order equations, this method is known as vanishing viscosity method, whenever the approximating equations are got adding the term −ε∆uε, for ε > 0. For more information about the vanishing viscosity method, one can see [84] (Sec. 1.4 and Sec.
8), [54] (Sec. II.7.3), [43] (Theorem 3.1.) or [16, 104, 47].
Nevertheless, the vanishing viscosity method is not the unique one in order to get existence by approximation of the original Hamilton-Jacobi equation. In [12], the existence for the stationary-Dirichlet problem and the evolutive-Cauchy problem is proved approximating the original Hamiltonian by kernels of convolution (Theorems III.1 and IV.1, respectively).
In [67], for the first time, H. Ishii used the Perron’s method to solve nonlinear first-order equations. This method has been introduced by O.
Perron ([97]) in order to find solutions of the Laplace equation and consists in building a solution as the supremum of suitable subsolutions. It is very useful in the theory of v. solutions, since the supremum of v. subsolutions is still a v. subsolution (Proposition 2.13). Therefore, one needs only to verify that a such supremum satisfies the v. supersolution-condition.
The Perron’s method can be sketched, as follows.
Let F a nonempty family of v. subsolutions of the Hamilton-Jacobi equation (2.3), such that
1. if v ∈ F is not a v. solution of (2.3), then there is a w ∈ F such that w(x) > v(x), for some x ∈ Ω,
2. if u(x) = sup{v(x)|v ∈ F }, for any x ∈ Ω, then u ∈ F . Set, for any x ∈ Ω,
u(x) = sup{v(x)|v ∈ F }, then u is a v. solution of (2.3).
When the Hamiltonian has the form ut + F (u)x, the corresponding
equations are called scalar conservation laws. We don’t say anything about this particular case, since, in general, a different notion of weak solutions is used (see, for example, [54] for some information about the conservation laws and entropy solutions).
Neither we speak about existence of continuous v. solutions when the Hamiltonian is only measurable w.r.t. the space-variable. This theory is developed by F. Camilli, A. Siconolfi and others and needs a different and non-pointwise way in order to test v. solutions (see [28], for the stationary case, and [29], for the evolutive one).
As we have seen, investigating the eikonal example, the definition of v. solutions is good in order to get uniqueness. Classical results for unique-ness when the Hamiltonian satisfies some coercive or Lipschitz assumptions, w.r.t. the gradient-variable, can be found in [10, 14, 44, 48, 54, 84] and, only for the evolutive case, in [43, 69], too. In particular, we want to quote a very recent comparison principle by A. Cutr´ı and G. Da Lio ([49]), holding for evolutive Hamiltonians satisfying a very general H¨ormander condition (next to the model studied in Sec. 2.4.2). However this comparison principle gives the uniqueness of the v. solution only in the continuous case, but it cannot be applied when (as in the model that we will study) the solutions are only lower semicontinuous.
To this purpose, in the next subsection, we are going to introduce the definition of v. solutions in the semicontinuous case.