V4E2 - Numerical Simulation

V4E2 - Numerical Simulation

Sommersemester 2018 Prof. Dr. J. Garcke Teaching assistant: B. Paparella

Tutor: Marko Rajkovi´c

Exercise sheet 1.

Let H:R^d×R×R^d→Rbe a Hamiltonian and Ω⊂R^d be an open domain. We consider the problem

H(x, u, Du) = 0, ∀x∈Ω.

and we assume H to satisfy our usual assumptions:

[A1]H(·,·,·) is uniformly continuous on Ω×R×R^d [A2]H(x, u,·) is convex on R^d

[A3]H(x,·, p) is monotone on R.

The definition ofviscosity solution for this problem was given in the lecture.

Exercise 1. (A first example) Check that

u(x) =

(x , 0< x≤ ¹₂ 1−x , ¹₂ < x <1

is a viscosity solution of H(x, u, Du) :=|u⁰(x)| −1 = 0, x∈(0,1). Is u a viscosity solution of

−|u⁰(x)|+ 1 = 0 in (0,1)?

(4 Punkte) Exercise 2. (H and change of sign - !!! UPDATED - there was a typo here !!!)

Check that u is a viscosity solution for H(x, w, Dw) = 0 iff −u is a viscosity solution for

−H(x,−w,−Dw) = 0.

(4 Punkte) An alternative way for defining viscosity solutions is provided with the help of sub- and super- differentials. We will see this better during the lecture, but for the moment it is useful to develop some manuality with the definitions.

Definition 1 (Sub- and Super- differentials). Let Ω be an open set inR^d andv: Ω→R. The super-differential D⁺v(x) of v atx∈Ω, is defined as the set

D⁺v(x) :=n

p∈R^d: lim sup

y→x y∈Ω

v(y)−v(x)−p·(y−x)

|y−x| ≤0o .

The sub-differential D⁻v(x) ofv atx∈Ω, is defined as the set:

D⁻v(x) :=

n

q ∈R^d: lim inf_y→x

y∈Ω

v(y)−v(x)−q·(y−x)

|y−x| ≥0 o

.

Exercise 3. (An example) a) Let

v1(x) :=|x|.

Compute D⁺v₁(0) andD⁻v₁(0).

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b) Let

v2(x) :=

(0 , x≤0

1

2bx²+ax , x >0.

Compute D⁺v2(0).

(5 Punkte) Exercise 4. (Subdifferential for convex functions)

Prove: If u:R^d→R is convex (i.e.u(λx+ (1−λ)y)≤λu(x) + (1−λ)u(y), for any x, y∈R, λ∈[0,1]), then its sub-differential atx in the sense of convex analysis is the set

∂cu(x) :={p∈R^d:u(y)≥u(x) +p·(y−x), ∀y∈R^d}.

Show that if u is convex then∂_cu(x) =D⁻u(x).

(4 Punkte)

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