MA 3065 Final: hw1 (1)
11 Jan 2019
Department
Course
Professor
Math 8590 – Homework 1 (Due Friday Sept 21)
Please hand in your solution to 1 problem from those below.
1. Let u∈USC(Rn)and define
A=nx∈Rn:∃ϕ∈C∞(Rn), u −ϕhas a local max at xo.
Show that Ais dense in Rn. [Hint: Let x0∈Rnand ε > 0, and consider the maximum
of u−ϕover B(x0,1), where ϕ(x) := |x−x0|2
ε. Send ε→0. You should recall that upper
semicontinuous functions assume their maximums over compact sets (why?).]
2. Show that u(x) = xis a viscosity solution of u′= 1 on the interval (0,1], but is not
a viscosity solution of u′= 1 on the interval [0,1). [Hint: Examine the subsolution
condition at x= 0. This exercise shows that smooth solutions need not be viscosity
solutions at boundary points.]
3. Let u: (0,1) →Rbe continuous.
(a) Show that uis nondecreasing on (0,1) if and only if uis a viscosity solution of
u′≥0on (0,1). [Hint: For the hard direction, suppose that u′≥0in the viscosity
sense on (0,1), but uis not nondecreasing on (0,1). Show that there exists 0<
x1< x2< x3<1such that u(x3)< u(x2)< u(x1). Construct a test function
ϕ∈C∞(R)with ϕ′<0such that ϕtouches ufrom below somewhere in the interval
(x1, x3). Drawing a picture might help.]
(b) Show that uis convex on (0,1) if and only if uis a viscosity solution of −u′′ ≤0on
(0,1). Show that in general, convex functions are not viscosity solutions of u′′ ≥0.
[Hint: The hint for the hard direction is similar to part (a). Suppose that −u′′ ≤0
on (0,1) but uis not convex on (0,1). Then there exists 0< x1< x2<1and
λ∈(0,1) such that
u(λx1+ (1 −λ)x2)> λu(x1) + (1 −λ)u(x2).
Construct a test function ϕ∈C∞(R)with ϕ′′ <0such that ϕtouches ufrom
above somewhere in the interval (x1, x2).]
4. Let U⊂Rnbe open. Suppose that u∈C(U)satisfies
u(x) = −
ZB(x,ε)
u dy +o(ε2)as ε→0+
for every x∈U. Note this is an asymptotic version of the mean value property. Show
that uis a viscosity solution of
−∆u= 0 in U.
[Hint: Show that for every ϕ∈C∞(Rn)
−∆ϕ(x) = 2(n+ 2)−
ZB(x,ε)
ϕ(x)−ϕ(y)
ε2dy +o(1) as ε→0+.
Document Summary
Math 8590 homework 1 (due friday sept 21) Please hand in your solution to 1 problem from those below: let u usc(rn) and de ne. A = nx rn : c (rn), u has a local max at xo. [hint: let x0 rn and > 0, and consider the maximum of u over b(x0, 1), where (x) := |x x0|2. [hint: examine the subsolution condition at x = 0. [hint: for the hard direction, suppose that u 0 in the viscosity sense on (0, 1), but u is not nondecreasing on (0, 1). Show that there exists 0 < x1 < x2 < x3 < 1 such that u(x3) < u(x2) < u(x1). C (r) with < 0 such that touches u from below somewhere in the interval (x1, x3). Drawing a picture might help. ] (b) show that u is convex on (0, 1) if and only if u is a viscosity solution of u 0 on (0, 1).
