Hand in y our solutions un til W ednesda y , Decem b er 20, 2.15 pm (PO b o x of y our T A in V3-128)

Hand in y our solutions un til W ednesda y , Decem b er 20, 2.15 pm (PO b o x of y our T A in V3-128)

total points: 20

Prof. Dr. Moritz Kaßmann Fakultät für Mathematik

Wintersemester 17/18 Universität Bielefeld

Partial Differential Equations II Exercise sheet X, December 14

Every exercise is worth 10 points. You receive the maximal points (20 points) if you solve two exercises correctly.

In the lecture we study Hölder-regularity estimates for solutions to partial differential equations of second order. Exercise X.1 explains that there is a natural limit to such results.

Exercise X.1 (10 points)

Let B₁ be the unit ball in R^d, d ≥ 2. Assume 0 < α < 1. Define u : B₁ → R by u(x) =x1|x|^α−1.

(a) Show u∈W^1,2(B₁)∩C^α(B₁).

(b) Show u /∈C^β(B₁) if β > α.

Next, defineaij :B1→R by

aij(x) =δij +(1−α)(d−1 +α) α(d−2 +α)

x_ix_j

|x|² .

(c) Show thata= (aij)ij defines an elliptic operator Lu= div(a∇u).

(d) Show thatu is a weak solution toLu= 0 inB1.

(e) How does the above relate to the regularity results proved in the lecture.

Exercise X.2 (10 points)

This exercise provides a sufficient condition for Hölder regularity based on the growth of localized energy functionals.

(a) Prove the following result:

Let B1 be the unit ball inR^d, d≥2,1 < p < d. Let u∈W^1,p(B1). Assume that for some constants α∈(0,1),c₀ ≥1 and for every ballB_r(x₀)bB₁

Z

Br(x0)

|∇u|^pdx≤c₀r δ

d−p+αp

,

whereδ = 1− |x₀|. Then there is a constantc1 ≥1 such that

|u(x)−u(y)| ≤c₁|x−y|^α if x, y∈B_r(0)for 0< r <1. Hint: Use the mean value theorem.

(b) On which quantities doesc1 depend?

Exercise X.3 (10 points)

We investigate the question whether weak solutions toLu=f withfa continuous function and La nice elliptic operator always have solutions uthat are twice differentiable.

LetR∈(0,1)and B_R=B_R(0)⊂R². Defineu:B_R→Rby u(x, y) = (x²−y²)(−logp

x²+y²)¹² .

(a) Show u∈C(B_R)∩C^∞(B_R\ {0}).

(b) Show ∆u=f inBR\ {0} for a functionf ∈C(BR).

(c) Showu /∈C²(BR).

(d) Show that the problem∆u=f in B_Rwith f as in (b) has a weak solution but no classical solution.¹

1Not required, voluntary problem

2