I am studying for qualifying exams. The following problem was on my final. I am unsure if I got it right or not.

Let be the unit ball in and satisfy

where is smooth with for and for or Determine which one of the

following cases holds:

Solution

Let's consider two cases: Case I and Case II

Case I:Assume and let so Since is continuous, on

Additionally, we see that and therefore on

By the Weak Minimum Principle

since on A condtradiction, since we

assumed and therefore Thus,

Case II:On the other hand, let's now assume and let

so Since is continuous,

on Additionally, we see that and therefore on By the

Weak Maximum Principle

since on A condtradiction, since we

assumed and therefore Thus,

Combining the results for Case I and Case II we see that and therefore