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