Let a solution of . We can find such that (if it's , we consider ). Since is compact and , we have . Let such that ( can't be in because on ). Since is a critical point, we have and by the Taylor's theorem we have where and is the Hessian matrix ( ). Since , we have . We deduce that for all and each eigenvalue of is . Hence
and we get that . We have to show that we can have , but it's a "well-known result" about harmonic functions.