On a counterexample to a conjecture of Saint Venant

The elliptic boundary value problem % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi% slcqGHuoarcaWG1bGaeyypa0dccaGae8hiaaIaaGymaiab-bcaGiab% -bcaGiab-bcaGiaabMgacaqGUbGaaeiiaiabfM6axjaabYcaaeaaae% aacaWG1bGaeyypa0JaaGimaiab-bcaGiab-bcaGiab-bcaGiaab+ga% caqGUbGaaeiiaiabgkGi2kabfM6axjaabYcaaaaa!4E11!\\begin{gathered}- \Delta u = 1 {\text{in }}\Omega {\text{,}} \hfill \\\hfill \\u = 0 {\text{on }}\partial \Omega {\text{,}} \hfill \\\end{gathered}\]is considered. The Saint Venant's conjecture for convex plane domains , having symmetry about two orthogonal axes, is that the maximum of |u| occurs only at the points on which are nearest to the origin. G. Sweers constructed one such domain and claimed that either the conjecture fails for or for ={(x, y);u(x, y) >}, which again is convex. We give a totally different proof of this claim. Our proof brings out clearly the reason for the failure of the conjecture and also allows us to construct many more such domains.