Abstract: | Varying conditions on the weight function f, the author is interested in whether the extremal of the ratio $inf frac{{left| {nabla u} right|_{Lp} left( Omega right)}}{{left| {fu} right|_{Lp} left( Omega right)}}{lambda }_{pq} left( {Omega ,f} right)$ remains symmetric. Here, f is positive almost everywhere in $u in mathop {W_p^1 }limits^{text{O}} left( Omega right)$ and the infimum is taken over all functions $u in mathop {W_p^1 left( Omega right)}limits^{text{O}}$ . Bibliography: 23 titles. |