AN EIGENVALUE ESTIMATE FOR A ROBIN p-LAPLACIAN IN C1 DOMAINS

Let Omega subset of R-n be a bounded C-1 domain and p > 1. For alpha > 0, define the quantity Lambda(alpha) = inf(u is an element of W1,p (Omega), u not equivalent to 0) (integral(Omega) vertical bar del u vertical bar(p) dx - alpha integral(partial derivative Omega) vertical bar u vertical bar(p) ds)/integral(Omega)vertical bar u vertical bar(p) dx with ds being the hypersurface measure, which is the lowest eigenvalue of the p-Laplacian in Omega with a non-linear alpha-dependent Robin boundary condition. We show the asymptotics Lambda(alpha) = (1 - p)alpha(P/(P-1) )+ o(alpha(P/(P-1))) as alpha tends to +infinity. The result was only known for the linear case p = 2 or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for C-1,C-lambda domains.