Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in the case $k=0$, i.e., for Lipschitz domains, this map is bounded between $H^{s+1/2}\to H^{s-1/2}$ for $s\in[-\frac12,\frac12]$, and in the cases $k\geq1$, the DtN map is bounded between $H^{s+1/2}\to H^{s-1/2}$ for $s\in[-k,k]$. These results are for example in McLean's book on strongly elliptic systems, and as you can see there seem to be room for improvement for the $k\geq1$ case. Namely one would expect the boundedness to occur for $s\in[-k-\frac12,k+\frac12]$. If it is true can you please provide a reference or give a sketch of an argument?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.