Operators of order 2 $ n $ with interior degeneracy
Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli · Discrete and Continuous Dynamical Systems - S · 2019
We consider a differential operator of order 2$ n $ of the type $ A_n u = (-1)^n (a u^{(n)})^{(n)} $, where $ a(x)>0 $ in $ [0, 1]\setminus\{x_0\} $ and $ a(x_0) = 0 $. We show that, for any $ n\in{\mathbb{N}} $, the operator $ -A_n $ generates a contractive analytic semigroup of angle $ \pi/2 $ on $ L^2 (0, 1) $. Note that the domain of $ A_n $ depends on the type of degeneracy of $ a $. Our theorems extend some previous results in [3] where $ n = 1 $.