Generation of analytic semigroups by a pair of generalized mixed linear regular ordinary differential operators with interface condition

Abstract

In this paper, we establish with suitable assumptions the analyticity of semigroups generated by a pair of generalized mixed linear regular differential operators

\(L_{(n,n)}u(x) \; := \; (L_{1n}u_1(x), L_{2n}u_2(x))\)

= \(\left(\displaystyle \sum_{0≤k≤n} p_k(x) \Big(\frac{d}{dx}\Big)^k u_1(x), \;  \displaystyle \sum_{0≤k≤n} q_k(x) \Big(\frac{d}{dx}\Big)^k u_2(x) \right)\)

with involving an interface condition in the setting of complex Hilbert space X = L²([a,b]) × L²([b,c]).  We obtain quite general results that extend previous works by the authors ([3], [10]).  The key for showing the generation analytic semigroups will be an inequality of the form

Re⟨(L(n,n) − ρI)u, u⟩_X + δ|Im⟨(L_(n,n) − ρI)u, u⟩_X | ≤ 0, \(\forall\) u ∈ D(L(n,n))

for some constant ρ > 0.