The Laplacian with Robin boundary conditions involving signed measures
Akhlil Khalid · Gulf Journal of Mathematics · 2018
In this work we propose to study the general Robin boundary value problem involving signed smooth measures on an arbitrary domain Ω of ℝd. A Kato class of measures is defined to make the associated form (ℰμ, ℱμ) a closed one. Moreover, the associated operator Δμ is a realization of the Laplacian on L2(Ω). In particular, when |μ| is locally infinite everywhere on ∂Ω, Δμ is the laplacian with Dirichlet boundary conditions. On the other hand, we will prove that the semigroup (e-tΔμ)t≥0 is sandwiched between (e-tΔμ+)t≥0 and (e-tΔ-μ-)t≥0 and we will see that the converse is also true.