Dirichlet-to-Neumann semigroup with respect to a general second order eigenvalue problem

Abstract

In this paper we study a Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on L 2 (∂Ω) given by ϕ → ∂νu where u is a weak solution ofUnder suitable assumptions on the matrix-valued function a, on the vector fields b and c, and on the function d, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated Dirichlet-to-Neumann semigroups.

Dynamical boundary conditions in a non-cylindrical domain for the Laplace equation

“…By the divergence theorem, the above definition coincides with the usual conormal derivative if P (t, x, D)u = 0 and if we impose sufficiently regularity to u and to the coefficients. Our definition of weak conormal derivative can be found in a similar way in [1,4,10].…”

“…We are then able to show that the operators satisfy the Tanabe-Sobolevskii conditions. On L 2 (∂Ω), we assume Hölder continuity of the forms with an exponent α ∈] 1 2 , 1] and prove that the operators satisfy the Yagi conditions. Our assumptions also allow us to study the asymptotic behavior at infinity of the Problem (1.4).…”

“…Our approach follows W. Arendt and A. Elst [3,4], see also [5,6], where form methods were used to deal with the Dirichlet-to-Neumann problem defined by the Laplace operator. Their technique has been also used for general second order problems by J. Abreu and E. Capelato [1], and E. M. Ouhabaz [10]. The novelty here is to study non-autonomous problems in bounded domains combining form methods for the Dirichlet-to-Neumann problem with the Tanabe-Sobolevskii and the Yagi conditions [25,26,28].…”