Dirichlet-to-Neumann Maps on Trees
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Dirichlet-to-Neumann Maps on Trees ´ Frevenza1,2 · Julio D. Rossi1,2 Leandro M. Del Pezzo1,2 · Nicolas Received: 25 May 2019 / Accepted: 4 November 2019 / © Springer Nature B.V. 2019
Abstract In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a “gradient” with a “normal vector”) and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann map is given by g → cg (here c is an explicit constant). Notice that this is a local operator of order one. We also consider linear undirected mean value formulas (taking into account not only the successors but the ancestor and the successors of a given node) and prove a similar result. For this kind of mean value formula we include some existence and uniqueness results for the associated Dirichlet problem. Finally, we give an alternative definition of the Dirichlet-to-Neumann map (taking into account differences along a given branch of the tree). With this alternative definition, for a certain range of parameters, we obtain that the Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the classical Laplacian in the Euclidean space). Keywords Dirichlet-to-Neumann map · Mean value formulas · Equations on trees Mathematics Subject Classification (2010) 35J05 · 35R30 · 31E05 · 37E25
1 Introduction Informally, the Dirichlet-to-Neumann map works as follows: given a function g on ∂, solve the Dirichlet problem for the Laplacian with this datum inside the domain and then Julio D. Rossi
[email protected] Leandro M. Del Pezzo [email protected] Nicol´as Frevenza [email protected] 1
CONICET, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina
2
Departamento de Matem´atica, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina
L.M. Del Pezzo et al.
compute the normal derivative of the solution on ∂ to obtain the operator (g). Our main goal in this paper is to study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. The study of Dirichlet-to-Neumann maps for partial differential equations (PDEs) has a rich history in the literature. For the classical second order operator div(a(x)Du) = 0 the Dirichlet-to-Neumann map is related to the widely studied Calderon’s inverse problem, that is, knowing the Dirichlet-to-Neumann map, g → a (g), find the coefficient a(x) (see for instance [5] and the survey [21]). This problem has a well known application in electrical impedance tomography. The Dirichlet-to-Neumann map is also related to fractional powers of the Laplacian. For the classical Laplacian in a half space it is well known that the Dirichlet-to-Neumann map gives the fractional Laplacian (with power 1/2), that is a nonlocal operator, see [4]. Let us include now a brief comment on pre
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