Higher-Order Operators on Networks: Hyperbolic and Parabolic Theory
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Integral Equations and Operator Theory
Higher-Order Operators on Networks: Hyperbolic and Parabolic Theory Federica Gregorio and Delio Mugnolo Dedicated to Professor Silvia Romanelli on the occasion of her 70th birthday. Abstract. We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions. Mathematics Subject Classification. 35K35, 35L35, 34B45, 47D06. Keywords. Linear operator semigroups, Polyharmonic operators, Dynamic boundary conditions, Extension theory, Quadratic forms.
1. Introduction Double-beam systems are a classical subject of theoretical mechanics, see e.g. [9,43]: they consist of two beams mediated by a viscoelastic material layer. On the mathematical level, this is modeled by strong couplings between the equations, usually complemented by identical, homogeneous boundary conditions—say, clamped or hinged. In the last two decades, coupled systems consisting of networks of (almost) one-dimensional beams have aroused more The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author has been partially supported by the Deutsche Forschungsgemeinschaft (Grant 397230547). This article/publication is based upon work from COST Action CA18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology). The authors wish to thank the anonymous referee for careful reading and useful comments. 0123456789().: V,-vol
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and more interest: unlike in double-beam systems, all interactions take place in the ramification points. Our aim in this note is twofold: we first discuss some properties of beam equations ∂4u ∂2u = − ∂t2 ∂x4 on networks of one-dimensional elements, with a focus on the solution properties that depend on rather general transmission conditions in the nodes. To this purpose, we propose a variational treatment of beam equations developing a formalism that happens then to be easily extendible to the study of parabolic equations driven by elliptic operators of arbitrary even order, again with general combinations of stationary and dynamic boundary conditions. The analysis of evolution equations on networks has become a very popular topic since Lumer introduced in [32] a theoretical framework to study heat equations on ramified structures; but in fact, time-dependent Schr¨ odinger equations on networks have been studied by quantum chemists since the 1940s and perhaps earlier, see the refer
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