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Theoretical Foundations of Incorporating Local Boundary Conditions Into Nonlocal Problems

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Theoretical Foundations of Incorporating Local Boundary Conditions Into Nonlocal Problems Vol. XX (XXXX) No. X THEORETICAL FOUNDATIONS OF INCORPORATING LOCAL BOUNDARY CONDITIONS INTO NONLOCAL PROBLEMS Burak Aksoylu ∗ Department of Mathematics, TOBB University of Economics and Technology Sogutozu Caddesi No:43, Ankara, 06560, Turkey & Department of Mathematics, Wayne State University 656 W. Kirby, Detroit, MI 48202, USA. email: [email protected] Horst Reinhard Beyer Department of Mathematics, TOBB University of Economics and Technology Sogutozu Caddesi No:43, Ankara, 06560, Turkey & Universidad Polit´ecnicade Uruapan Carretera Carapan, 60120 Uruapan Michoac´an,M´exico& Theoretical Astrophysics, IAAT, Eberhard Karls University of T¨ubingen T¨ubingen72076, Germany. email: [email protected] Fatih Celiker y Department of Mathematics, Wayne State University 656 W. Kirby, Detroit, MI 48202, USA. email: [email protected] (Received 2016) We study nonlocal equations from the area of peridynamics on bounded domains. We present four main results. In our recent paper, we discover that, on R, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, as main result 1, we construct an abstract convolution operator on bounded domains which is a generalization of the standard convolution based on integrals. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. As main result 2, we prove that the solution operator can be uniquely decomposed into a Hilbert-Schmidt operator and a multiple of the identity operator. As main result 3, we prove that Hilbert-Schmidt operators provide a smoothing of the input data in the sense a square integrable function is mapped into a function that is smooth up to boundary of the domain. As main result 4, for the homogeneous nonlocal wave equation, we prove that continuity is preserved by time evolution. Namely, the solution is discontinuous if and only if the initial data is discontinuous. As a consequence, discontinuities remain stationary. Keywords: Nonlocal wave equation, nonlocal operator, peridynamics, boundary condition, Hilbert-Schmidt operator, operator theory. Mathematics Subject Classification (2000): 35L05, 74B99. ∗ Burak Aksoylu was supported in part by the National Science Foundation DMS 1016190 grant, European Commission Marie Curie Career Integration 293978 grant, and Scientific and Technological Research Council of Turkey (TUB¨ ITAK)_ MFAG 115F473 grant. Research visit of Horst R. Beyer was supported in part by the TUB¨ ITAK_ 2221 Fellowship for Visiting Scientist Program. Sabbatical visit of Fatih Celiker was supported in part by the TUB¨ ITAK_ 2221 Fellowship for Scientist on Sabbatical Leave Program. y Fatih Celiker was supported in part by the National Science Foundation DMS 1115280 grant. [1] B. Aksoylu, H.R. Beyer, and F. Celiker 2 1. Introduction There are indications that a development of nonlocal theories is necessary for description of certain natural phenomena. It is part of the folklore in physics that the point particle model, which is the root for locality in physics, is the cause of unphysical singular behavior in the description of phenomena. On the other hand, all fundamental theories of physics are local. There are many more alternatives for formulating nonlocal theories compared to local ones. Therefore, the task of formulating a viable nonlocal theory consistent with experiment seems much harder than that of a local one. In any case, for such formulation, an understanding of the capacities of nonlocal theories appears inevitable. Considering wave phenomena only partially successfully described by a classical wave equation, it seems reasonable to expect that a more successful model can be obtained by employing the functional calculus of self-adjoint operators, i.e., by replacing the classical governing operator A by a suitable function f(A). We call f the regulating function. By choosing different regulating functions, we can define governing operators tailored to the needs of the underlying application. Since classical boundary conditions (BC) is an integral part of the classical operator, these BC are automatically inherited by f(A). In this way, we vision to model wave phenomena by using appropriate f(A) and, as a consequence, need to study the effect of f(A) on the solutions. One advantage of our approach is that every symmetry that commutes with A also commutes with f(A). As a result, required invariance with respect to classical symmetries such as translation, rotation and so forth is preserved. The choice of regulating functions appropriate for the physical situation at hand is an object for future research. We are interested in studying instances of successful modeling by nonlocal theories of phenomena that cannot be captured by local theories. There are noteworthy developments in the area of nonlocal modeling. For instance, crack propagation [1] and viscoelastic damping [2] are modeled by peridynamics (PD) and fractional derivatives, respectively, both of which are nonlocal. PD is a nonlocal extension of continuum mechanics developed by Silling [1], is capable of quantitatively predicting the dynamics of propagating cracks, including bifurcation. Its effectiveness has been established in sophisticated applications such as Kalthoff-Winkler experiments of the fracture of a steel plate with notches [3,4], fracture and failure of composites, nanofiber networks, and polycrystal fracture [5{8]. Further applications are in the context of multiscale modeling, where PD has been shown to be an upscaling of molecular dynamics [9, 10] and has been demonstrated as a viable multiscale material model for length scales ranging from molecular dynamics to classical elasticity [11]. Also see the review and news articles [12{15] for a comprehensive discussion, and the recent book [16]. Disturbances in solids propagate in form of waves. The wave equation is the basic model for the descrip- tion of the evolution of deformations. The driving application is PD, whose equation of motion corresponds exactly to the nonlocal wave equation under consideration. The same operator is also employed in nonlo- cal diffusion [12, 17, 18]. Similar classes of operators are used in numerous applications such as population models, particle systems, phase transition, coagulation, etc. Nonlocal operators have also been used in image processing [19]. In addition, we witness a major effort to meet the need for mathematical theory for PD applications and related nonlocal problems addressing, for instance, conditioning analysis, domain decomposition and variational theory [20{22], discretization [22{25], nonlinear PD [26, 27], convergence of solutions [28{32]. Since PD is a nonlocal theory, one might expect only the appearance of nonlocal BC, and, indeed, so far the concept of local BC does not apply to PD. Instead, external forces must be supplied through the loading force density b [1]. On the other hand, we demonstrate that the anticipation that local BC are incompatible with nonlocal operators is not quite correct. Our approach presents an alternative to nonlocal BC and we hope that the ability to enforce local BC provides a remedy for surface effects seen in PD; see [16, Chap. 4, 5, 7, and 12] and [33, 34]. We also hope that the applicability of PD can be extended to problems that require local BC such as contact, shear, and traction. There are four main results in this paper. Throughout the paper, we label the related results with the following identifiers. • Main Result 1: On a bounded domain Ω, utilizing the eigenbasis of the classical operator which allows us to incorporate local BC, we construct a convolution operator C∗ that possesses the Hilbert- Schmidt property. • Main Result 2: The solution operator of the nonlocal wave equation utt(x; t) + (c − C∗)u(x; t) = 0 B. Aksoylu, H.R. Beyer, and F. Celiker 3 can be uniquely decomposed into a Hilbert-Schmidt operator and a multiple of the identity operator. • Main Result 3: Hilbert-Schmidt operators provide a smoothing of the input data in the sense an L2(Ω) function is mapped into a function that is smooth up to boundary of Ω. • Main Result 4: The solution to the homogeneous nonlocal wave equation is discontinuous if and only if the initial data is discontinuous. Hence, discontinuities remain stationary. The rest of the article is structured as follows. Several aspects of constructing convolutions are studied in Sec. 2.. In Sec. 2.1., we introduce the proposed convolution definition on Hilbert spaces. The concept of regulating function is central to our study. In Sec. 2.2., we demonstrate that the regulating function for the bounded domain is similar to that for the unbounded domain. Hence, our construction of convolutions is an appropriate generalization. In Sec. 2.3., we study properties of operators which comprise Main Result 1, i.e., the convolution operator enjoys the Hilbert-Schmidt property. In Sec. 2.4., we provide diagonalizations of the classical governing operator A and functions, f(A), of A. The governing nonlocal operator is a function of A. On the other hand, the representation of the solution of the initial value problem corresponding to f(A) contains bounded functions of f(A). Hence, in Sec. 2.5., we prove that bounded functions of f(A) are bounded functions of A. When we refer to operators that are (bounded) functions of functions of A, we mean solution operators as in Sec. 2.5.. In Sec. 2.6., we choose an instance of f(A) that corresponds to the PD governing operator in the unbounded domain case. We present important properties such as boundedness and self-adjointness of the PD governing operator, and provide the conditions that make it a function of A. We also present the form the solution operators take with this concrete choice of f(A).
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