Dirichlet’s principle and wellposedness of solutions for a nonlocal \(p\)-Laplacian system. (English) Zbl 1291.35104
Summary: We prove Dirichlet’s principle for a nonlocal \(p\)-Laplacian system which arises in the nonlocal setting of peridynamics when \(p=2\). This nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain. Our analysis uses nonlocal versions of integration by parts techniques that resemble the classical Green and Gauss identities. The nonlocal energy functional associated with this “elliptic” type system exhibits a general kernel which could be weakly singular. The coercivity of the system is shown by employing a nonlocal Poincaré’s inequality. We use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy, from which we obtain the wellposedness of this steady state diffusion system.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 74B05 | Classical linear elasticity |
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