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Continuous Dependence of Solutions to Nonlocal Systems with Heterogeneous Kernels of Interaction
Nonlocal models have gained interest due to their flexibility in handling discontinuities by recording long range interactions through a kernel which gives additional flexibility. To be physically relevant, mathematical models must guarantee existence and uniqueness of solutions, as well as continuity with respect to the data. Thus, small changes in data or parameters will lead to appropriate changes in the solution. In this dissertation we concentrate on this stability for the nonlocal Laplacian. We consider the stability of the solution with regards to changes in the forcing term, the collar (nonlocal boundary) term, and the kernel(s) (a part of the operator itself that depicts the interactions between points). The biharmonic operator appears in modeling deformations and damage in beams and plates. We extend these continuous dependence results to these nonlocal higher order operators in a version of the nonlocal biharmonic that iterates the nonlocal Laplacian.
Buczkowski, Nicole, "Continuous Dependence of Solutions to Nonlocal Systems with Heterogeneous Kernels of Interaction" (2022). ETD collection for University of Nebraska - Lincoln. AAI29322736.