$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals

Qiang Du; Tadele Mengesha; Xiaochuan Tian; Qiang Du; Tadele Mengesha; Xiaochuan Tian

doi:10.3934/mine.2023097

Mathematics in Engineering

2023, Volume 5, Issue 6: 1-31. doi: 10.3934/mine.2023097

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Research article

$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals

1.
Department of Applied Physics and Applied Mathematics, and the Data Science Institute, Columbia University, New York, NY 10027, USA
2.
Department of Mathematics, University of Tennessee Knoxville, Tennessee 37996, USA
3.
Department of Mathematics, University of California, San Diego, CA, USA

Received: 13 July 2023 Revised: 10 August 2023 Accepted: 14 August 2023 Published: 14 September 2023

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ \Omega $ that is either a bounded domain in $ \mathbb{R}^{d} $ or $ \mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.
- $ L^p $ compactness,
- system of singular integral equations,
- nonlocal equations
Citation: Qiang Du, Tadele Mengesha, Xiaochuan Tian. $ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals[J]. Mathematics in Engineering, 2023, 5(6): 1-31. doi: 10.3934/mine.2023097

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Abstract

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ \Omega $ that is either a bounded domain in $ \mathbb{R}^{d} $ or $ \mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.

References

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