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Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator

  • Congming Li
  • Ling Wang
  • Jiongduo Xie
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Abstract

In this paper, we study the fundamental solution of the mixed local-nonlocal operator $ -Δ+(-Δ)^s,$ $ 0<s<1. $ We derive precise asymptotic expansions of the fundamental solution and its gradient, up to the first nontrivial correction term, both near the origin and at infinity. The expansions describe the transition between the local and nonlocal diffusion scales and include the critical regimes in which logarithmic terms occur. As applications of these asymptotics, we establish a distributional Bôcher-type theorem for nonnegative supersolutions with an isolated singularity. We further obtain quantitative positive and antisymmetric maximum principles on punctured balls.

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Cite

BibTeXRIS
BibTeX
@misc{li2026asymptotic,
  title = {{Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator}},
  author = {Congming Li and Ling Wang and Jiongduo Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01403},
  eprint = {2609.01403},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01403},
}
RIS
TY  - UNPB
TI  - Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator
AU  - Congming Li
AU  - Ling Wang
AU  - Jiongduo Xie
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01403
UR  - https://arxiv.org/abs/2609.01403
AB  - In this paper, we study the fundamental solution of the mixed local-nonlocal operator $ -Δ+(-Δ)^s,$ $ 0<s<1. $ We derive precise asymptotic expansions of the fundamental solution and its gradient, up to the first nontrivial correction term, both near the origin and at infinity. The expansions describe the transition between the local and nonlocal diffusion scales and include the critical regimes in which logarithmic terms occur. As applications of these asymptotics, we establish a distributional Bôcher-type theorem for nonnegative supersolutions with an isolated singularity. We further obtain quantitative positive and antisymmetric maximum principles on punctured balls.
ER  -