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Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing

  • Shuhei Kitano
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Abstract

Let n≥1, 0<s<1, c>0, and let A be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of min{u, (−Δ)s u−(c−⟨ Ax,x⟩)}=0 in ℝn has the form u(x)=Kmax{1−⟨ Bx,x⟩,0}1+s for some K>0 and some positive definite symmetric matrix B. In particular, its positivity set is an ellipsoid. For s=1/2 and n≥2, this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.

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Cite

BibTeXRIS
BibTeX
@misc{kitano2026ellipsoidal,
  title = {{Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing}},
  author = {Shuhei Kitano},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00703},
  eprint = {2609.00703},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00703},
}
RIS
TY  - UNPB
TI  - Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing
AU  - Shuhei Kitano
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00703
UR  - https://arxiv.org/abs/2609.00703
AB  - Let n≥1, 0<s<1, c>0, and let A be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of min{u, (−Δ)^s u−(c−⟨ Ax,x⟩)}=0 in ℝ^n has the form u(x)=Kmax{1−⟨ Bx,x⟩,0}^1+s for some K>0 and some positive definite symmetric matrix B. In particular, its positivity set is an ellipsoid. For s=1/2 and n≥2, this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.
ER  -