Critical Robin Problems on Manifolds with Noncompact Boundary

Abstract

We work on connected, noncompact n-dimensional Riemannian manifolds (ℳ, g), n ≥ 3, that are complete with respect to d_g and have noncompact boundary and bounded geometry. We study the existence of positive solutions in C^∞(ℳ) ∩ L^{2^*}(ℳ) ∩ L^∞(ℳ) to the Yamabe problem: { −Δ_g u+(n−2)/(4(n−1)) R_g u & =𝒞₀ u^{(n+2)/(n−2)} in ℳ, (∂ u)/(∂ ν)+(n−2)/(2(n−1)) H_g u & =𝒞₁ u^n/(n−2) on ∂ ℳ, . where 𝒞₀=Y_𝒞(ℳ), 𝒞₁=Y_𝒞(ℳ) 𝒞 d̅^2/(n−2), the parameter 𝒞 ≥ 0 is fixed, and Y_𝒞(ℳ):=inf _{u ∈ C_c^∞(ℳ) backslash{0}} E_g(u)/((|u|_L^{2^*}(ℳ)^{2^*}+𝒞|u|_{L^{2^#}(∂ ℳ)}^{2^*})^{2 / 2^*}) . Here E_g is the associated quadratic energy. We assume Y_𝒞(ℳ)<Y_{𝒞, ∞}(ℳ) and Y_{𝒞, ∞}(ℳ)>0, where Y_{𝒞, ∞} is the corresponding variational level at infinity. We approach this problem using nonnegative solutions u_j ∈ W_∂^1,2(B_g(𝒪, j)), j ∈ ℕ, j ≥ j₀, for which we prove that sup _{j ≥ j₀}|u_j|_{L^∞(B_g(𝒪, j−1))}<∞ . For 𝒞>0, we also prove that, along a subsequence, |u_j|_{L² _{(B_g(𝒪, j) ∩ ∂ ℳ)}} → d̅, with 0<d̅ ≤ 𝒞^{−1 / 2^*}. For 𝒞=0, we set d̅=0.

Optimal insulation and concentration breaking for nonlinear Robin boundary value problems
Francesco Della Pietra, Francescantonio Oliva
arXiv6 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEs Optimisation & controlarXiv
Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
Wei Jiang, Xianlong Pan
arXiv1 Sep 2026PreprintCC BY-SA 4.0Full textNumerical analysis & scientific computingarXiv
Neumann Boundary Value Problems on p-Adic Analytic Manifolds
Patrick Erik Bradley
HAL1 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEsHALarXiv
Positive scalar curvature on products of noncompact manifolds
Lizhi Chen, Kuntao Jin, Milan Jovanovic
arXiv25 Sep 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv
Sub-Laplacian Comparison Theorem and Gradient Estimates on Complete Noncompact Contact Riemannian Manifolds
Guilin Yang, Biqiang Zhao, Weihao Zheng
arXiv22 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{apaza2026critical,
  title = {{Critical Robin Problems on Manifolds with Noncompact Boundary}},
  author = {Juan Pablo Alcon Apaza},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40603},
  eprint = {2610.40603},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40603},
}

RIS

TY  - UNPB
TI  - Critical Robin Problems on Manifolds with Noncompact Boundary
AU  - Juan Pablo Alcon Apaza
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40603
UR  - https://arxiv.org/abs/2610.40603
AB  - We work on connected, noncompact n-dimensional Riemannian manifolds (ℳ, g), n ≥ 3, that are complete with respect to d_g and have noncompact boundary and bounded geometry. We study the existence of positive solutions in C^∞(ℳ) ∩ L^2^*(ℳ) ∩ L^∞(ℳ) to the Yamabe problem: { −Δ_g u+(n−2)/(4(n−1)) R_g u & =𝒞_0 u^(n+2)/(n−2) in ℳ, (∂ u)/(∂ ν)+(n−2)/(2(n−1)) H_g u & =𝒞_1 u^n/(n−2) on ∂ ℳ, . where 𝒞_0=Y_𝒞(ℳ), 𝒞_1=Y_𝒞(ℳ) 𝒞 d̅^2/(n−2), the parameter 𝒞 ≥ 0 is fixed, and Y_𝒞(ℳ):=inf _u ∈ C_c^∞(ℳ) backslash{0} E_g(u)/((|u|_L^2^*(ℳ)^2^*+𝒞|u|_L^2^#(∂ ℳ)^2^*)^2 / 2^*) . Here E_g is the associated quadratic energy. We assume Y_𝒞(ℳ)<Y_𝒞, ∞(ℳ) and Y_𝒞, ∞(ℳ)>0, where Y_𝒞, ∞ is the corresponding variational level at infinity. We approach this problem using nonnegative solutions u_j ∈ W_∂^1,2(B_g(𝒪, j)), j ∈ ℕ, j ≥ j_0, for which we prove that sup _j ≥ j_0|u_j|_L^∞(B_g(𝒪, j−1))<∞ . For 𝒞>0, we also prove that, along a subsequence, |u_j|_L^2 _(B_g(𝒪, j) ∩ ∂ ℳ) → d̅, with 0<d̅ ≤ 𝒞^−1 / 2^*. For 𝒞=0, we set d̅=0.
ER  -

Abstract

Related

Cite