Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction
Abstract
We study heat semigroups on bounded Lipschitz domains Ω⊂ ℝd with dimension d>2 under non-local Robin boundary conditions. The boundary operator B ∈ ℒ( L2(∂Ω)) is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator C ∈ ℒ(L2(∂Ω)) such that |Bu| ≤ C|u| for every u∈ L2(∂Ω), Cbf 1∈ L∞(∂Ω). Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from L2(Ω) to L∞(Ω) has short-time order t−d/4. The main step is the construction of a positive eigenfunction ϕ of a dominating elliptic operator such that 0 < δ ≤ ϕ ≤ M almost everywhere in Ω. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an L∞-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.
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BibTeX
@misc{schwerdt2026ultracontractivity,
title = {{Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction}},
author = {Christoph Schwerdt},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30763},
eprint = {2608.30763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30763},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction
AU - Christoph Schwerdt
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30763
UR - https://arxiv.org/abs/2608.30763
AB - We study heat semigroups on bounded Lipschitz domains Ω⊂ ℝ^d with dimension d>2 under non-local Robin boundary conditions. The boundary operator B ∈ ℒ( L^2(∂Ω)) is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator C ∈ ℒ(L^2(∂Ω)) such that |Bu| ≤ C|u| for every u∈ L^2(∂Ω), Cbf 1∈ L^∞(∂Ω). Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from L^2(Ω) to L^∞(Ω) has short-time order t^−d/4. The main step is the construction of a positive eigenfunction ϕ of a dominating elliptic operator such that 0 < δ ≤ ϕ ≤ M almost everywhere in Ω. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an L^∞-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.
ER -