Pith Number

pith:UADAUEFP

pith:2026:UADAUEFP64KPMGY2P3ABTRSIGZ

not attested not anchored not stored refs pending

Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions

Massimo Grossi, Mengyao Chen, Qi Li

Positive solutions to the Robin problem -Δu = u^p approach a constant as β tends to zero, with uniform blow-up for p < 1, convergence to a fixed constant for p = 1, and uniform decay to zero for p > 1.

arxiv:2604.10139 v2 · 2026-04-11 · math.AP

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\usepackage{pith}
\pithnumber{UADAUEFP64KPMGY2P3ABTRSIGZ}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

For all p ≥ 0 the solution u_β behaves like a constant as β → 0; specifically, u_β blows up uniformly if 0 ≤ p < 1, converges to a constant if p = 1, and converges uniformly to zero if p > 1. In the critical regime p ≥ (N+2)/(N-2) on a ball and 0 < β < 2/(p-1) a radial positive solution exists.

C2weakest assumption

The proofs assume that positive solutions exist for each fixed β > 0 (or that the problem is well-posed in the appropriate function space) and that the domain is smooth; the existence statement for the critical case further assumes the domain is a ball and restricts β to a specific interval.

C3one line summary

As beta goes to zero, solutions to -Δu = u^p with Robin conditions behave like constants that blow up for p<1, stay finite for p=1, and go to zero for p>1; existence of radial solutions is shown for balls in the critical regime when beta is small.

Receipt and verification

First computed	2026-06-10T01:10:00.786370Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a0060a10aff714f61b1a7ec019c648365c014a802749e86b80beebd9e4805afa

Aliases

arxiv: 2604.10139 · arxiv_version: 2604.10139v2 · doi: 10.48550/arxiv.2604.10139 · pith_short_12: UADAUEFP64KP · pith_short_16: UADAUEFP64KPMGY2 · pith_short_8: UADAUEFP

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/UADAUEFP64KPMGY2P3ABTRSIGZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: a0060a10aff714f61b1a7ec019c648365c014a802749e86b80beebd9e4805afa

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "86fde8234d48cf8a537371295cbf4846c5def8f843f6c61b1595f50fe1887b5e",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-04-11T10:08:31Z",
    "title_canon_sha256": "6ed69b2e168cf87ee95feb8c0bd9b1c6a53c430f206405db50c7734bcceeeda2"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.10139",
    "kind": "arxiv",
    "version": 2
  }
}