On solutions to a class of degenerate equations with the Grushin operator

Laura Abatangelo; Alberto Ferrero; Paolo Luzzini

doi:10.1016/j.jde.2025.113666

On solutions to a class of degenerate equations with the Grushin operator

Laura Abatangelo
, Alberto Ferrero
, Paolo Luzzini

Dipartimento di Scienze e Innovazione Tecnologica

Risultato della ricerca: Contributo su rivista › Articolo in rivista › peer review

Abstract

The Grushin Laplacian −Δ_α is a degenerate elliptic operator in R^h+k that degenerates on {0}×R^k. We consider weak solutions of −Δ_αu=Vu in an open bounded connected domain Ω with V∈W^1,σ(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of R^h+k. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of Ω. As an application we derive strong unique continuation properties for solutions.

Lingua originale	Inglese
Numero di articolo	113666
Rivista	Journal of Differential Equations
Volume	445
DOI	https://doi.org/10.1016/j.jde.2025.113666
Stato di pubblicazione	Pubblicato - 15 nov 2025

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10.1016/j.jde.2025.113666

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title = "On solutions to a class of degenerate equations with the Grushin operator",

abstract = "The Grushin Laplacian −Δα is a degenerate elliptic operator in Rh+k that degenerates on \{0\}×Rk. We consider weak solutions of −Δαu=Vu in an open bounded connected domain Ω with V∈W1,σ(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of Rh+k. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of Ω. As an application we derive strong unique continuation properties for solutions.",

keywords = "Almgren monotonicity formula, Grushin operator, Unique continuation property",

author = "Laura Abatangelo and Alberto Ferrero and Paolo Luzzini",

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T1 - On solutions to a class of degenerate equations with the Grushin operator

AU - Abatangelo, Laura

AU - Ferrero, Alberto

AU - Luzzini, Paolo

PY - 2025/11/15

Y1 - 2025/11/15

N2 - The Grushin Laplacian −Δα is a degenerate elliptic operator in Rh+k that degenerates on {0}×Rk. We consider weak solutions of −Δαu=Vu in an open bounded connected domain Ω with V∈W1,σ(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of Rh+k. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of Ω. As an application we derive strong unique continuation properties for solutions.

AB - The Grushin Laplacian −Δα is a degenerate elliptic operator in Rh+k that degenerates on {0}×Rk. We consider weak solutions of −Δαu=Vu in an open bounded connected domain Ω with V∈W1,σ(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of Rh+k. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of Ω. As an application we derive strong unique continuation properties for solutions.

KW - Almgren monotonicity formula

KW - Grushin operator

KW - Unique continuation property

UR - https://www.scopus.com/pages/publications/105012228229

U2 - 10.1016/j.jde.2025.113666

DO - 10.1016/j.jde.2025.113666

M3 - Article

SN - 0022-0396

VL - 445

JO - Journal of Differential Equations

JF - Journal of Differential Equations

M1 - 113666

ER -

On solutions to a class of degenerate equations with the Grushin operator

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