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

Laura Abatangelo, Alberto Ferrero, Paolo Luzzini

Abstract

The Grushin Laplacian $- Δ_α$ is a degenerate elliptic operator in $\mathbb{R}^{h+k}$ that degenerates on $\{0\} \times \mathbb{R}^k$. We consider weak solutions of $- Δ_αu= Vu$ in an open bounded connected domain $Ω$ with $V \in W^{1,σ}(Ω)$ and $σ> Q/2$, where $Q = h + (1+α)k$ is the so-called homogeneous dimension of $\mathbb{R}^{h+k}$. 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.

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

Abstract

The Grushin Laplacian is a degenerate elliptic operator in that degenerates on . We consider weak solutions of in an open bounded connected domain with and , where is the so-called homogeneous dimension of . By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of . As an application we derive strong unique continuation properties for solutions.

Paper Structure

This paper contains 15 sections, 33 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Asymptotic behavior of solutions at the origin
  3. Unique continuation principles
  4. Functional setting
  5. The Grushin operator in spherical coordinates
  6. The monotonicity formula
  7. End of the proof of Theorem \ref{['t:main']}
  8. Proof of Corollary \ref{['c:1']}
  9. Proof of Corollary \ref{['c:2']}
  10. Proof of Corollary \ref{['c:positive-measure']}
  11. Regularity results and properties of weighted Sobolev spaces
  12. Properties of the Sobolev spaces $W^{1,p}_\alpha (\Omega)$
  13. Local $W^{2,p}_\alpha$-regularity results
  14. Appendix
  15. Proof of Proposition \ref{['p:Pohozaev']}

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^{h+k}$ a bounded connected open set containing $0$. Let $\alpha \in \mathbb{N}$ and let $u \in W^{1,2}_{\alpha,\, {\rm loc}}(\Omega)$ be a nontrivial weak solution to eq:u with $V \in W^{1,\sigma}_{{\rm loc}}(\Omega)$ and $\sigma > Q/2$. Then there exists $j\in\mathbb{ in $W^{1,2}_\alpha (B_r^\alpha)$ and uniformly in $B_r^\alpha$ for any $r>0$, where $\mu_j$ is an

Theorems & Definitions (66)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.2
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 56 more