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Unique continuation from conical boundary points for fractional equations

Alessandra De Luca, Veronica Felli, Stefano Vita

TL;DR

This work develops a sharp, local description of boundary behavior for fractional elliptic equations with conical boundary points and a Hardy-type singular potential. By smoothing the corner and employing an approximation scheme together with a Pohozaev-type identity, the authors establish an Almgren-type monotonicity formula for the Caffarelli–Silvestre extension and perform a detailed blow-up analysis. The asymptotics are quantized by a weighted spherical eigenproblem on the cone cap, yielding a leading profile of the form $|z|^\gamma\psi(z/|z|)$ with $\gamma$ linked to eigenvalues $\mu_j$ via $\gamma_j=\sqrt{((N-2s)/2)^2+\mu_j}-(N-2s)/2$, and they deduce strong unique continuation from the boundary. The results extend boundary-unique-continuation theory to nonlocal fractional operators in domains with corner singularities and Hardy-type potentials, providing a precise framework for blow-up profiles and their spectral content.

Abstract

We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev type inequality. Then, the asymptotics of solutions at the conical point follow by an Almgren type monotonicity formula, blow-up analysis and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollary.

Unique continuation from conical boundary points for fractional equations

TL;DR

This work develops a sharp, local description of boundary behavior for fractional elliptic equations with conical boundary points and a Hardy-type singular potential. By smoothing the corner and employing an approximation scheme together with a Pohozaev-type identity, the authors establish an Almgren-type monotonicity formula for the Caffarelli–Silvestre extension and perform a detailed blow-up analysis. The asymptotics are quantized by a weighted spherical eigenproblem on the cone cap, yielding a leading profile of the form with linked to eigenvalues via , and they deduce strong unique continuation from the boundary. The results extend boundary-unique-continuation theory to nonlocal fractional operators in domains with corner singularities and Hardy-type potentials, providing a precise framework for blow-up profiles and their spectral content.

Abstract

We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev type inequality. Then, the asymptotics of solutions at the conical point follow by an Almgren type monotonicity formula, blow-up analysis and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollary.
Paper Structure (12 sections, 30 theorems, 257 equations, 3 figures)

This paper contains 12 sections, 30 theorems, 257 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Weak solutions
  4. A fractional Hardy inequality on cones
  5. Other inequalities
  6. Spherical eigenvalues
  7. Approximation scheme and Pohozaev inequality
  8. Approximation scheme
  9. A Pohozaev-type inequality
  10. Almgren-type monotonicity formula and blow-up analysis
  11. Almgren-type monotonicity formula
  12. Blow-up analysis

Key Result

Theorem 1.1

Let $U\in H^1(B^+_1,t^{1-2s})$ be a non-trivial weak solution to eq1EXT. Then there exists an eigenvalue $\mu_{j_0}$ of problem probautov such that gammauguale--eq:gamma-h0 is satisfied, with $\gamma= \lim_{r\to 0^+} \mathcal{N}(r)$ and $\mathcal{N}$ being as in Nintro. Moreover, if $m\in\mathbb N\s where $(\beta_{j_0}, \dots, \beta_{j_0+m-1}) \in \mathbb{R}^m\setminus\{0\}$ and, for every $j=j_0,

Figures (3)

  • Figure 1: Sections of $\mathcal{C}$ (in dark blue) and of its smooth approximation $\mathcal{C}_n$ (in light blue).
  • Figure 2: Section of $\Omega_n$.
  • Figure 3: Section of $\Omega_{r,n,\rho,\delta}$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1: Cone spanned by $\omega$
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 52 more