---
abstract: 'In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.'
author:
- Veronica Felli
- Alberto Ferrero
title: 'Unique continuation and classification of blow-up profiles for elliptic systems with Neumann boundary coupling and applications to higher order fractional equations'
---

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Introduction and statement of the main results {#s:introduction}
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The present paper is devoted to the study of unique continuation from a boundary point and classification of blow-up profiles for elliptic systems with Neumann boundary coupling. Systems of such a kind arise from higher order extensions of the fractional Laplacian, as first observed in [@Y], where the well known Caffarelli-Silvestre extension procedure characterizing the fractional Laplacian as the Dirichlet-to-Neumann map in one extra spatial dimension was generalized to higher powers of the Laplacian. More precisely in [@Y] (see also [@CY]) it is proved that, if $s\in(1,2)$ and $f\in H^s({{\mathbb R}}^N)$, then $$\label{eq:44b}
(-\Delta)^s f=K_{s}\lim_{t\to 0^+}t^{b}\frac{\partial(\Delta_b U)}{\partial
  t}$$ where $b=3-2s$, $K_{s}$ is a constant depending only on $s$, $\Delta_bU=\Delta U+\frac{b}{t}\frac{\partial
  U}{\partial t}$ and $U$ is the unique 