Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

By M. Fall, Veronica Felli. 2013

180 citations recorded by the source corpus.

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.

Subfields: Applied Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/dd44ac3650449526b586d237de6cea1560047742

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Enrico Valdinoci
2. C. Bucur
3. Nicola Garofalo
4. E. Lenzmann
5. Jesse Railo
6. Keijo Monkkonen
7. Giovanni Covi
8. Rupert L. Frank

In Mechanics of Materials

1. Jesse Railo
2. Keijo Monkkonen
3. Giovanni Covi
4. Enrico Valdinoci
5. E. Lenzmann
6. Angkana Rüland
7. Agnid Banerjee
8. Rupert L. Frank

Citations in the corpus

• Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations (2017, methodology): “ parabolic equations with sufficiently regular coefficients. For nonlocal elliptic equations with principal part(−∆)sa strong uniquecontinuation theorem was obtained by Fall and Felli, see Theorem 1.3 in [FF]. Their analysis combines the approach in [GL1], [GL2] with the Caffarelli-Silvestre extension me”
• The fractional Calderón problem: Low regularity and stability (2017, methodology): “, we will employ three balls inequalities and Lebeau-Robbiano type interpolation inequalities based on Carleman estimates from [Ru15] together with a new Carleman estimate (Proposition5.7). See also [FF14], [Yu17] for the frequency function approach towards unique continuation for fractional equatio”
• Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
• All functions are locally $s$-harmonic up to a small error (2014, unknown): “For instance, any function that vanishes on an open subset of B1 cannot be extended to a function that is s-harmonic in B1, unless it vanishes identically, in view of the Unique Continuation Principle (see [5]).”
• On higher order extensions for the fractional Laplacian (2013, methodology): “Fall and Felli [3] have recently used the Almgren frequency formula to analyze unique continuation for fractional-order elliptic equations, although their analysis is also restricted to γ < 1.”
• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “There are also strong unique continuation results for s ∈ (0, 1) if one assumes more regularity from the function [22, 72].”
• Uniqueness of Radial Solutions for the Fractional Laplacian (2013, background): “) = V(x)uhas not been understood so far in rudimentary terms. Nonlinear Case. For nonlinear problems of the form (1.5), the known nonperturbative uniqueness results can be summarized as follows. (See [22, 18] for some perturbative uniqueness results when sis close to 1.) • Benjamin–Ono equation: In ”