All functions are locally $s$-harmonic up to a small error

By Serena Dipierro, Ovidiu Savin, Enrico Valdinoci. 2014

98 citations recorded by the source corpus.

We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f76bccf85ce1d92208420d336bc7a62053bc2d15

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

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Nonlocal elliptic equations in bounded domains: a survey (2015, background): “Barriers and L∞ bounds 10 6.”
Global regularity for the free boundary in the obstacle problem for the fractional Laplacian (2015, background): “To show that non-degeneracy does not holds in bounded domains even if one assumes the obstacle φ to be C∞ and uniformly concave, consider a s-harmonic function u in B1 which satisfies D2u≤−Id in B1/2 (the existence of such a function follows for instance from [12]).”
The fractional Calderón problem: Low regularity and stability (2017, methodology): “ approximation property, stating that any f 2L2( ) can be approximated by functions uj where (( ) s+ q)u= 0 in and one can control the support of u. This strong approximation result was rst proved in [DSV17] for the fractional Laplacian and Ck norms. Related results for other equations and norms are”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “However, such inequality fails when the solution changes sign as shown by Kassmann [27] in the case p = 2, see also Dipierro-Savin-Valdinoci [21].”
Nonlocal Diffusion and Applications (2015, background): “See [43] for the entire and more general proof.”
A logistic equation with nonlocal interactions (2016, methodology): “Br eo 2keokL∞(Rn) M. Now, Theorem 1.7 follows plainly from Proposition 6.1. 7. Fitting the resources in a nonlocal setting and proof of Theorem 1.8 Now we prove Theorem 1.8, by exploiting a result in [DSV15], joined to a minimization argument. More precisely, we make use of Theorem 1.1 in [DSV15], w”
The Bernstein Technique for Integro-Differential Equations (2020, result): “1Probably these additional difficulties arising in the nonlocal setting are not only a technical issue: as a matter of fact, in striking contrast with the classical case, the data at infinity can produce arbitrary modifications on the solutions of a nonlocal equation; see [22].”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, result): “nctions in the unit ball when s 2R+ nZ (compare to theorem 1.6). Here sharmonicity simply means that ( ) su= 0 in some domain . The s-harmonic approximation in the case s2(0;1) was already studied in [16]; similar higher regularity approximation results are proved in [10, 27] for the fractional Schr”