Local Density of Solutions to Fractional Equations1 Alessandro Carbotti2 Serena Dipierro3 and Enrico Valdinoci4 October 4 2022

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Local Density of Solutions to Fractional Equations1

Alessandro Carbotti2, Serena Dipierro3, and Enrico Valdinoci4

October 4, 2022

1Supported by the Australian Research Council Discovery Project 170104880 NEW “Nonlocal

Equations at Work”, the DECRA Project DE180100957 “PDEs, free boundaries and applications”

and the Fulbright Foundation. The authors are members of INdAM/GNAMPA.

2Dipartimento di Matematica e Fisica, Università del Salento, Via Per Arnesano, 73100 Lecce,

Italy. alessandro.carbotti@unisalento.it

3Department of Mathematics and Statistics, University of Western Australia, 35 Stirling High-

way, Crawley WA 6009, Australia. serena.dipierro@uwa.edu.au

4Department of Mathematics and Statistics, University of Western Australia, 35 Stirling High-

way, Crawley WA 6009, Australia. enrico.valdinoci@uwa.edu.au

arXiv:2210.00427v1 [math.AP] 2 Oct 2022

Contents

1 Introduction: why fractional derivatives? 9

2 Main results 47

3 Boundary behaviour of solutions of time-fractional equations 53

3.1 Sharp boundary behaviour for the time-fractional eigenfunctions . . . . . . . 53

3.2 Sharp boundary behaviour for the time-fractional harmonic functions . . . . 55

4 Boundary behaviour of solutions of space-fractional equations 59

4.1 Green representation formulas and solution of (−∆)su=fin B1with homo-

geneousDirichletdatum ............................. 59

4.1.1 Solving (−∆)su=fin B1for discontinuous fvanishing near ∂B1. . 59

4.1.2 Solving (−∆)su=fin B1for fHölder continuous near ∂B1..... 63

4.2 Existence and regularity for the ﬁrst eigenfunction of the higher order frac-

tionalLaplacian.................................. 65

4.3 Boundary asymptotics of the ﬁrst eigenfunctions of (−∆)s.......... 71

4.4 Boundary behaviour of s-harmonic functions . . . . . . . . . . . . . . . . . . 81

5 Proof of the main result 85

5.1 A result which implies Theorem 2.1 ....................... 85

5.2 A pivotal span result towards the proof of Theorem 5.1 ............ 86

5.3 Every function is locally Λ−∞-harmonic up to a small error, and completion

of the proof of Theorem 5.1 ........................... 108

5.3.1 Proof of Theorem 5.1 when fisamonomial .............. 108

5.3.2 Proof of Theorem 5.1 when fis a polynomial . . . . . . . . . . . . . 110

5.3.3 Proof of Theorem 5.1 for a general f.................. 111

Appendices

A Some applications 115

Index 123

Preface

The study of nonlocal operators of fractional type possesses a long tradition, motivated

both by mathematical curiosity and by real world applications. Though this line of research

presents some similarities and analogies with the study of operators of integer order, it also

presents a number of remarkable diﬀerences, one of the greatest being the recently discovered

phenomenon that all functions are (locally) fractionally harmonic (up to a small error). This

feature is quite surprising, since it is in sharp contrast with the case of classical harmonic

functions, and it reveals a genuinely nonlocal peculiarity.

More precisely, it has been proved in [DSV17] that given any Ck-function fin a bounded

domain Ωand given any  > 0, there exists a function fwhich is fractionally harmonic in Ω

and such that the Ck-distance in Ωbetween fand fis less than .

Figure 1: All functions are fractional harmonic, at diﬀerent scales (scale of the original function).

Interestingly, this kind of results can be also applied at any scale, as shown in Figures 1,2

and 3. Roughly speaking, given any function, without any special geometric prescription, in

a given bounded domain (as in Figure 1), one can “complete” the function outside the domain

in such a way that the resulting object is fractionally harmonic. That is, one can endow

the function given in the bounded domain with a number of suitable oscillations outside the

domain in order to make an integro-diﬀerential operator of fractional type vanish. This idea

is depicted in Figure 2. As a matter of fact, Figure 2must be considered just a “qualitative”

picture of this method, and does not have any demand of being “realistic”. On the other

hand, even if Figure 2did not provide a correct fractional harmonic extension of the given

function outside the given domain, the result can be repeated at a larger scale, as in Figure 3,

adding further remote oscillations in order to obtain a fractional harmonic function.

In this sense, this type of results really says that whatever graph we draw on a sheet

of paper, it is fractionally harmonic (more rigorously, it can be shadowed with an arbitrary

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LocalDensityofSolutionstoFractionalEquations1AlessandroCarbotti2,SerenaDipierro3,andEnricoValdinoci4October4,20221SupportedbytheAustralianResearchCouncilDiscoveryProject170104880NEWNonlocalEquationsatWork,theDECRAProjectDE180100957PDEs,freeboundariesandapplicationsandtheFulbrightFoundation.Theau...

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Local Density of Solutions to Fractional Equations1 Alessandro Carbotti2 Serena Dipierro3 and Enrico Valdinoci4 October 4 2022

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