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.Comment: To appear in J. Eur. Math. Soc. (JEMS

Dipierro, Serena

Savin, Ovidiu

Valdinoci, Enrico

English

arXiv

Serena Dipierro

Ovidiu Savin

Enrico Valdinoci

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

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Journal of the European Mathematical Society

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

We show that we can approximate locally every function with a fractional harmonic function in that vanishes outside a compact set. 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

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We show that we can approximate locally every function with a fractional harmonic function in that vanishes outside a compact set.  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

Weierstraß-Institutfür Angewandte Analysis und StochastikLeibniz-Institut im Forschungsverbund Berlin e. V.Preprint ISSN 2198-5855All functions are locally s-harmonicup to a small errorSerena Dipierro1, Ovidiu Savin2, Enrico Valdinoci3submitted: April 14, 20141 Maxwell Institute for Mathematical Sciencesand School of MathematicsUniversity of EdinburghJames Clerk Maxwell BuildingKing’s BuildingsEdinburgh EH9 3JZScotlandUnited KingdomE-Mail: serena.dipierro@ed.ac.uk2 Department of MathematicsColumbia University2990 BroadwayNew York NY 10027USAE-Mail: savin@math.columbia.edu3 Weierstraß-InstitutMohrenstrasse 3910117 BerlinGermanyE-Mail: enrico.valdinoci@wias-berlin.deNo. 1944Berlin 20142010 Mathematics Subject Classification. 35R11, 60G22, 35A35, 34A08.Key words and phrases. Density properties, approximation, s-harmonic functions.The first author has been supported by EPSRC grant EP/K024566/1 “Monotonicity formula methods for nonlinearPDEs”. The second author has been supported by NSF grant DMS-1200701. The third author has been supportedby ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities” andPRIN grant 201274FYK7 “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari”.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Leibniz-Institut im Forschungsverbund Berlin e. V.Mohrenstraße 3910117 BerlinGermanyFax: +49 30 20372-303E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/1ABSTRACT. We show that we can approximate every function f ∈ Ck(B1) with a s-harmonicfunction in B1 that vanishes outside a compact set.That is, s-harmonic functions are dense in Ckloc. This result is clearly in contrast with therigidity of harmonic functions in the classical case and can be viewed as a purely nonlocalfeature.1. INTRODUCTIONIt is a well-known fact that harmonic functions are very rigid. For instance, in dimension 1, theyreduce to a linear function and, in any dimension, they never possess local extrema.The goal of this paper is to show that the situation for fractional harmonic functions is completelydifferent, namely one can fix any function in a given domain and find a s-harmonic functionarbitrarily close to it.Heuristically speaking, the reason for this phenomenon is that while classical harmonic functionsare determined once their trace on the boundary is fixed, in the fractional setting the operatorsees all the data outside the domain. Hence, a careful choice of these data allows a s-harmonicfunction to “bend up and down” basically without any restriction.The rigorous statement of this fact is in the following Theorem 1.1. For this, we recall that,given s ∈ (0, 1), the fractional Laplace operator of a function u is defined (up to a normalizingconstant) as(−∆)su(x) :=∫Rn2u(x)− u(x+ y)− u(x− y)|y|n+2s dy.We refer to [4,6,8,9] for other equivalent definitions, motivations and applications.Theorem 1.1. Fix k ∈ N. Then, given any function f ∈ Ck(B1) and any  > 0, thereexist R > 1 and u ∈ Hs(Rn) ∩ Cs(Rn) such that{(−∆)su = 0 in B1,u = 0 in Rn \BRand‖f − u‖Ck(B1) 6 .As usual, in Theorem 1.1, we have denoted by Ck(B1) the space of all the functions f : B1 →R that possess an extension f˜ ∈ Ck(B1+µ) (i.e. f˜ = f in B1), for some µ > 0.We also mention that an important rigidity feature for classical harmonic functions is imposedby Harnack inequality: namely if u is harmonic and non-negative in B1 then u(x) and u(y)are comparable for any x, y ∈ B1/2. A striking difference with the nonlocal case is that thistype of Harnack inequality fails for the fractional Laplacian (namely it is necessary to requirethat u is non-negative in the whole of Rn and not only in B1, see e.g. Theorem 2.2 in [5]). Asan application of Theorem 1.1, we point out that one can construct examples of s-harmonicfunctions with a “wild” behavior, that oscillate as many times as we want, and reach interiorextrema basically at any prescribed point. In particular, one can construct s-harmonic functionsto be used as barriers basically without any geometric restriction.As a final observation, we would like to stress that, while Theorem 1.1 reveals a purely non-local phenomenon, a similar result does not hold for any nonlocal operator. For instance, it isnot possible to replace “s-harmonic functions” with “nonlocal minimal surfaces” in the statementof Theorem 1.1, that is it is not true that any graph may be locally approximated by nonlocal2minimal surfaces. Indeed, the uniform density estimates satisfied by the nonlocal minimal sur-faces prescribe a severe geometric restriction that prevent the formation of sharp edges andthin spikes.We refer to [2] for the definition of nonlocal minimal surfaces and for their density properties: asa matter of fact, one of the consequence of Theorem 1.1 is that density properties do not holdtrue for s-harmonic functions, so s-harmonic functions and nonlocal minimal surfaces may havevery different behaviors.The rest of the paper is organized as follows: in Section 2 we collect some preliminary results,such as a (probably well-known) generalization of the Stone-Weierstrass Theorem and the con-struction of a s-harmonic function inB1 that has a well-defined growth from the boundary. Then,in Section 3, we construct a s-harmonic function with an arbitrarily large number of derivativesprescribed. This is, in a sense, already the core of our argument, since these types of propertiesare typical for the fractional case and do not hold for classical harmonic functions. Also, fromthis result, the proof of Theorem 1.1 will follow via a scaling and approximation method.2. PRELIMINARY OBSERVATIONSIn this section we collect some auxiliary results that will be needed in the rest of the paper.First of all, we recall a version of the Stone-Weierstrass Theorem for smooth functions. We givea quick proof of it since in general this result is presented only in the continuous setting.Lemma 2.1. For any f ∈ Ck(B1) and any  > 0 there exists a polynomial P such that ‖f −P‖Ck(B1) 6 .Proof. Without loss of generality we may suppose that f ∈ Ck0 (B2). Also, given  > 0 as inthe statement of Lemma 2.1, we fix R > 0 such that(1)∫Rn\BRe−|x|2dx 6 .Then, we fix η > 0, to be taken arbitrarily small (possibly in dependence of  and R, which arefixed once and for all), and we take Jη ∈ N large enough such that(2)∑j>Jη(−1)jj! ηj6 e−1/η.Let alsoQ(x) := (piη)−n/2Jη∑j=0(−1)j |x|2jj! ηj,P (x) :=∫Rnf(y)Q(x− y) dy,and G(x) := (piη)−n/2e−|x|2/η.We remark that Q is a polynomial in x, hence so is P . Moreover, by a Taylor expansion,G(x) = Q(x) + (piη)−n/2∑j>Jη(−1)j |x|2jj! ηjand so, using (2), we conclude that, for any x ∈ B3,(3) |G(x)−Q(x)| 6 e−1/√η,3provided that η is sufficiently small.Now we recall (1) and we observe that, for any α ∈ Nn with |α| 6 k and any x ∈ B1,|Dα(G ∗ f)(x)−Dαf(x)|=∣∣∣∣∫RnG(y)(Dαf(x− y)−Dαf(x))dy∣∣∣∣6 pi−n/2∫Rne−|z|2∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣ dz6 2pi−n/2  ‖f‖Ck(Rn) + pi−n/2∫BRe−|z|2∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣ dz6 C(+Rn supz∈BR∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣) ,for some C > 0. Now, if η is sufficiently small, we have thatsup|x−y|6√η R∣∣∣Dαf(x)−Dαf(y)∣∣∣ 6 R−n,thus we conclude that(4) |Dα(G ∗ f)(x)−Dαf(x)| 6 C,for any α ∈ Nn with |α| 6 k and any x ∈ B1, for a suitable C > 0.Furthermore, using (3) we see that, for any α ∈ Nn with |α| 6 k and any x ∈ B1,|Dα(G ∗ f)(x)−DαP (x)| = |Dα(G ∗ f)(x)−Dα(Q ∗ f)(x)|=∣∣∣∣∫B3(G(y)−Q(y))Dαf(x− y) dy∣∣∣∣6 C ‖f‖Ck(Rn) e−1/√η6 ,as long as η is small enough. From this and (4) we obtain‖f − P‖Ck(Rn) 6 ‖f − (G ∗ f)‖Ck(Rn) + ‖(G ∗ f)− P‖Ck(Rn) 6 C,for some C > 0, which is the desired result, up to renaming . Now, we construct a s-harmonic function in B1 that has a well-defined growth from the bound-ary:Lemma 2.2. Let ψ¯ ∈ C∞(R, [0, 1]) such that ψ¯(t) = 0 for any t ∈ R \ (2, 3) and ψ¯(t) > 0for any t ∈ (2, 3).Let ψ0(x) := ψ¯(|x|) and ψ ∈ Hs(Rn) ∩ Cs(Rn) be the solution of{(−∆)sψ = 0 in B1,ψ = ψ0 in Rn \B1.Then, if x ∈ ∂B1−, we have that(5) ψ(x) = κ s + o(s)as → 0+, for some κ > 0.4Proof. We notice that the function ψ ∈ Hs(Rn) may be constructed by the direct method ofthe calculus of variations, and also ψ ∈ Cs(Rn), see e.g. [7].Also, we use the Poisson Kernel representation (see e.g. [1,6]) to write, for any x ∈ B1,ψ(x) = c∫Rn\B1ψ0(y) (1− |x|2)s(|y|2 − 1)s |x− y|n dy= c (1− |x|2)s∫ 32[∫Sn−1ρn−1ψ¯(ρ)(ρ2 − 1)s |x− ρω|n dω]dρ,for some c > 0. Now we take x ∈ B1, with |x| = 1− , and we obtainψ(x) = c (2− 2)s∫ 32[∫Sn−1ρn−1ψ¯(ρ)(ρ2 − 1)s |(1− )e1 − ρω|n dω]dρ= 2s c s∫ 32[∫Sn−1ρn−1ψ¯(ρ)(ρ2 − 1)s |e1 − ρω|n dω]dρ+ o(s)= κ s + o(s),for some κ > 0, as desired. We observe that alternative proofs of Lemma 2.2 may be obtained from a boundary Harnackinequality in the extended problem and from explicit barriers, see [3,7].By blowing up the functions constructed in Lemma 2.2 we obtain the existence of a sequenceof s-harmonic functions approaching (x · e)s+, for a fixed unit vector e, as stated below:Corollary 2.3. Fixed e ∈ ∂B1, there exists a sequence of functions ve,j ∈ Hs(Rn)∩Cs(Rn)such that (−∆)sve,j = 0 in B1(e), ve,j = 0 in Rn \B4j(e), andve,j(x)→ κ(x · e)s+ in L1(B1(e)),as j → +∞, for some κ > 0.Proof. Let ψ be as in Lemma 2.2 andve,j(x) := jsψ(j−1x− e).The s-harmonicity of ve,j and the property of its support can be derived from the ones of ψ. Wenow prove the convergence. For this, given x ∈ B1(e) we write pj := j−1x − e and j :=1− |pj| = 1− |j−1x− e|. We remark that1 > |x− e|2 = |x|2 − 2x · e+ 1,which implies that(6) |x|2 < 2x · e, and x · e > 0 for all x ∈ B1(e).As a consequence|pj|2 = |j−1x− e|2 = j−2|x|2 + 1− 2j−1x · e = 1− 2j−1(x · e)+ + o(j−1) (x · e)2+and soj = j−1 (1 + o(1)) (x · e)+.Therefore, using (5), we haveve,j(x) = jsψ(pj)= js(κsj + o(sj))= js(κj−s(x · e)s+ + o(j−s))= κ (x · e)s+ + o(1).5Integrating over B1(e) we obtain the desired convergence. 3. SPANNING THE DERIVATIVE OF A FUNCTION AND PROOF OF THEOREM 1.1The main result of this section is that we can find a s-harmonic function with an arbitrarilylarge number of derivatives prescribed. For this, we use the standard norm notation for a givenmultiindex α = (α1, . . . , αn) ∈ Nn, according to which|α| := α1 + · · ·+ αn.Theorem 3.1. For any β ∈ Nn there exist R > r > 0, p ∈ Rn, v ∈ Hs(Rn)∩Cs(Rn) suchthat {(−∆)sv = 0 in Br(p),v = 0 in Rn \BR(p),(7)Dαv(p) = 0 for any α ∈ Nn with |α| 6 |β| − 1,(8)Dαv(p) = 0 for any α ∈ Nn with |α| = |β| and α 6= β,(9)and Dβv(p) = 1.(10)Proof. We denote by Z the set containing the couples (v, x) of all functions v ∈ Hs(Rn) ∩Cs(Rn) and points x ∈ Br(p) that satisfy (7) for some R > r > 0 and p ∈ Rn.We letN :=|β|∑j=0nj.To any (v, x) ∈ Z we can associate a vector in RN by listing all the derivatives of v up toorder |β| evaluated at x, that is (Dαv(x))|α|6|β|∈ RN .We claim that the vector space spanned by this construction exhausts RN (if we prove this,then we obtain (8)–(10) by writing the vector with entry 1 when α = β and 0 otherwise as linearcombination of the above functions).Thus we argue by contradiction, assuming that the vector space above does not exhaust RNbut lies in a subspace. That is, there exists c = (cα)|α|6|β| ∈ RN \ {0} such that(11)∑|α|6|β|cαDαv(x) = 0for any (v, x) ∈ Z . In particular, fixed any ξ ∈ Rn \ {0} and letting e := ξ/|ξ|, we havethat (11) holds true when v := ve,j and x ∈ B1(e), as warranted by Corollary 2.3.6Accordingly, for every ϕ ∈ C∞0 (B1(e)),0 = limj→+∞∫Rn∑|α|6|β|cαDαve,j(x)ϕ(x) dx= limj→+∞∫Rn∑|α|6|β|(−1)|α|cα ve,j(x)Dαϕ(x) dx= κ∫Rn∑|α|6|β|(−1)|α|cα (x · e)s+Dαϕ(x) dx= κ∫Rn∑|α|6|β|cαDα(x · e)s+ ϕ(x) dx.Consequently, for any x ∈ B1(e),(12)∑|α|6|β|cαDα(x · e)s+ = 0.Recalling (6), we observe that, for any x ∈ B1(e),Dα(x · e)s+ = s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ eα11 . . . eαnn .So we write (12) as(13)∑|α|6|β|c˜α(x) ξα = 0,for any x ∈ B1(e) and any ξ ∈ Rn \ {0} (and hence for any ξ ∈ Rn by continuity),where c˜α(x) := s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ cα.We remark that, for a fixed x ∈ B1(e), equation (13) says that a polynomial in the variable ξ isidentically equal to 0. Therefore all its coefficients must vanish, namely(14) s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ cα = 0for any x ∈ B1(e) and any |α| 6 |β|.Notice that none of the terms s, (s − 1), . . . , (s − |α| + 1) vanish since s is not an integer.Using this and (6), we deduce from (14) that cα = 0 for any |α| 6 |β|, that is c = 0, againstour assumptions. We stress that Theorem 3.1 reflects a purely nonlocal feature. Indeed, in the local case (i.e.when s = 1) the statement of Theorem 3.1 would be clearly false when |m| > 2, since thesum of the pure second derivatives of any harmonic function must vanish and cannot sum upto 1.With the aid of Theorem 3.1, we can now complete the proof of Theorem 1.1:Proof of Theorem 1.1. By Lemma 2.1, we can reduce ourselves to the case in which f is apolynomial. Consequently, the linearity of the fractional Laplace operator allows us to reduce tothe case in which f is a monomial, sayf(x) =xββ!for some β ∈ Nn. Then we take v as in Theorem 3.1 and we defineu(x) := η−|β|v(ηx+ p),7with η ∈ (0, 1/2) to be taken conveniently small in the sequel (in dependence of  that is fixedin the statement of Theorem 1.1).Let also g(x) := u(x)−f(x) = u(x)−(β!)−1xβ . By Theorem 3.1 we know thatDαg(0) = 0for any α ∈ Nn with |α| 6 |β|. Furthermore, if |α| > |β|+ 1,|Dαg(x)| = η|α|−|β||Dαv(ηx+ p)| 6 C|α| η‖v‖C|α|(B1/2(p)),for any x ∈ B1, for some C|α| > 0. As a consequence, defining k′ := k + |β| + 1 and fixedany γ ∈ Nn with |γ| 6 k′ − 1 and any x ∈ B1, we obtain by a Taylor expansion thatDγg(x) =∑|β|+16|γ|+|α|6k′−1Dγ+αg(0)α!xα +∑|γ|+|α|=k′k′α!∫ 10(1− t)k′−1Dγ+αg(tx) dt xαand so |Dγg(x)| 6 Cη, with C > 0 possibly depending also on v.Since this is valid for any x ∈ B1 we obtain that‖u− f‖Ck(B1) = ‖g‖Ck(B1) 6 ‖g‖Ck′−1(B1) 6 Cη,for some C > 0, which implies the statement of Theorem 1.1 as long as η ∈ (0, C−1). REFERENCES[1] Claudia Dalia Bucur, Representation formulas for solutions of fractional Laplacian equations, Preprint (2014).[2] L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010),no. 9, 1111–1144, DOI 10.1002/cpa.20331. MR 2675483 (2011h:49057)[3] Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Par-tial Differential Equations 32 (2007), no. 7-9, 1245–1260, DOI 10.1080/03605300600987306. MR 2354493(2009k:35096)[4] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolevspaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573, DOI 10.1016/j.bulsci.2011.12.004. MR 2944369[5] Moritz Kassmann, A new formulation of Harnack’s inequality for nonlocal operators, C. R. Math. Acad. Sci.Paris 349 (2011), no. 11-12, 637–640, DOI 10.1016/j.crma.2011.04.014 (English, with English and Frenchsummaries). MR 2817382 (2012g:31016)[6] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translatedfrom the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR0350027 (50 #2520)[7] Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to theboundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302, DOI 10.1016/j.matpur.2013.06.003. MR3168912[8] Luis Enrique Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, PhDThesis, University of Texas at Austin, 2005.[9] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series,No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)

We show that we can approximate every function f  08 Ck (B1) by an s-harmonic function in B1 that vanishes outside a compact set. That is, s-harmonic functions are dense in Cloc k. 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

S. Dipierro

O. Savin

E. Valdinoci

AIR Universita degli studi di Milano

We show that we can approximate every function f Ck (B1) with a
s-harmonic function in B1 that vanishes outside a compact set. That is,
s-harmonic functions are dense in Ck 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

Repositorium für Naturwissenschaften und Technik

Weierstraß-Institutfür Angewandte Analysis und StochastikLeibniz-Institut im Forschungsverbund Berlin e. V.Preprint ISSN 2198-5855All functions are locally s-harmonicup to a small errorSerena Dipierro1, Ovidiu Savin2, Enrico Valdinoci3submitted: April 14, 20141 Maxwell Institute for Mathematical Sciencesand School of MathematicsUniversity of EdinburghJames Clerk Maxwell BuildingKing’s BuildingsEdinburgh EH9 3JZScotlandUnited KingdomE-Mail: serena.dipierro@ed.ac.uk2 Department of MathematicsColumbia University2990 BroadwayNew York NY 10027USAE-Mail: savin@math.columbia.edu3 Weierstraß-InstitutMohrenstrasse 3910117 BerlinGermanyE-Mail: enrico.valdinoci@wias-berlin.deNo. 1944Berlin 20142010 Mathematics Subject Classification. 35R11, 60G22, 35A35, 34A08.Key words and phrases. Density properties, approximation, s-harmonic functions.The first author has been supported by EPSRC grant EP/K024566/1 “Monotonicity formula methods for nonlinearPDEs”. The second author has been supported by NSF grant DMS-1200701. The third author has been supportedby ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities” andPRIN grant 201274FYK7 “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari”.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Leibniz-Institut im Forschungsverbund Berlin e. V.Mohrenstraße 3910117 BerlinGermanyFax: +49 30 20372-303E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/1ABSTRACT. We show that we can approximate every function f ∈ Ck(B1) with a s-harmonicfunction in B1 that vanishes outside a compact set.That is, s-harmonic functions are dense in Ckloc. This result is clearly in contrast with therigidity of harmonic functions in the classical case and can be viewed as a purely nonlocalfeature.1. INTRODUCTIONIt is a well-known fact that harmonic functions are very rigid. For instance, in dimension 1, theyreduce to a linear function and, in any dimension, they never possess local extrema.The goal of this paper is to show that the situation for fractional harmonic functions is completelydifferent, namely one can fix any function in a given domain and find a s-harmonic functionarbitrarily close to it.Heuristically speaking, the reason for this phenomenon is that while classical harmonic functionsare determined once their trace on the boundary is fixed, in the fractional setting the operatorsees all the data outside the domain. Hence, a careful choice of these data allows a s-harmonicfunction to “bend up and down” basically without any restriction.The rigorous statement of this fact is in the following Theorem 1.1. For this, we recall that,given s ∈ (0, 1), the fractional Laplace operator of a function u is defined (up to a normalizingconstant) as(−∆)su(x) :=∫Rn2u(x)− u(x+ y)− u(x− y)|y|n+2sdy.We refer to [4,6,8,9] for other equivalent definitions, motivations and applications.Theorem 1.1. Fix k ∈ N. Then, given any function f ∈ Ck(B1) and any ε > 0, thereexist R > 1 and u ∈ Hs(Rn) ∩ Cs(Rn) such that{(−∆)su = 0 in B1,u = 0 in Rn \BRand‖f − u‖Ck(B1) 6 ε.As usual, in Theorem 1.1, we have denoted by Ck(B1) the space of all the functions f : B1 →R that possess an extension f̃ ∈ Ck(B1+µ) (i.e. f̃ = f in B1), for some µ > 0.We also mention that an important rigidity feature for classical harmonic functions is imposedby Harnack inequality: namely if u is harmonic and non-negative in B1 then u(x) and u(y)are comparable for any x, y ∈ B1/2. A striking difference with the nonlocal case is that thistype of Harnack inequality fails for the fractional Laplacian (namely it is necessary to requirethat u is non-negative in the whole of Rn and not only in B1, see e.g. Theorem 2.2 in [5]). Asan application of Theorem 1.1, we point out that one can construct examples of s-harmonicfunctions with a “wild” behavior, that oscillate as many times as we want, and reach interiorextrema basically at any prescribed point. In particular, one can construct s-harmonic functionsto be used as barriers basically without any geometric restriction.As a final observation, we would like to stress that, while Theorem 1.1 reveals a purely non-local phenomenon, a similar result does not hold for any nonlocal operator. For instance, it isnot possible to replace “s-harmonic functions” with “nonlocal minimal surfaces” in the statementof Theorem 1.1, that is it is not true that any graph may be locally approximated by nonlocal2minimal surfaces. Indeed, the uniform density estimates satisfied by the nonlocal minimal sur-faces prescribe a severe geometric restriction that prevent the formation of sharp edges andthin spikes.We refer to [2] for the definition of nonlocal minimal surfaces and for their density properties: asa matter of fact, one of the consequence of Theorem 1.1 is that density properties do not holdtrue for s-harmonic functions, so s-harmonic functions and nonlocal minimal surfaces may havevery different behaviors.The rest of the paper is organized as follows: in Section 2 we collect some preliminary results,such as a (probably well-known) generalization of the Stone-Weierstrass Theorem and the con-struction of a s-harmonic function inB1 that has a well-defined growth from the boundary. Then,in Section 3, we construct a s-harmonic function with an arbitrarily large number of derivativesprescribed. This is, in a sense, already the core of our argument, since these types of propertiesare typical for the fractional case and do not hold for classical harmonic functions. Also, fromthis result, the proof of Theorem 1.1 will follow via a scaling and approximation method.2. PRELIMINARY OBSERVATIONSIn this section we collect some auxiliary results that will be needed in the rest of the paper.First of all, we recall a version of the Stone-Weierstrass Theorem for smooth functions. We givea quick proof of it since in general this result is presented only in the continuous setting.Lemma 2.1. For any f ∈ Ck(B1) and any ε > 0 there exists a polynomial P such that ‖f −P‖Ck(B1) 6 ε.Proof. Without loss of generality we may suppose that f ∈ Ck0 (B2). Also, given ε > 0 as inthe statement of Lemma 2.1, we fix R > 0 such that(1)∫Rn\BRe−|x|2dx 6 ε.Then, we fix η > 0, to be taken arbitrarily small (possibly in dependence of ε and R, which arefixed once and for all), and we take Jη ∈ N large enough such that(2)∑j>Jη(−1)jj! ηj6 e−1/η.Let alsoQ(x) := (πη)−n/2Jη∑j=0(−1)j |x|2jj! ηj,P (x) :=∫Rnf(y)Q(x− y) dy,and G(x) := (πη)−n/2e−|x|2/η.We remark that Q is a polynomial in x, hence so is P . Moreover, by a Taylor expansion,G(x) = Q(x) + (πη)−n/2∑j>Jη(−1)j |x|2jj! ηjand so, using (2), we conclude that, for any x ∈ B3,(3) |G(x)−Q(x)| 6 e−1/√η,3provided that η is sufficiently small.Now we recall (1) and we observe that, for any α ∈ Nn with |α| 6 k and any x ∈ B1,|Dα(G ∗ f)(x)−Dαf(x)|=∣∣∣∣∫RnG(y)(Dαf(x− y)−Dαf(x))dy∣∣∣∣6 π−n/2∫Rne−|z|2∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣ dz6 2π−n/2 ε ‖f‖Ck(Rn) + π−n/2∫BRe−|z|2∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣ dz6 C(ε+Rn supz∈BR∣∣∣Dαf(x−√η z)−Dαf(x)∣∣∣) ,for some C > 0. Now, if η is sufficiently small, we have thatsup|x−y|6√η R∣∣∣Dαf(x)−Dαf(y)∣∣∣ 6 R−nε,thus we conclude that(4) |Dα(G ∗ f)(x)−Dαf(x)| 6 Cε,for any α ∈ Nn with |α| 6 k and any x ∈ B1, for a suitable C > 0.Furthermore, using (3) we see that, for any α ∈ Nn with |α| 6 k and any x ∈ B1,|Dα(G ∗ f)(x)−DαP (x)| = |Dα(G ∗ f)(x)−Dα(Q ∗ f)(x)|=∣∣∣∣∫B3(G(y)−Q(y))Dαf(x− y) dy∣∣∣∣6 C ‖f‖Ck(Rn) e−1/√η6 ε,as long as η is small enough. From this and (4) we obtain‖f − P‖Ck(Rn) 6 ‖f − (G ∗ f)‖Ck(Rn) + ‖(G ∗ f)− P‖Ck(Rn) 6 Cε,for some C > 0, which is the desired result, up to renaming ε. Now, we construct a s-harmonic function in B1 that has a well-defined growth from the bound-ary:Lemma 2.2. Let ψ̄ ∈ C∞(R, [0, 1]) such that ψ̄(t) = 0 for any t ∈ R \ (2, 3) and ψ̄(t) > 0for any t ∈ (2, 3).Let ψ0(x) := ψ̄(|x|) and ψ ∈ Hs(Rn) ∩ Cs(Rn) be the solution of{(−∆)sψ = 0 in B1,ψ = ψ0 in Rn \B1.Then, if x ∈ ∂B1−ε, we have that(5) ψ(x) = κ εs + o(εs)as ε→ 0+, for some κ > 0.4Proof. We notice that the function ψ ∈ Hs(Rn) may be constructed by the direct method ofthe calculus of variations, and also ψ ∈ Cs(Rn), see e.g. [7].Also, we use the Poisson Kernel representation (see e.g. [1,6]) to write, for any x ∈ B1,ψ(x) = c∫Rn\B1ψ0(y) (1− |x|2)s(|y|2 − 1)s |x− y|ndy= c (1− |x|2)s∫ 32[∫Sn−1ρn−1ψ̄(ρ)(ρ2 − 1)s |x− ρω|ndω]dρ,for some c > 0. Now we take x ∈ B1, with |x| = 1− ε, and we obtainψ(x) = c (2ε− ε2)s∫ 32[∫Sn−1ρn−1ψ̄(ρ)(ρ2 − 1)s |(1− ε)e1 − ρω|ndω]dρ= 2s c εs∫ 32[∫Sn−1ρn−1ψ̄(ρ)(ρ2 − 1)s |e1 − ρω|ndω]dρ+ o(εs)= κ εs + o(εs),for some κ > 0, as desired. We observe that alternative proofs of Lemma 2.2 may be obtained from a boundary Harnackinequality in the extended problem and from explicit barriers, see [3,7].By blowing up the functions constructed in Lemma 2.2 we obtain the existence of a sequenceof s-harmonic functions approaching (x · e)s+, for a fixed unit vector e, as stated below:Corollary 2.3. Fixed e ∈ ∂B1, there exists a sequence of functions ve,j ∈ Hs(Rn)∩Cs(Rn)such that (−∆)sve,j = 0 in B1(e), ve,j = 0 in Rn \B4j(e), andve,j(x)→ κ(x · e)s+ in L1(B1(e)),as j → +∞, for some κ > 0.Proof. Let ψ be as in Lemma 2.2 andve,j(x) := jsψ(j−1x− e).The s-harmonicity of ve,j and the property of its support can be derived from the ones of ψ. Wenow prove the convergence. For this, given x ∈ B1(e) we write pj := j−1x − e and εj :=1− |pj| = 1− |j−1x− e|. We remark that1 > |x− e|2 = |x|2 − 2x · e+ 1,which implies that(6) |x|2 < 2x · e, and x · e > 0 for all x ∈ B1(e).As a consequence|pj|2 = |j−1x− e|2 = j−2|x|2 + 1− 2j−1x · e = 1− 2j−1(x · e)+ + o(j−1) (x · e)2+and soεj = j−1 (1 + o(1)) (x · e)+.Therefore, using (5), we haveve,j(x) = jsψ(pj)= js(κεsj + o(εsj))= js(κj−s(x · e)s+ + o(j−s))= κ (x · e)s+ + o(1).5Integrating over B1(e) we obtain the desired convergence. 3. SPANNING THE DERIVATIVE OF A FUNCTION AND PROOF OF THEOREM 1.1The main result of this section is that we can find a s-harmonic function with an arbitrarilylarge number of derivatives prescribed. For this, we use the standard norm notation for a givenmultiindex α = (α1, . . . , αn) ∈ Nn, according to which|α| := α1 + · · ·+ αn.Theorem 3.1. For any β ∈ Nn there exist R > r > 0, p ∈ Rn, v ∈ Hs(Rn)∩Cs(Rn) suchthat {(−∆)sv = 0 in Br(p),v = 0 in Rn \BR(p),(7)Dαv(p) = 0 for any α ∈ Nn with |α| 6 |β| − 1,(8)Dαv(p) = 0 for any α ∈ Nn with |α| = |β| and α 6= β,(9)and Dβv(p) = 1.(10)Proof. We denote by Z the set containing the couples (v, x) of all functions v ∈ Hs(Rn) ∩Cs(Rn) and points x ∈ Br(p) that satisfy (7) for some R > r > 0 and p ∈ Rn.We letN :=|β|∑j=0nj.To any (v, x) ∈ Z we can associate a vector in RN by listing all the derivatives of v up toorder |β| evaluated at x, that is (Dαv(x))|α|6|β|∈ RN .We claim that the vector space spanned by this construction exhausts RN (if we prove this,then we obtain (8)–(10) by writing the vector with entry 1 when α = β and 0 otherwise as linearcombination of the above functions).Thus we argue by contradiction, assuming that the vector space above does not exhaust RNbut lies in a subspace. That is, there exists c = (cα)|α|6|β| ∈ RN \ {0} such that(11)∑|α|6|β|cαDαv(x) = 0for any (v, x) ∈ Z . In particular, fixed any ξ ∈ Rn \ {0} and letting e := ξ/|ξ|, we havethat (11) holds true when v := ve,j and x ∈ B1(e), as warranted by Corollary 2.3.6Accordingly, for every ϕ ∈ C∞0 (B1(e)),0 = limj→+∞∫Rn∑|α|6|β|cαDαve,j(x)ϕ(x) dx= limj→+∞∫Rn∑|α|6|β|(−1)|α|cα ve,j(x)Dαϕ(x) dx= κ∫Rn∑|α|6|β|(−1)|α|cα (x · e)s+Dαϕ(x) dx= κ∫Rn∑|α|6|β|cαDα(x · e)s+ ϕ(x) dx.Consequently, for any x ∈ B1(e),(12)∑|α|6|β|cαDα(x · e)s+ = 0.Recalling (6), we observe that, for any x ∈ B1(e),Dα(x · e)s+ = s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ eα11 . . . eαnn .So we write (12) as(13)∑|α|6|β|c̃α(x) ξα = 0,for any x ∈ B1(e) and any ξ ∈ Rn \ {0} (and hence for any ξ ∈ Rn by continuity),where c̃α(x) := s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ cα.We remark that, for a fixed x ∈ B1(e), equation (13) says that a polynomial in the variable ξ isidentically equal to 0. Therefore all its coefficients must vanish, namely(14) s (s− 1) . . . (s− |α|+ 1) (x · e)s−|α|+ cα = 0for any x ∈ B1(e) and any |α| 6 |β|.Notice that none of the terms s, (s − 1), . . . , (s − |α| + 1) vanish since s is not an integer.Using this and (6), we deduce from (14) that cα = 0 for any |α| 6 |β|, that is c = 0, againstour assumptions. We stress that Theorem 3.1 reflects a purely nonlocal feature. Indeed, in the local case (i.e.when s = 1) the statement of Theorem 3.1 would be clearly false when |m| > 2, since thesum of the pure second derivatives of any harmonic function must vanish and cannot sum upto 1.With the aid of Theorem 3.1, we can now complete the proof of Theorem 1.1:Proof of Theorem 1.1. By Lemma 2.1, we can reduce ourselves to the case in which f is apolynomial. Consequently, the linearity of the fractional Laplace operator allows us to reduce tothe case in which f is a monomial, sayf(x) =xββ!for some β ∈ Nn. Then we take v as in Theorem 3.1 and we defineu(x) := η−|β|v(ηx+ p),7with η ∈ (0, 1/2) to be taken conveniently small in the sequel (in dependence of ε that is fixedin the statement of Theorem 1.1).Let also g(x) := u(x)−f(x) = u(x)−(β!)−1xβ . By Theorem 3.1 we know thatDαg(0) = 0for any α ∈ Nn with |α| 6 |β|. Furthermore, if |α| > |β|+ 1,|Dαg(x)| = η|α|−|β||Dαv(ηx+ p)| 6 C|α| η‖v‖C|α|(B1/2(p)),for any x ∈ B1, for some C|α| > 0. As a consequence, defining k′ := k + |β| + 1 and fixedany γ ∈ Nn with |γ| 6 k′ − 1 and any x ∈ B1, we obtain by a Taylor expansion thatDγg(x) =∑|β|+16|γ|+|α|6k′−1Dγ+αg(0)α!xα +∑|γ|+|α|=k′k′α!∫ 10(1− t)k′−1Dγ+αg(tx) dt xαand so |Dγg(x)| 6 Cη, with C > 0 possibly depending also on v.Since this is valid for any x ∈ B1 we obtain that‖u− f‖Ck(B1) = ‖g‖Ck(B1) 6 ‖g‖Ck′−1(B1) 6 Cη,for some C > 0, which implies the statement of Theorem 1.1 as long as η ∈ (0, C−1ε). 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All functions are locally $s$ -harmonic up to a small error

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All functions are locally sss-harmonic up to a small error

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