Boundary properties of fractional objects: Flexibility of linear equations
and rigidity of minimal graphs

Serena Dipierro; Ovidiu Savin; Enrico Valdinoci

doi:10.1515/crelle-2019-0045

Published by De Gruyter April 16, 2020

Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs

Serena Dipierro , Ovidiu Savin and Enrico Valdinoci

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2019-0045

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Abstract

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in ℝ3, i.e. nonlocal minimal surfaces with a graphical structure.

We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.

This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.

We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

Funding statement: The first and third authors are member of INdAM and are supported by the Australian Research Council Discovery Project DP170104880 NEW “Nonlocal Equations at Work”. The first author’s visit to Columbia has been partially funded by the Fulbright Foundation and the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”. The second author is supported by the National Science Foundation grant DMS-1500438. The third author’s visit to Columbia has been partially funded by the Italian Piano di Sostegno alla Ricerca “Equazioni nonlocali e applicazioni”.

A A classical counterpart of Theorem 1.1

In this appendix, we show that Theorem 1.1 possesses a classical counterpart for the Laplace equation, namely:

Theorem A.1.

Let n⩾2, k∈N and f∈Ck⁢(B1′¯). Then, for every ε>0 there exist fε∈Ck⁢(Rn-1) and uε∈C⁢(Rn) such that

{Δ⁢uε=0in B1∩{xn>0},uε=0in B1∩{xn=0},

limxn↘0⁡uε⁢(x′,xn)xn=fε⁢(x′) for all ⁢x′∈B1′,

and

∥fε-f∥Ck⁢(B1′¯)⩽ε.

We observe that formally Theorem A.1 corresponds to Theorem 1.1 with σ:=1. The arguments that we proposed for σ∈(0,1) can be carried out also when σ=1, and thus prove Theorem A.1. Indeed, for the classical Laplace equation, one can obtain Lemma 2.1 by taking, for instance, the real part of the holomorphic function ℂ∋z↦zk, with k∈ℕ, and then use this homogeneous solution in the proof of Lemma 2.2. Then, once Lemma 2.2 is proved with σ:=1, one can exploit the proof of Theorem 1.1 presented in Section 2 with σ:=1 and obtain Theorem A.1.

However, in the classical case there is also an explicit polynomial expansion which recovers Lemma 2.2 with σ:=1, that is when (2.3) is replaced by

{Δ⁢u=0in Br∩{xn>0},u=0in Br∩{xn=0}.

Hence, to establish Theorem A.1, we focus on the following argument of classical flavor (which is not reproducible for σ∈(0,1) and thus requires new strategies in the case of Theorem 1.1).

Proof of Lemma 2.2 with σ:=1.

Given

κ=(κγ)γ∈ℕn-1|γ|⩽k

in ℝNk, where Nk is defined in (2.5), we aim at finding a function u∈C⁢(B1¯) such that

(A.1){Δ⁢u=0in B1∩{xn>0},u=0in B1∩{xn=0},

and for which there exists ϕ∈Ck⁢(B1′¯) such that

(A.2)limxn↘0⁡u⁢(x′,xn)xn=ϕ⁢(x′) for all ⁢x′∈B1′,

with, in the notation of (2.4),

(A.3)𝒟k⁢ϕ⁢(0)=κ.

To this end, we define ϕ to be the polynomial

ϕ⁢(x′):=∑γ∈ℕn-1|γ|⩽kκγγ!⁢(x′)γ.

In this way, (A.3) is automatically satisfied. Furthermore, we observe that Δx′j⁢ϕ vanishes identically when 2⁢j>k, and therefore we can set

u⁢(x′,xn):=∑j∈ℕj⩽k/2(-1)j⁢Δx′j⁢ϕ⁢(x′)(2⁢j+1)!⁢xn2⁢j+1=∑j=0+∞(-1)j⁢Δx′j⁢ϕ⁢(x′)(2⁢j+1)!⁢xn2⁢j+1.

We observe that

limxn↘0⁡u⁢(x′,xn)xn=limxn↘0⁡∑j∈ℕj⩽k/2(-1)j⁢Δx′j⁢ϕ⁢(x′)(2⁢j+1)!⁢xn2⁢j=ϕ⁢(x′),

which establishes (A.2).

Also, we see that u⁢(x′,0)=0, and moreover, using the notation J:=j-1,

Δ⁢u⁢(x)=Δx′⁢u⁢(x)+∂xn2⁡u⁢(x)

=∑j=0+∞(-1)j⁢Δx′j+1⁢ϕ⁢(x′)(2⁢j+1)!⁢xn2⁢j+1+∑j=1+∞(-1)j⁢Δx′j⁢ϕ⁢(x′)(2⁢j-1)!⁢xn2⁢j-1

=∑j=0+∞(-1)j⁢Δx′j+1⁢ϕ⁢(x′)(2⁢j+1)!⁢xn2⁢j+1+∑J=0+∞(-1)J+1⁢Δx′J+1⁢ϕ⁢(x′)(2⁢J+1)!⁢xn2⁢J+1

=0,

which gives (A.1). ∎

References

[1] R. Bañuelos and K. Bogdan, Symmetric stable processes in cones, Potential Anal. 21 (2004), no. 3, 263–288. 10.1023/B:POTA.0000033333.72236.dcSearch in Google Scholar

[2] B. Barrios, A. Figalli and X. Ros-Oton, Global regularity for the free boundary in the obstacle problem for the fractional Laplacian, Amer. J. Math. 140 (2018), no. 2, 415–447. 10.1353/ajm.2018.0010Search in Google Scholar

[3] B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609–639. 10.2422/2036-2145.201202_007Search in Google Scholar

[4] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43–80. 10.4064/sm-123-1-43-80Search in Google Scholar

[5] J. P. Borthagaray, W. Li and R. H. Nochetto, Finite element discretizations of nonlocal minimal graphs: Convergence, Nonlinear Anal. 189 (2019), Article ID 111566. 10.1016/j.na.2019.06.025Search in Google Scholar

[6] C. Bucur, Local density of Caputo-stationary functions in the space of smooth functions, ESAIM Control Optim. Calc. Var. 23 (2017), no. 4, 1361–1380. 10.1051/cocv/2016056Search in Google Scholar

[7] C. Bucur, L. Lombardini and E. Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 3, 655–703. 10.1016/j.anihpc.2018.08.003Search in Google Scholar

[8] X. Cabré, E. Cinti and J. Serra, Stable s-minimal cones in ℝ3 are flat for s∼1, J. reine angew. Math. (2019), 10.1515/crelle-2019-0005. 10.1515/crelle-2019-0005Search in Google Scholar

[9] X. Cabré and M. Cozzi, A gradient estimate for nonlocal minimal graphs, Duke Math. J. 168 (2019), no. 5, 775–848. 10.1215/00127094-2018-0052Search in Google Scholar

[10] L. Caffarelli, D. De Silva and O. Savin, Obstacle-type problems for minimal surfaces, Comm. Partial Differential Equations 41 (2016), no. 8, 1303–1323. 10.1080/03605302.2016.1192646Search in Google Scholar

[11] L. Caffarelli, S. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models 10 (2017), no. 1, 141–170. 10.3934/krm.2017006Search in Google Scholar

[12] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. 10.1002/cpa.20331Search in Google Scholar

[13] L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843–871. 10.1016/j.aim.2013.08.007Search in Google Scholar

[14] A. Carbotti, S. Dipierro and E. Valdinoci, Local density of solutions to fractional equations, De Gruyter Studies in Mathematics 74, De Gruyter, Berlin 2019. 10.1515/9783110664355Search in Google Scholar

[15] A. Carbotti, S. Dipierro and E. Valdinoci, Local density of caputo-stationary functions of any order, Complex Var. Elliptic Equ. (2018), 10.1080/17476933.2018.1544631. 10.1080/17476933.2018.1544631Search in Google Scholar

[16] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, J. Differential Geom. 112 (2019), no. 3, 447–504. 10.4310/jdg/1563242471Search in Google Scholar

[17] M. Cozzi, A. Farina and L. Lombardini, Bernstein–Moser-type results for nonlocal minimal graphs, Comm. Anal. Geom., to appear. 10.4310/CAG.2021.v29.n4.a1Search in Google Scholar

[18] M. Cozzi and A. Figalli, Regularity theory for local and nonlocal minimal surfaces: An overview, Nonlocal and nonlinear diffusions and interactions: New methods and directions, Lecture Notes in Math. 2186, Springer, Cham (2017), 117–158. 10.1007/978-3-319-61494-6_3Search in Google Scholar

[19] J. Dávila, M. del Pino and J. Wei, Nonlocal s-minimal surfaces and Lawson cones, J. Differential Geom. 109 (2018), no. 1, 111–175. 10.4310/jdg/1525399218Search in Google Scholar

[20] S. Dipierro, O. Savin and E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 86. 10.1007/s00526-016-1020-9Search in Google Scholar

[21] S. Dipierro, O. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 957–966. 10.4171/JEMS/684Search in Google Scholar

[22] S. Dipierro, O. Savin and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal. 272 (2017), no. 5, 1791–1851. 10.1016/j.jfa.2016.11.016Search in Google Scholar

[23] S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, J. Geom. Anal. 29 (2019), no. 2, 1428–1455. 10.1007/s12220-018-0045-zSearch in Google Scholar

[24] S. Dipierro, O. Savin and E. Valdinoci, Nonlocal minimal graphs in the plane are generically sticky, preprint (2019), https://arxiv.org/abs/1904.05393; Comm. Math. Phys., to appear. 10.1007/s00220-020-03771-8Search in Google Scholar

[25] S. Dipierro and E. Valdinoci, Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness, Recent developments in nonlocal theory, De Gruyter, Berlin (2018), 165–209. 10.1515/9783110571561-006Search in Google Scholar

[26] A. Farina and E. Valdinoci, Flatness results for nonlocal minimal cones and subgraphs, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 4, 1281–1301. 10.2422/2036-2145.201708_019Search in Google Scholar

[27] A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. reine angew. Math. 729 (2017), 263–273. 10.1515/crelle-2015-0006Search in Google Scholar

[28] G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE 7 (2014), no. 7, 1649–1682. 10.2140/apde.2014.7.1649Search in Google Scholar

[29] G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. 10.1016/j.aim.2014.09.018Search in Google Scholar

[30] N. V. Krylov, On the paper “All functions are locally s-harmonic up to a small error” by Dipierro, Savin, and Valdinoci, preprint (2018), https://arxiv.org/abs/1810.07648. Search in Google Scholar

[31] L. Lombardini, Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces, Interfaces Free Bound. 20 (2018), no. 2, 261–296. 10.4171/IFB/402Search in Google Scholar

[32] P. J. Méndez-Hernández, Exit times from cones in 𝐑n of symmetric stable processes, Illinois J. Math. 46 (2002), no. 1, 155–163. 10.1215/ijm/1258136146Search in Google Scholar

[33] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302. 10.1016/j.matpur.2013.06.003Search in Google Scholar

[34] X. Ros-Oton and J. Serra, Boundary regularity estimates for nonlocal elliptic equations in C1 and C1,α domains, Ann. Mat. Pura Appl. (4) 196 (2017), no. 5, 1637–1668. 10.1007/s10231-016-0632-1Search in Google Scholar

[35] A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems 34 (2018), no. 4, Article ID 045003. 10.1088/1361-6420/aaac5aSearch in Google Scholar

[36] M. Sáez and E. Valdinoci, On the evolution by fractional mean curvature, Comm. Anal. Geom. 27 (2019), no. 1, 211–249. 10.4310/CAG.2019.v27.n1.a6Search in Google Scholar

[37] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 33–39. 10.1007/s00526-012-0539-7Search in Google Scholar

[38] S. Terracini, G. Tortone and S. Vita, On s-harmonic functions on cones, Anal. PDE 11 (2018), no. 7, 1653–1691. 10.2140/apde.2018.11.1653Search in Google Scholar

Received: 2019-07-03

Revised: 2019-12-15

Published Online: 2020-04-16

Published in Print: 2020-12-01