Regularity for a fractional p-Laplace equation
aa r X i v : . [ m a t h . A P ] A ug REGULARITY FOR A FRACTIONAL p -LAPLACEEQUATION ARMIN SCHIKORRA, TIEN-TSAN SHIEH, AND DANIEL SPECTOR
Abstract.
In this note we consider regularity theory for a frac-tional p -Laplace operator which arises in the complex interpolationof the Sobolev spaces, the H s,p -Laplacian. We obtain the natu-ral analogue to the classical p -Laplacian situation, namely C s + αloc -regularity for the homogeneous equation. Introduction and main result
In recent years equations involving what we will call the distributional W s,p -Laplacian, defined for test functions ϕ as( − ∆) sp u [ ϕ ] := Z R d Z R d | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d + sp dy dx, have received a lot of attention, e.g. [3, 6, 7, 12, 14, 15, 19]. The W s,p -Laplacian ( − ∆) sp appears when one computes the first variationof certain energies involving the W s,p semi-norm[ u ] W s,p ( R d ) := (cid:18)Z R d Z R d | u ( x ) − u ( y ) | p | x − y | d + sp dy dx (cid:19) p , (1.1)which was introduced by Gagliardo [10] and independently by Slo-bodeckij [25] to describe the trace spaces of Sobolev maps.Regularity theory for equations involving this fractional p -Laplace op-erator is a very challenging open problem and only partial resultsare known: C ,αloc -regularity for suitable right-hand-side data was ob-tained by Di Castro, Kuusi and Palatucci [6, 7]; A generalization ofthe Gehring lemma was proven by Kuusi, Mingione and Sire [14, 15]; A.S. is supported by DFG-grant SCHI-1257-3-1 and the DFG-Heisenberg fellow-ship. D.S. is supported by the Taiwan Ministry of Science and Technology underresearch grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2. Part ofthis work was written while A.S. was visiting NCTU with support from the TaiwanMinistry of Science and Technology through the Mathematics Research PromotionCenter.
A stability theorem similar to the Iwaniec stability result for the p -Laplacian was established by the first-named author [21]. The currentstate-of-the art with respect to regularity theory is higher Sobolev-regularity by Brasco and Lindgren [3].Aside from their origins as trace spaces, the fractional Sobolev spaces W s,p ( R d ) := (cid:8) u ∈ L p ( R d ) : [ u ] W s,p ( R d ) < + ∞ (cid:9) also arise in the real interpolation of L p and ˙ W ,p . If one alternativelyconsiders the complex interpolation method, one is naturally led toanother kind of fractional Sobolev space H s,p ( R d ), where taking theplace of the differential energy (1.1) one can utilize the semi-norm(1.2) [ u ] H s,p ( R d ) := (cid:18)Z R d | D s u | p (cid:19) p . Here D s = ( ∂ s ∂x s , . . . , ∂ s ∂x sd ) is the fractional gradient for ∂ s u∂x si ( x ) := c d,s Z R d u ( x ) − u ( y ) | x − y | d + s x i − y i | x − y | dy, i = 1 , . . . , d. Composition formulae for the fractional gradient have been studied inthe classical work [11], while more recently they have been consideredby a number of authors [1, 4, 5, 20, 22, 24]. While it is common in theliterature (for example in [17]) to see H s,p ( R d ) equipped with the L p -norm of the fractional Laplacian ( − ∆) s (see Section 2 for a definition),we here utilize (1.2) because it preserves the structural properties ofthe spaces for s ∈ (0 ,
1) more appropriately. In particular, for s = 1 wehave D = D (the constant c d,s tends to zero as s tends to one), whilefor s ∈ (0 ,
1) the fractional Sobolev spaces defined this way support afractional Sobolev inequality in the case p = 1 (see [23]). Let us alsoremark that for p = 2 these spaces are the same, W s, = H s, , but for p = 2 this is not the case.Returning to the question of a fractional p -Laplacian, in the contextof H s,p ( R d ) computing the first variation of energies involving the H s,p semi-norms (1.2) yields an alternative fractional version of a p -Laplacian, we shall call it the H s,p -Laplaciandiv s ( | D s u | p − D s u ) = d X i =1 ∂ s ∂x si ( | D s u | p − ∂ s u∂x si ) . EGULARITY FOR A FRACTIONAL p -LAPLACE EQUATION 3 Somewhat surprisingly while the regularity theory for the homogeneousequation of the W s,p -Laplacian( − ∆) sp u = 0is far from being understood, the regularity for the H s,p -Laplacian(1.3) div s ( | D s u | p − D s u ) = 0actually follows the classical theory, which is the main result we provein this note: Theorem 1.1.
Let Ω ⊂ R d be open, p ∈ (2 − d , ∞ ) and s ∈ (0 , .Suppose u ∈ H s,p ( R d ) is a distributional solution to (1.3) , that is (1.4) Z R d | D s u | p − D s u · D s ϕ = 0 ∀ ϕ ∈ C ∞ c (Ω) . Then u ∈ C s + αloc (Ω) for some α > only depending on p . The key observation for Theorem 1.1 is that v := I − s u , where I − s denotes the Riesz potential, actually solves an inhomogeneous classical p -Laplacian equation with good right-hand side. Proposition 1.2.
Let u be as in Theorem 1.1. Then v := I − s u satis-fies − div( | Dv | p − Dv ) ∈ L ∞ loc (Ω) . Therefore, Theorem 1.1 follows from the regularity theory of the clas-sical p -Laplacian: By Proposition 1.1, v is a distributional solutionto div( | Dv | p − Dv ) = µ and µ is sufficiently integrable whence v ∈ C ,αloc (Ω) [8, 9, 26] (see alsothe excellent survey paper by Mingione [18]). In particular, one canapply the potential estimates by Kuusi and Mingione [13, Theorem1.4, Theorem 1.6] to deduce that Dv ∈ C ,αloc (Ω), which implies that u ∈ C s + αloc (Ω).Let us also remark, that the reduction argument used for Proposi-tion 1.2 extends the class of fractional partial differential equationsintroduced in [24], which will be treated in a forthcoming work. ARMIN SCHIKORRA, TIEN-TSAN SHIEH, AND DANIEL SPECTOR Proof of Proposition 1.2
With ( − ∆) σ we denote the fractional Laplacian( − ∆) σ f ( x ) := ˜ c d,σ Z R d f ( x ) − f ( y ) | x − y | d + σ dy, and with I σ its inverse, the Riesz potential. Let v := I − s u where u satisfies (1.4), so that(2.1) Z R d | Dv | p − Dv · D s ϕ = 0 for all ϕ ∈ C ∞ c (Ω).Now let Ω ⋐ Ω be an arbitrary open set compactly contained in Ω,and let φ be a test function supported in Ω . Pick an open set Ω so that Ω ⋐ Ω ⋐ Ω and a cutoff function η , supported in Ω andconstantly one in Ω . Then in particular one can take ϕ := η ( − ∆) − s φ as a test function in (2.1) to obtain Z R d | Dv | p − Dv · D s ( η ( − ∆) − s φ ) = 0 . That is, Z R d | Dv | p − Dv · Dφ = Z R d | Dv | p − Dv · D s ( η c ( − ∆) − s φ ) . where η c := (1 − η ). We set T ( φ ) := D s ( η c ( − ∆) − s φ )Now we show that by the disjoint support of η c and φ we have(2.2) k T ( φ ) k L p ( R d ) ≤ C Ω , Ω ,d,s,p k φ k L ( R d ) . Once we have this, the claim is proven as H¨older’s inequality and real-izing the L ∞ norm via duality implies − div( | Dv | p − Dv ) ∈ L ∞ loc (Ω) . To see (2.2), we use the disjoint support arguments as in [2, LemmaA.1] [16, Lemma 3.6.]: First we see that since η c ( x ) φ ( x ) ≡ T ( φ ) = ˜ c d, − s D s Z R d η c ( x ) φ ( y ) | x − y | N +1 − s dy. Now taking a cutoff-function ζ supported in Ω , ζ ≡ we have T ( φ ) = ˜ c d, − s D s Z R d η c ( x ) ζ ( y ) φ ( y ) | x − y | N +1 − s dy = ˜ c d, − s Z R d k ( x, y ) φ ( y ) dy, EGULARITY FOR A FRACTIONAL p -LAPLACE EQUATION 5 where k ( x, y ) := D sx κ ( x, y ) := D sx η c ( x ) ζ ( y ) | x − y | N +1 − s . The positive distance between the supports of η c and ζ implies thatthese kernels k , κ are a smooth, bounded, integrable (both, in x and in y ), and thus by a Young-type convolution argument we obtain (2.2).One can also argue by interpolation, (cid:13)(cid:13)(cid:13)(cid:13)Z R d κ ( x, y ) φ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ k φ k L ( R d ) , as well as (cid:13)(cid:13)(cid:13)(cid:13)Z R d D x κ ( x, y ) φ ( y ) dy (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ k φ k L ( R d ) . Interpolating this implies the desired result that (cid:13)(cid:13)(cid:13)(cid:13)Z R d D sx κ ( x, y ) φ ( y ) dy (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ k φ k L ( R d ) . Thus (2.2) is established and the proof of Proposition 1.2 is finished. (cid:3)
References
1. P. Biler, C. Imbert, and G. Karch,
The nonlocal porous medium equation:Barenblatt profiles and other weak solutions , Arch. Ration. Mech. Anal. (2015), no. 2, 497–529. MR 32944092. S. Blatt, Ph. Reiter, and A. Schikorra,
Harmonic analysis meets critical knots.Critical points of the M¨obius energy are smooth , Trans. Amer. Math. Soc. (2016), no. 9, 6391–6438. MR 34610383. L. Brasco and E. Lindgren,
Higher sobolev regularity for the fractional p -laplaceequation in the superquadratic case , Adv.Math. (2015).4. L. Caffarelli, F. Soria, and J.-L. V´azquez, Regularity of solutions of the frac-tional porous medium flow , J. Eur. Math. Soc. (JEMS) (2013), no. 5, 1701–1746. MR 30822415. L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractionalpotential pressure , Arch. Ration. Mech. Anal. (2011), no. 2, 537–565.MR 28475346. A. Di Castro, T. Kuusi, and G. Palatucci,
Local behaviour of fractional p -minimizers , preprint (2014).7. , Nonlocal harnack inequalities , J. Funct. Anal. (2014), 1807–1836.8. E. DiBenedetto, C α local regularity of weak solutions of degenerate ellipticequations , Nonlinear Anal. (1983), no. 8, 827–850.9. L. C. Evans, A new proof of local C ,α regularity for solutions of certain degen-erate elliptic p.d.e , J. Differential Equations (1982), no. 3, 356–373. ARMIN SCHIKORRA, TIEN-TSAN SHIEH, AND DANIEL SPECTOR
10. E. Gagliardo,
Caratterizzazioni delle tracce sulla frontiera relative ad alcuneclassi di funzioni in n variabili , Rend. Sem. Mat. Univ. Padova (1957),284–305. MR 0102739 (21 On some composition formulas , Proc. Amer. Math. Soc. (1959),433–437. MR 010778812. J. Korvenp¨a¨a, T. Kuusi, and G. Palatucci, The obstacle problem for nonlin-ear integro-differential operators , Calc. Var. Partial Differential Equations (2016), no. 3, Art. 63, 29. MR 350321213. T. Kuusi and G. Mingione, Universal potential estimates , J. Funct. Anal. (2012), no. 10, 4205–4269.14. T. Kuusi, G. Mingione, and Y. Sire,
A fractional Gehring lemma, with appli-cations to nonlocal equations , Rend. Lincei - Mat. Appl. (2014), 345–358.15. , Nonlocal self-improving properties , Analysis & PDE (2015), 57–114.16. L. Martinazzi, A. Maalaoui, and A. Schikorra, Blow-up behaviour of a frac-tional adams-moser-trudinger type inequality in odd dimension , Comm.P.D.E(accepted) (2015).17. V. Maz’ya,
Sobolev spaces with applications to elliptic partial differential equa-tions , augmented ed., Grundlehren der Mathematischen Wissenschaften [Fun-damental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg,2011. MR 277753018. G. Mingione,
Recent advances in nonlinear potential theory , pp. 277–292,Springer International Publishing, Cham, 2014.19. A. Schikorra,
Integro-differential harmonic maps into spheres , Comm. PartialDifferential Equations (2015), no. 3, 506–539. MR 328524320. , L p -gradient harmonic maps into spheres and SO ( N ), Differential In-tegral Equations (2015), no. 3-4, 383–408. MR 330656921. , Nonlinear commutators for the fractional p-laplacian and applications ,Mathematische Annalen (2015), 1–26.22. , ε -regularity for systems involving non-local, antisymmetric opera-tors , Calc. Var. Partial Differential Equations (2015), no. 4, 3531–3570.MR 342608623. A. Schikorra, D. Spector, and J. Van Schaftingen, An L -type estimate for Rieszpotentials , Rev. Mat. Iberoamer. (accepted) (2014).24. T.-T. Shieh and D. Spector, On a new class of fractional partial differentialequations , Adv. Calc. Var. (2015), no. 4, 321–336. MR 340343025. L. N. Slobodecki˘ı, S. L. Sobolev’s spaces of fractional order and their applicationto boundary problems for partial differential equations , Dokl. Akad. Nauk SSSR(N.S.) (1958), 243–246. MR 010632526. N. N. Ural ′ ceva, Degenerate quasilinear elliptic systems , Zap. Nauˇcn. Sem.Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (1968), 184–222. Armin Schikorra, Mathematisches Institut, Abt. f¨ur Reine Mathe-matik, Albert-Ludwigs-Universit¨at, Eckerstraße 1, 79104 Freiburg imBreisgau, Germany [email protected]
EGULARITY FOR A FRACTIONAL p -LAPLACE EQUATION 7 Tien-Tsan Shieh, National Center for Theoretical Sciences, 2F Astronomy-Mathematics Building, National Taiwan University, No. 1, Sec. 4,Roosevelt Rd., Taipei City 106, Taiwan, R.O.C. [email protected]