On a new class of fractional partial differential equations II

On a new class of fractional partial differential equations II

Tien-Tsan Shieh and Daniel E. Spector

Dedicated to Irene Fonseca, on the occasion of her 60th birthday, with esteem and affection

Abstract.In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish anL¹ Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler-Lagrange equations obtained as conditions of minimality. In addition we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

Keywords.Fractional Gradient, Fractional Hardy Inequality, Fractional Partial Differential Equations, Interpolation, Dirichlet forms.

2010 Mathematics Subject Classification.26A33,35R11,49J45,35JXX.

1 Introduction

In the preceding paper of the same name [42], the authors undertook the exposition of the Riesz fractional gradient and its systematic study from the perspective of the calculus of variations. Here we recall that for s ∈ (0,1) one can define in d-dimensional Euclidean space the fractional gradient by

D^su(x):=c_d,s ˆ

R^d

u(x)−u(y)

|x−y|^d+s

x−y

|x−y| dy, (1)

The first author is partially supported by National Science Council of Taiwan under research grant NSC 101-2115-M-009-014-MY3. The second author is supported by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2.

arXiv:1611.07204v1 [math.FA] 22 Nov 2016

providedu has sufficient smoothness and decay such that (1) is well-defined as a Lebesgue integral. The fractional gradient is the canonical example¹ of the non- local gradients considered by Mengesha and the second author in [33], where lo- calization results were obtained for these non-local gradients and integral function- als defined in terms of them, while it can be contrasted with the more well-known fractional Laplacian

(−∆)^s/2u(x) :=c˜_d,s ˆ

R^d

u(x)−u(y)

|x−y|^d+s dy (2) as a curl free vector with the same differential order. The latter has been the subject of extensive study, making its way into the canon of literature in both harmonic analysis [45] and the study of fractional derivatives [34], while recently there has been a renewed interest in it from the standpoint of fractional partial differential equations [5, 9–11, 14, 15, 28, 30, 31, 35].

As the fractional gradient has not received such prominent attention, the pur- pose of the preceding paper was to introduce it as a fundamental object of study.

In particular, we showed that with such a definition one can continuously inter- polate the class of minimization problems in the calculus of variations with linear dependence in the field from differential order zero to one: ForΩ⊂R^dopen, we established existence of minimizers of integral functionals of the form

Fs(u) = ˆ

R^df(x, D^su)−guχ_Ω dx, (3) for suitable hypothesis on f, g. We further showed that such minimizers are so- lutions to corresponding Euler-Lagrange equations, a form of fractional partial differential equations arising from the conditions of minimality.

Remark 1.1.A consensus as to what constitutes a general fractional elliptic equa- tion has not been made, though any candidate should contain the fractional Laplace’s equation as its most basic example. Various theories to this effect have been pur- sued in a number of directions, see for example a “fully nonlinear divergence form theory" of Caffarelli and Silvestre [9, 10], a "divergence form elliptic complex in- terpolation theory" of Caffarelli and Stinga [11], and a "divergence form elliptic real interpolation theory" with contributions by a number of authors: Brasco and Lindgren [5], Di Castro, Kuusi, and Palatucci [14, 15], Korvenpaa, Kuusi, and Palatucci [28], Kuusi, Mingione and Sire [30, 31] and Schikorra [39].

1 Technically the non-local gradients in [33] were defined with integration over a bounded do- main, though with suitable modification much of the analysis can be extended to integration over all of space.

As far as the authors are aware, there was no formal name for the object (1) preceding our paper, though it may be recognizable² through the relation

D^su(x) ≡R(−∆)^s/2u(x) (4) forRthe vector valued Riesz transform:

Rf(x) = c_d p.v.

ˆ

R^df(y) x−y

|x−y|^d+1 dy.

In this respect it is prudent to make here a more thorough review of papers utilizing the fractional gradient that have come to our attention since the publication of [42].

The earliest reference we are aware of concerning an equivalent definition to (1) is the 1959 paper of Horváth [23], while it is implicit in the 1961 paper of Sobolev and Nikol’ski˘ı- see p. 148 in [43]. From the standpoint of applications, a bounded domain analogue of (1) can be subsumed in the non-local continuum mechanics theory developed by Edelen and Laws [17, Equation (3.1), p. 27] and Edelen, Green and Laws [16, Equation (3.1), p. 38]. In these several papers, the standard local PDE - local conservation of mass, balance of momentum, and balance of moment of momentum- are equipped with a global balance of energy and global variational postulate in the constitutive equations and global balance of energy and global Clausius-Duhem inequality, respectively. In particular, the constitutive assumption in Equation (3.2) in [17] assumes the internal energy depends on the non-local substate variables which in turn depend on the deformation gradient from the deduction following Equation (3.8) on p. 28, while in Equation (3.2) in [16] the Helmholtz free energy is assumed to depend on a similar non-local substate variable depending on the deformation gradient. By taking this non-local substate variable to be a convolution with a restricted Riesz potential, one finds a local version of (1). In more contemporary work it has appeared in the papers of Caffarelli and Vazquez [7], Caffarelli, Soria, and Vazquez [6] and Biler, Imbert and Karch [3] in the context of a non-local porous medium equation as the gradient of the Riesz potential of the density - a “fractional potential pressure" (in particular the formula (1) has been recorded in [3]). The fractional gradient’s appearance as a boundary-type operator in the spirit of Caffarelli and Silvestre’s result [8] has been obtained by Stinga and Torrea in [46], while in [36, 37] Schikorra has considered a vector-valued analogue in the context of harmonic maps, establishing regularity for critical points of conformal energies of the fractional gradient. Finally let us mention a second order analogue - a fractional Hessian - has been considered by Guillen and Schwab in [22].

2 L. Caffarelli gave the second author such a description in a discussion in 2014 in Haifa, Israel.

As developed in the preceding paper, a motivation for considering either (1) or (2) can be found in the desire for a theory which gives rise to spaces with good functional properties - compactness, embeddings, etc. Indeed this was precisely the aim of Sobolev and Nikol’ski˘ı’s paper [43]. In fact, in one dimension both (1) and (2) can be obtained explicitly from linear combinations of the Liouville fractional derivatives they suggest for such a theory:

d^s

dx^s₊u(x) = 1 Γ(−s)

ˆ _∞

0

u(x+h)−u(x)

|h|^1+s

h

|h| dh and

d^s

dx^s₋u(x) = −1 Γ(−s)

ˆ 0

−∞

u(x+h)−u(x)

|h|^1+s

h

|h| dh.

In particular,

( d^s

dx^s₊ − d^s

dx^s₋)u(x) =cD^su(x) ( d^s

dx^s₊ + d^s

dx^s₋)u(x) =c(˜ −∆)^s/2u(x),

while the relation (4) also holds in this one dimensional example with the Hilbert transform in place of the Riesz transforms.

The purpose of this paper is to continue to advance the theory begun in the pa- per [42] and built upon in the subsequent papers developing inequalities for the fractional gradient [38] and regularity of solutions of fractional PDE defined in terms of it [40, 41]. One aspect of this development is to establish a number of new results for the fractional gradient. This includes anL¹ Hardy inequality, fur- ther regularity results for fractional PDE, the existence of minimizers for certain integral functionals of the fractional gradient with nonlinear dependence in the field, and also the existence of solutions to corresponding Euler-Lagrange equa- tions obtained as conditions of minimality. Yet despite the several papers we have written and the current advances in the theory we present here, given the general lack of study of the fractional gradient as a fundamental object, there is still much to be explored. Therefore a second aspect of this paper is to present some open problems we have formulated in the course of our research. The answers to some of these questions would fill in details currently missing in our understanding of fractional phenomena that have been settled in the integer setting. The answer to others are of interest because they would establish connections with areas of clas- sical interest such as complex interpolation or the theory of Dirichlet forms. In any case, answers to these questions would certainly provide new insight and tools that would be useful in future work.

1.1 AnL¹Hardy Inequality for the Fractional Gradient

The first item we address in this paper is the existence of anL¹Hardy inequality for the fractional gradient. This question of inequalities for the fractional gradient in the L¹ regime was taken up in [38] in the case of Sobolev’s inequality. Let us here recall that while in this endpoint the Sobolev inequality fails for the fractional Laplacian (cf. [45, p.119]), in [38] the following result was demonstrated: For any s∈(0,1)there exists a constantC =C(s, d) >0 such that

kuk_L^d/(d−s)_(R^d₎ ≤CkD^sukL¹(R^d;R^d),

for allusuch thatD^su∈L¹(R^d;R^d). One observes a similar difficulty for a frac- tional Hardy inequality in theL¹endpoint, since while it is a simple consequence of a result of Stein and Weiss [44] that for 1< p <+∞one has the inequality

ˆ

R^d

|u|^p

|x|^sp dx≤C ˆ

R^d|D^su|^p dx, (5) the same counterexample as in Sobolev’s inequality shows that one cannot have (5) whenp=1 with the fractional Laplacian on the right hand side. Nonetheless, the first result we show in this paper is

Theorem 1.2.For alls∈(0,1)one has (d−1)Γ(^s₂)Γ(^d−1₂ ) π^(2−s)/22^1−sΓ(^d−₂^s)

ˆ

R^d

|u|

|x|^s dx≤ ˆ

R^d|D^s|u||dx.

for allusuch thatD^s|u| ∈L¹(R^d;R^d).

The argument is surprisingly simple, as it follows the proof of the classical Hardy inequality. In fact, we would have such an inequality for all u such that D^su∈L¹(R^d;R^d)if one could answer

Open Problem 1.3.Does there exist a constant C > 0 (possibly depending on d ands) such that

ˆ

R^d|D^s|u||dx≤C ˆ

R^d|D^su| dx for allusuch thatD^su∈L¹(R^d;R^d)?

The answer to such a question for 1 < p < +∞follows easily from complex interpolation, since one has (see, for example, Bergh and Löfström [2, p. 153, (7)])

(L^p(R^d), W^1,p(R^d))_[s] =H^s,p(R^d),

which combined with the sublinearity of the mapz 7→ |z|and its boundedness on L^p(R^d)and W^1,p(R^d) allows one to invoke the result of Calderón and Zygmund [12]. When p = 1 the reliance of the above interpolation argument on retracts would yield interpolation of the Hardy spaceH¹(R^d)and a Hardy-Sobolev space, the result of which would be an estimate involving both the fractional gradient and the fractional Laplacian. Yet such an inequality can already be deduced from the L¹Hardy inequality utilizing Gagliardo semi-norms (see, e.g. Frank and Seiringer [19]) as the spaceW^s,1(R^d)can be seen to be embedded in the space of all locally integrable functions whose fractional Laplacian is in H¹(R^d) by integrating the basic inequality

|D^su(x)|+|(−∆)^s/2u(x)| ≤C ˆ

R^d

|u(x)−u(y)|

|x−y|^d+s dy,

which follows from the definitions (1) and (2). Given this difficulty when p = 1, one is interested to understand if the space of functions such that D^su ∈ L¹(R^d;R^d)is an interpolation space. In fact, one wonders

Open Problem 1.4.IsW^1,1(R^d) an interpolation space?

Even considering the real interpolation ofL¹(R^d)and ˙W^2,1(R^d)one has (cf. [2, p.

147])

L¹(R^d),W˙ ^2,1(R^d)

1/2,q =B_1q¹ (R^d)

whereB_1q¹ (R^d)are Besov spaces and not the classical Sobolev spaceW^1,1(R^d).

1.2 Regularity for fractional PDE

Secondly, as was observed in the recent papers of the authors in collaboration with Armin Schikorra [40, 41], the question of regularity for fractional partial dif- ferential equations in this framework follows the classical regularity. For example, in [40] we extended the technique of Iwaniec and Sbordone [24] to obtain the fol- lowing result: Let Ω ⊂ R^d be open. Suppose A : R^d → R^d×d is a function of

vanishing mean oscillation and uniformly elliptic, i.e.

sup

Q Q|A−

Q

A|dx <+∞,

|Q|→0,∞lim Q|A−

Q

A|dx=0 and

λ|ξ|² ≤A(x)ξ ·ξ ≤Λ|ξ|², (6) for allx, ξ ∈ R^d and some 0< λ ≤Λ <+∞. Further supposeG ∈L^p(R^d;R^d) and

ˆ

R^dA(x)D^su(x)·D^sϕ(x) dx= ˆ

R^dG·D^sϕ (7) for all ϕ ∈ C_c^∞(Ω). Then D^su ∈ L^p_loc(Ω) and for anyK ⊂⊂ Ω there exists a constantC =C(K,Ω, A, s, p)>0 such that

kD^sukL^p(K;R^d) ≤C

kGkL^p(R^d;R^d) +k(−∆)^s/2ukL²(R^d)

.

More recently, in [41] we introduced a reduction argument that amounts to lift- ing the fractional PDE to a classical equation to obtain regularity of the homoge- nous equation for anH^s,p-Laplacian: Let Ω ⊂ R^d be open,p ∈ (2− ¹_d,∞) and s∈(0,1]. Supposeu∈H^s,p(R^d)satisfies

ˆ

R^d|D^su|^p−2D^su·D^sϕ=0 ∀ϕ ∈C_c^∞(Ω). (8) Thenu∈C_loc^s+α(Ω) for someα > 0 only depending onp.

In fact, this technique extends to a larger class of homogeneous equations for which regularity is known in the classical inhomogeneous case, a result we now develop. Following [29], we assume thata : R^d×R^d → R^d satisfies the growth, ellipticity, and continuity assumptions:

|a(x, ξ)|+|∂_ξa(x, ξ)|(|ξ|²+s²)^1/2 ≤L(|ξ|²+s²)^(p−1)/2

ν(|ξ|²+s²)^(p−2)/2|ζ|² ≤∂_ξa(x, ξ)ζ ·ζ (9) for allx, ξ, ζ ∈R^d;∂_ξais assumed to be continuous inξ ifp≥2 and continuous away from the origin ifp <2;ais assumed to be measurable inx;ν, L, sare fixed

parameters with 0< ν ≤Lands ≥0. Then a functionvis a weak solution to the equation

−diva(x, Dv) =µ ifv ∈W_loc^1,p(Ω)and

ˆ

Ωa(x, Dv)·Dϕ= ˆ

Ωϕ dµ (10)

for allϕ ∈ C_c^∞(Ω). Further define the averaged renormalized modulus of conti- nuity ofx7→ a(x,·)as

ω(r):=



 sup

z∈R^d,B(x,r)⊂Ω B(x,r)

|a(y, ξ)−ffl

B(x,r)a(w, ξ) dw| (|ξ|²+s)^p−1

!2

dy





1/2

.

Ifp≥2, we assume the Dini-Hölder condition S :=sup

r

ˆ r 0

[ω(ρ)]^2/p ρ^α˜

dρ

ρ <+∞ (11)

for some ˜α < αM, the maximal Hölder regularity ofDv satisfying the homoge- neous equation

−diva(Dv) = 0.

Ifp∈(2− ¹_d,2]we assume that S :=sup

r

ˆ r 0

[ω(ρ)]^σ ρ^α˜

dρ

ρ < +∞ (12)

for some σ < 1. Then by Theorems 1.4 and 1.6 in [29], we find that for µ ∈ L^∞_loc(Ω) and v satisfying (10) one has Dv ∈ C_loc^α (Ω). We here apply the same reduction argument as in [41] to obtain the following regularity result.

Theorem 1.5.Suppose thatp∈(2−¹_d,∞)and thata(x, ξ)satisfies(9). Ifp≥2, further assume that a satisfies (11), while if p < 2 assume that a satisfes (12).

Moreover, we additionally assume that for all |x| sufficiently large |a(x, ξ)| ≤ L|ξ|^p−1. Then for anyu∈H_g^s,p(Ω)that satisfies

ˆ

R^da(x, D^su)·D^sϕ dx=0 for allϕ∈C_c^∞(Ω), one hasu∈C_loc^s+α(Ω).

Returning to the linear equation whenA is only assumed to be bounded, mea- surable, and satisfy (6), this argument allows us to show

Theorem 1.6.SupposeA: R^d→ R^d×dis bounded, measurable, and satisfies(6).

Further supposeG∈L^p(R^d;R^d)for somep > d/sand ˆ

R^dA(x)D^su·D^sϕ(x) dx= ˆ

R^dG·D^sϕ dx.

for allϕ∈C_c^∞(Ω). ThenI_1−su∈C_loc^α (Ω).

However, this regularity does not match that obtained in [41] forA ∈ V M O, and so one wonders

Open Problem 1.7.Is it true that for every s ∈ (0,1), any u satisfying (7) with G∈L^p(R^d;R^d)for somep > d/sandAonly bounded and measurable is Hölder continuous of some exponentα > 0 (which possibly depends upons)?

This can be compared with the classical setting, where it was De Giorgi [13]

who proved the Hölder regularity of solutions to elliptic equations with bounded and measurable coefficients. Let us recall his result here, which for convenience of reference and exposition we follow the formulation of Kinderlehrer and Stam- pacchia [27, p. 66]. The regularity result of De Giorgi says that anyuthat satisfies

ˆ

R^dA(x)Du·Dϕ(x)dx= ˆ

R^dG·Dϕ dx

for allϕ ∈ C_c^∞(Ω) is necessarily locally Hölder continuous. The main idea un- derlying the proof, and relevant to our considerations, is that one should test the equation withϕ = (u−k)⁺ and show that it decreases the energy, i.e.

ˆ

R^dA(x)Dϕ·Dϕ(x) dx≤ ˆ

R^dA(x)Du·Dϕ(x) dx.

This inequality allows one to leverage the classical Sobolev inequality against the equation, a ‘reverse’ Sobolev inequaliy, which produces the desired result, first that the solution is bounded and then that it is Hölder continuous.

Our interest in this lowering of energy with respect to such test functions stems from the relationship of the bilinear operators in question and the notion of Dirich- let forms, an idea we now explore. For any tensorA :R^d →R^d×d, define the map B_s : H^s(R^d)×H^s(R^d) →Rby

B_s[u, v]:=

ˆ

R^dA(x)D^su·D^sv(x) dx.

ThenB_s is a bilinear form on the Hilbert spaceH^s(R^d), while if we additionally assume A is symmetric, bounded and elliptic (satisfies the lower bound in (6)), thenB_ssatisfies

1. B_s[u, u] ≥0 for allu∈H^s(R^d).

2. B_s[u, v] ≤CkD^sukL²(R^d;R^d)kD^svkL²(R^d;R^d).

3. {u∈H^s(R^d) :u≡0 inΩ^c}equipped with the scalar product

< u, v >:=B_s[u, v]

is a Hilbert space.

These are three of the four defining conditions of a Dirichlet form [21, p.4-5], and therefore one wonders

Open Problem 1.8.If we definev :=min{max{u,0},1}, can one show B_s[v, v]≤B_s[u, u]?

An affirmative answer to this question would imply thatBsis indeed a Dirichlet form and so, in particular, by a formula of Beurling and Deny (see, for example, [21, p. 48,51]), one obtains the representation

Bs[u, v] = Xd i,j=1

ˆ

R^d

∂u

∂xi

∂v

∂xjdνij(x) +

ˆ

R^d

ˆ

R^d(u(x)−u(y))(v(x)−v(y))dJ(x, y) + ˆ

R^du(x)v(x)dk(x), for Radon measuresJ, k, ν_ij satisfyingJ symmetric and positive off the diagonal, kpositive, andν_ij such that for any compact setK

Xd i,j=1

ν_ij(K)ξ_iξ_j ≥0 ν_ij =ν_ji.

When {νij} are absolutely continuous with respect to the Lebesgue measure and the coercivity is non-degenerate, the first term on the right hand side of the Beurling-Deny represtation falls into the framework of the De Giorgi regularity theory, while a non-local analogue to this theory has been developed for the second

term by Kassmann [25, 26]. Precisely, if dJ(x, y) = k(x, y)dxdy is a locally integrable kernel that satisfies

k(x, y) = k(y, x)

λ≤ k(x, y)|x−y|^−d−α ≤Λ |x−y| ≤ 1 k(x, y) ≤M|x−y|^−d−η |x−y| >1

(13)

for someα∈(0,2), 0< λ ≤Λ<+∞, andη > 0 andusatisfies ˆ

R^d

ˆ

R^d(u(x)−u(y))(ϕ(x)−ϕ(y))k(x, y)dydx= ˆ

Ωf ϕ dx

forf sufficiently nice and all ϕ ∈ C_c^∞(Ω), thenu ∈ C_loc^α (Ω), theL^∞_loc estimates having been obtained previously by Fukushima [20].

A more direct approach that would immediately yield regularity is given in Open Problem 1.9.Given A : R^d → R^d×d, bounded, measurable and elliptic (satisfying (6)), can one findk_A that satisfies (13) such that

ˆ

R^dA(x)D^su·D^sϕ(x)dx= ˆ

R^d

ˆ

R^d(u(x)−u(y))(ϕ(x)−ϕ(y))k_A(x, y) dydx for allu, ϕ∈C_c^∞(Ω)?

Even a negative answer to Open Problems 1.8 and 1.9 would be interesting, in that it would give a family of examples of bilinear forms whose equations exhibit regularity properties (in the caseA ∈V M Oandpsufficiently large, for example) that are not Dirichlet forms.

1.3 Integral Functionals of the Fractional Gradient

Finally we here broaden the existence theory established in [42] to the case of possibly non-linear dependence in the field:

F_s(u) = ˆ

R^df(x, u, D^su) dx. (14) To this end we require a lower semicontinuity result for functionals with respect to strong-weak convergence on unbounded domains. In principle, we would like to apply Theorem 7.5 on p. 492 in [18]. However, with the introduction of an un- bounded domain, one finds the constant functions are no longer integrable, and so the reduction to the case where the integrand is bounded below cannot be applied

directly. If one assumes the integrand is non-negative, then the argument can be copied verbatim. We prefer to keep the general assumptions of the theorem, sup- plementing them with the simple additional assumption that outside a large ball, the integrandf has the lower bound

f(x, z, ξ) ≥α(x) +β(x)˜ ·ξ−C|z|^q,

for some α ∈ L¹(R^d), ˜β ∈ L^p/(p−1)(R^d) and C > 0. This only differs from the standard theorem in that ˜β does not depend on z. In particular, we prove the following

Theorem 1.10.Let 1 < p, q < ∞ and suppose that f : R^d ×R×R^d → Ris L^d× Bmeasurable function such that

f(x, z, ξ)≥ −C(|z|^q+|ξ|^p)−ω(x)

for L^d almost every x and for all (z, ξ) ∈ R×R^d, for some ω ∈ L¹(R^d) and C > 0. Assume that f(x,·,·) is lower semicontinuous in R×R^d for L^d almost everyx ∈R^d. Further assume that

i. f(x, z,·)is convex inR^dforL^d almost everyx ∈R^dand for allz ∈R; ii. ForL^dalmost everyx∈R^dand all(z, ξ)∈R×R^d,

f(x, z, ξ)≥α(x)−C|z|^q+β(x, z)·ξ

where α ∈ L¹(R^d), β : R^d × R → R^d is a L^d × B measurable func- tion such that β(x, z) ≡ β(x)˜ outside B(0, R) for some R > 0 with β˜ ∈ L^p/(p−1)(B(0, R)^c)andC > 0;

iii. For any sequences {u_n} ⊂ L^q(R^d) {v_n} ⊂ L^p(R^d;R^d) such that u_n → u strongly inL^q(R^d)andv_n → vweakly inL^p(R^d;R^d), and such that

supn

ˆ

B(0,R)f(x, u_n(x), v_n(x))dx <+∞,

then the sequence|β(x, u_n(x))|^p/(p−1) is equi-integrable inB(0, R).

Then the functional

(u, v) ∈L^q(R^d)×L^p(R^d;R^d) 7→

ˆ

R^df(x, u, v) dx

is sequentially lower semicontinuous with respect to strong convergence inL^q(R^d) and weak convergence inL^p(R^d;R^d).

Let us next recall the definition of the spaces on which we will show the exis- tence of minimizers. Here and in what follows, Ω ⊂ R^d is a bounded open set.

We first introduce the fractional Sobolev spaces without boundary values H^s,p(R^d):={u∈L^p(R^d): D^su∈L^p(R^d;R^d)},

and those with a given boundary valueg ∈H^s,p(R^d)

H_g^s,p(Ω):={u∈H^s,p(R^d) :u≡g inΩ^c}.

Remark 1.11.We have changed notation here in contrast to the preceding paper, whereL^s,p(R^d)andL^s,pg (Ω)were used to denote the two previous spaces, respec- tively.

Then the next result of this section is the following theorem on the existence of minimizers to the general integral functionals (14):

Theorem 1.12.Assume f : R^d × R × R^d is L^d × B measurable, f(x,·,·) is lower semicontinuous forL^dalmost everyx∈R^d, andf(x, z,·)is convex forL^d almost everyx∈R^dand allz ∈R. Further assume thatf satisfies the coercivity condition

f(x, z, ξ)≥a(x) +b|ξ|^p

for almost everyx ∈ R^d, for every (z, ξ) ∈ R×R^d, and for somea ∈ L¹(R^d), b > 0, and p > 1. Finally, assume thatF_s(u₀) is finite for some u₀ ∈ H_g^s,p(Ω).

Then there exist at least one minimizeru∈Hg^s,p(Ω)of the functionalF_s: F_s(u) ≤F_s(v)

for allv ∈Hg^s,p(Ω).

Accordingly, we obtain existence of solutions to corresponding Euler-Lagrange equations under further smoothness and growth assumptions onf.

Theorem 1.13.Assume that the functionf : R^d×R×R^dsatisfies the conditions of Theorem1.12. Additionally assume growth conditions

|f(x, z, ξ)| ≤ C(|z|^p +|ξ|^p) +γ₁(x),

|D_zf(x, z, ξ)| ≤ C(|z|^p−1+|ξ|^p−1) +γ₂(x),

|D_ξf(x, z, ξ)| ≤ C(|z|^p−1+|ξ|^p−1) +γ₃(x).

(15)

whereγ₁, γ₂, γ₃ ∈L^p/(p−1)(R^d). Further suppose thatuis a minimizer ofF_sover

∈Hg^s,p(Ω). Thenusatisfies ˆ

R^df_z(x, u, D^su)ϕ+D_ξf(x, u, D^su)·D^sϕ dx =0 for allϕ∈C_c^∞(Ω).

The plan of the papers is as follows. We first define some notation and prove an important tool - a compactness result - in Section 2. We then prove the Hardy inequality in Section 3 followed by the regularity results in Section 4. Finally we give the proofs of our results concerning integral functionals of the fractional gradient in Section 5.

2 Preliminaries

In this paper we work in d-dimensional Euclidean space, denoting by L^d the Lebesgue measure, which we often shorten todxin the integration formulas.

We denote byBthe Borelσ-algebra.

We writeB(x, r)for a ball centered atxwith radiusr > 0.

In the introduction we have utilized the constantsc_d,s,c˜_d,s to ensure that (D^su)b(ξ) =−2πiξ(2π|ξ|)^−1+su(ξˆ ),

((−∆)^s/2u)b(ξ) = (2π|ξ|)^su(ξ),ˆ where we use the convention

b u(ξ) =

ˆ

R^N u(x)e^−2πix·ξ dx In particular, one has

˜

c_d,s := 2^s−1sΓ(^d+s₂ ) π^d2Γ(1−s/2), c_d,s := (−d+1−s) 1

γ(1−s).

Here, the constantγ arises from the consideration ofα < 0, where the fractional Laplacian has as its inverse the Riesz potential I_−αu := (−∆)^α/2u, which has integral formula fors∈(0, d)

Iαu(x) = 1 γ(α)

ˆ

R^d

u(y)

|x−y|^d−α dy,

and

γ(α) = π^d/22^αΓ(α/2) Γ(^d−₂^α) .

An important step in the argument that one has existence of minimizers of in- tegral functionals with non-linear dependence in the field with respect to weak convergence in Hg^s,p(Ω) is the following compactness result that improves weak convergence in the fields to strong convergence.

Theorem 2.1.AssumeΩis a bounded open subset ofR^d, suppose s ∈(0,1) and 1 < p < ^d_s. Then for any sequence{u_n} ⊂H_g^s,p(Ω)such that

u_n →uweakly inH_g^s,p(Ω), we have that

u_n →ustrongly inL^q(Ω) for everyq ∈[1, p^∗). Here _p¹∗ = ¹_p − ^s_d.

Proof of Theorem2.1. Suppose u_m → uweakly in H_g^s,p(Ω). We will show that for any subsequence (which we will not relabel), there is a further subsequence which is Cauchy in L^q(Ω), and therefore the original sequence is strongly con- vergent. Without loss of generality, we replace the sequenceu_m −g with u_m ∈ H₀^s,p(Ω). Letηbe a standard mollifier. Set

u_m =η∗um

for >0 andm∈N. We may assume all functions{um}^∞_m=1have support inΩ⁰. We claim that

u_m →u_m uniformly inL^q(Ω)as→0.

By density of smooth functions with rapidly decreasing decay at infinity, we can representu_m = I_s(−∆)^s/2u_m. Definev_m = (−∆)^s/2u_m. Then boundedness of the Riesz transforms onL^p(R^d) for 1 < p < +∞, we have thatvm and |D^sum| have comparable norms inL^p(R^d). Thus, we estimate

u_m(x)−u_m(x) = ˆ

B(0,1)η(y)(u_m(x−y)−u_m(x))dy

= ˆ

B(0,1)η(y) ˆ

R^d

v_m(x−y−z)

|z|^d−s − v_m(x−z)

|z|^d−s dz dy

= ˆ

B(0,1)η(y) ˆ

R^d

1

|z−y|^d−s − 1

|z|^d−s

vm(x−z)dz dy.

Changing variablesz =z, we find˜

u_m(x)−u_m(x) = ^d ˆ

B(0,1)η(y) ˆ

R^d

1

|z˜−y|^d−s − 1

|z˜|^d−s

v_m(x−z)dz dy˜

= ^{d d−s}

ˆ

B(0,1)η(y) ˆ

R^d

1

|z˜−y|^d−s − 1

|z˜|^d−s

vm(x−z)˜ dz dy˜

=^s ˆ

B(0,1)η(y) ˆ

R^d

1

|z˜−y|^d−s − 1

|z˜|^d−s

v_m(x−z)˜ dz dy.˜

Thus, integrating over a bounded open setΩ⁰ which containsΩand we obtain

ˆ

Ω⁰|u_m(x)−u_m(x)|dx

≤^s ˆ

Ω⁰

ˆ

B(0,1)

ˆ

R^dη(y) 1

|z−y|^d−s − 1

|z|^d−s

|vm(x−z)|dz dy dx

=^s ˆ

B(0,1)

ˆ

R^dη(y) 1

|z−y|^d−s − 1

|z|^d−s

ˆ

Ω⁰|v_m(x−z)|dx dz dy

≤^s|Ω⁰|^1/p0 ˆ

B(0,1)

ˆ

R^dη(y) 1

|z−y|^d−s − 1

|z|^d−s

ˆ

Ω⁰|vm(x−z)|^pdx _1/p

dz dy

≤^s|Ω⁰|^1/p0 ˆ

B(0,1)

ˆ

R^dη(y) 1

|z−y|^d−s − 1

|z|^d−s dz dy

ˆ

R^d|v_m(x)|^pdx _1/p

≤C^s|Ω⁰|^1/p0 sup

y∈B(0,1)

ˆ

R^d

1

|z−y|^d−s − 1

|z|^d−s dz

!

kv_mkL^p(R^d)

Fory ∈B(0,1), we estimate the following two integrals

ˆ

B(0,2)

1

|z−y|^d−s − 1

|z|^d−s

dz≤2ˆ

B(0,2)

1

|z|^d−s dz <+∞

and ˆ

R^d\B(0,2)

1

|z−y|^d−s − 1

|z|^d−s dz =

ˆ

R^d\B(0,2)

ˆ ₁

0

d dt

1

|z−ty|^d−s dt dz

= ˆ

R^d\B(0,2)

ˆ ₁

0

(ty−z)·y

|z−ty|^s+2−s dt dz

≤ ˆ

R^d\B(0,2)

ˆ ₁

0

|y|

|z−ty|^d+1−s dt dz

≤ |y| ˆ 1

0

ˆ

R^d\B(0,2)

1

|z−ty|^d+1−s dz dt

≤ |y| ˆ 1

0

ˆ

R^d\B(0,1)

1

|z|^d+1−s dz dt <+∞.

Since {u_m} is weakly convergent, we know that it is a bounded sequence in Hg^s,p(Ω)and sokv_mkL^p(R^d) is bounded. Thus, we find that

ku_m−u_mkL¹(Ω) =O(^s).

On the other hand, the Sobolev inequality says that ku_m−u_mkL^p∗(Ω) ≤ ku_m−u_mkL^p∗(R^d)

≤CkD^s(u_m−u_m)kL^p(R^d) ≤CkD^su_mkL^p(R^d) <+∞. The previousL¹(Ω) bound and the interpolation inequality

ku_m−umkL^q(Ω) ≤ ku_m−umk^θ_L1(Ω)ku_m−umk^1−θ_Lp∗(Ω)

implies that one has, for any 1≤q < p^∗

ku_m−u_mkL^q(Ω) ≤C^sθ

where the constantC is independent ofm. Here, precisely ¹_q =θ+ ¹⁻_p∗^θ for some 0 < θ <1.

We would now like to invoke the Arzela-Ascoli theorem concerning the se- quence{u_m} of smooth functions restricted onΩfor every fixed. We therefore

prove that for each fixed > 0, the sequence {u_m} is uniformly bounded and equicontinuous onΩ. Forx∈Ω, we estimate

|u_m(x)| ≤ ˆ

B(x,)

η(x−y)|um(y)|dy

≤ kηkL^∞(R^d)ku_mkL¹(Ω⁰) ≤ C

^d <+∞

form∈N. Moreover, sinceη are smooth andumhave compact support, we have

∇u_m = ˆ

R^d∇η(x−y)u_m(y) dy and therefore

|∇u_m| ≤ k∇ηkL^∞(R^d)ku_mkL¹(Ω)

for m ∈ N. These estimates prove the claim of the uniformly boundedness and equicontinuity of the sequence{u_m}^∞_m=1for every fixed.

In the final step, we want to construct a subsequence {umk}^∞_k=1 ⊂ {um}^∞_m=1 such that

lim sup

j,k→∞ ku_mj −u_mkkL^q(Ω⁰) =0.

In order to show this, first we claim for fixedδ, there exists subsequence{umk}^∞_k=1 ⊂ {u_m}^∞_m=1 such that

lim sup

j,k→∞ ku_mj −u_mkkL^q(Ω⁰) ≤δ.

Forsmall enough, we have

ku_mk −u_mkkL^q(Ω⁰) ≤ δ 2.

Since {u_m} have support in some fixed bounded set Ω⁰, we apply the Arzela- Ascoli theorem to find a subsequence {u_mk}^∞_k=1 ⊂ {u_m}^∞_m=1 converging uni- formly inΩ⁰. This is

lim sup

j,k→∞ ku_mj −u_mkkL^q(Ω⁰) =0.

Therefore, we have lim sup

j,k→∞ku_mj −u_mkkL^q(Ω⁰)

≤ lim sup

j→∞ ku_mj −u_mjkL^q(Ω⁰)+lim sup

j,k→∞ ku_mj −u_mkkL^q(Ω⁰)

+lim sup

k→∞ ku_mk −u_mkkL^q(Ω⁰)

≤ δ 2 + δ

2 =δ.

Thus, choosing the sequence δ_n := _n¹ and a standard diagonalization argument, we may find a subsequence{u_mk}^∞_k=1 ⊂ {u_m}^∞_m=1 such that

lim sup

j,k→∞ ku_mj −u_mkkL^q(Ω⁰) =0.

This shows the sequence is Cauchy, which by completeness ofL^q(Ω)implies the strong convergence of the sequence to some function, which by uniqueness of the weak limit implies u_mj → u strongly in L^q(Ω) (and also all of R^d, since um =u≡0 inΩ^c).

3 Hardy’s Inequality

Proof of Theorem1.2. Letting Cd,s to denote the constant on the left hand side, we first show that

C_d,s ˆ

R^d

|u|

|x|^s dx= ˆ

R^d−D^s|u| · x

|x| dx,

from which the inequality follows by bringing the modulus into the integral. We have

C_d,s ˆ

R^d

|u|

|x|^s dx=C_d,sγ(d−s) ˆ

R^d|u| ·I_d−s dx

=C_d,sγ(d−s) ˆ

R^dI_1−s∗ |u| ·I_d−1 dx

=C_d,sγ(d−s) γ(d−1)

ˆ

R^dI_1−s∗ |u| · 1

|x| dx.

Now recalling that

div x

|x| = (d−1) 1

|x|,

we find

C_d,sγ(d−s) γ(d−1)

ˆ

R^dI_1−s∗ |u| · 1

|x| dx

= 1

(d−1)C_d,sγ(d−s) γ(d−1)

ˆ

R^dI_1−s∗ |u| ·div x

|x| dx

= 1

(d−1)C_d,sγ(d−s) γ(d−1)

ˆ

R^d−DI_1−s∗ |u| · x

|x| dx

= 1

(d−1)Cd,sγ(d−s) γ(d−1)

ˆ

R^d−D^s|u| · x

|x| dx,

and the claim is proven since the constantC_d,s is defined such that the coefficient of the right hand side is one.

4 Regularity

The following fundamental result underlies the regularity of homogeneous frac- tional equations for which the regularity is known in the corresponding non-fractional setting.

Proposition 4.1.LetΩ₁ ⊂⊂ Ω₂ ⊂⊂ Ω,φ ∈ C_c^∞(Ω₁), η ∈ C_c^∞(Ω) be such that η ≡1 onΩ2. Then denoting byη^c := (1−η), the operatorT defined by

T(φ) :=D^s(η^c(−∆)^1−s2 φ)

is bounded from functions φ ∈ L¹(R^d) with suppφ ⊂ Ω1 into L^p(R^d;R^d). In particular, one has the bound

kT(φ)kL^p(R^d;R^d) ≤ C_{Ω1,Ω2,d,s,p,η}kφkL¹(Ω1). (16) This proposition has been established in the paper [41], whose argument we repeat here for the convenience of the reader.

Proof. We will show that for T as defined above, one has the estimate

kT(φ)kL^q(R^d;R^d) ≤C_{Ω1,Ω2,d,s,p}kφkL¹(R^d). (17) We use the disjoint support arguments as in [4, Lemma A.1] [32, Lemma 3.6.]:

First we see that sinceη^c(x)φ(x)≡0, T(φ) = c˜_d,1−sD^s

ˆ

R^d

−η^c(x)φ(y)

|x−y|^d+1−s dy.

Now taking a cutoff-function ζ whose support has a positive distance from the boundary ofΩ2,ζ ≡1 onΩ1 we have

T(φ) = c˜_d,1−sD^s ˆ

R^d

−η^c(x)ζ(y)φ(y)

|x−y|^d+1−s dy =c˜_d,1−s ˆ

R^dk(x, y)φ(y) dy, where

κ(x, y):= −η^c(x)ζ(y)

|x−y|^d+1−s and k(x, y):=D_x^sκ(x, y).

The positive distance between the supports ofη^c and ζ implies that these kernels k,κare a smooth, bounded, integrable (both, inxand iny), and thus by a Young- type convolution argument we obtain (17). One can also argue by interpolation, as Minkowski’s inequality for integrals implies

ˆ

R^dD_x^sκ(·, y)φ(y)dy

L^p(R^d;R^d)

≤sup

y kD^s_xκ(·, y)k_L^p_(R^d_;R^d₎kφkL¹(R^d), while Theorem 2.4 on p. 886 of Adams and Meyers paper [1] can be applied to obtain

kD^s_xκ(·, y)k_L^p_(R^d_;R^d₎ ≤CkD_xκ(·, y)k^s_L^p_(R^d_;R^d₎kRκ(·, y)k^1−s_L^p_(R^d_;R^d₎. Then boundedness of the Riesz Transform and integrability of the kernels estab- lishes (17) and the proof is finished.

As a consequence, we deduce

Corollary 4.2.Let Ω₁ ⊂⊂ Ω₂ ⊂⊂ Ω, φ ∈ C_c^∞(Ω₁), η ∈ C_c^∞(Ω) be such that η ≡1 onΩ2. Then denoting byT^∗ the adjoint operator to

T(φ) :=D^s(η^c(−∆)^1−2sφ), one has

T^∗ :L^q(R^d;R^d) →L^∞(Ω1)

for every1 < q <+∞, with the operator norm ofT^∗depending onΩ₁,Ω₂, d, s, q andη.

Proof of Theorem1.5. Suppose thatu∈Hg^s,p(Ω)satisfies ˆ

R^da(x, D^su)·D^sϕ=0 (18)

for all ϕ ∈ C_c^∞(Ω). Define v := I_1−su, where I_σ is the Riesz potential, the inverse of(−∆)^σ/2. Now letΩ1 ⊂Ωbe an arbitrary open set compactly contained in Ω, and let φ be a test function supported in Ω₁. Pick an open set Ω₂ so that Ω1 ⊂ Ω2 ⊂ Ωand a cutoff functionη, supported inΩand constantly one in Ω2. Then in particular one can take

ϕ :=η(−∆)^1−s2 φ as a test function to obtain

ˆ

R^da(x, D^sv)·D^s(η(−∆)^1−s2 φ) = 0.

Thus,

ˆ

R^da(x, D^sv)·Dφ= ˆ

R^da(x, D^sv)·D^s(η^c(−∆)^1−2sφ) whereη^c := (1−η). We set

T(φ) :=D^s(η^c(−∆)^1−2sφ),

and from the assumptions onawe may apply Corollary 4.2 to deduce that T^∗·a(x, D^su) ∈L^∞_loc(Ω).

In other words,vis a solution to the equation ˆ

Ωa(x, Dv)·Dφ dx= ˆ

Ωφ dµ.

Thus by Theorems 1.4 and 1.6 in [29], we find that µ ∈ L^∞_loc(Ω) implies Dv ∈ C_loc^α (Ω). Now asDv =D^su∈C_loc^α (Ω), we obtainu∈C_loc^s+α(Ω).

Proof of Theorem1.6. Suppose thatu∈Hg^s,2(Ω)satisfies ˆ

R^dA(x)D^su·D^sϕ= ˆ

R^dG·D^sϕ (19) for all ϕ ∈ C_c^∞(Ω). Define v := I₁₋su, where Iσ is the Riesz potential, the inverse of(−∆)^σ/2. Now letΩ₁ ⊂Ωbe an arbitrary open set compactly contained in Ω, and let φ be a test function supported in Ω1. Pick an open set Ω2 so that Ω₁ ⊂ Ω₂ ⊂ Ωand a cutoff functionη, supported inΩand constantly one in Ω₂. Then in particular one can take

ϕ :=η(−∆)^1−2sφ

as a test function in (19) to obtain ˆ

R^dA(x)Dv·D^s(η(−∆)^1−2sφ) = ˆ

R^dG·D^s(η(−∆)^1−2sφ) That is,

ˆ

R^dA(x)Dv·Dφ= ˆ

R^dA(x)Dv ·D^s(η^c(−∆)^1−s2 φ) +G·D^s(η(−∆)^1−s2 φ)

= ˆ

R^dG·Dφ+ (A(x)Dv−G)·D^s(η^c(−∆)^1−2sφ) whereη^c := (1−η). We set

T(φ) :=D^s(η^c(−∆)^1−2sφ),

and by Corollary 4.2, we find thatv is a solution to the classical elliptic equation with bounded and measurable coefficients

ˆ

ΩA(x)Dv·Dφ dx= ˆ

ΩG·Dφ+T^∗·((A(x)Dv −G))φ.

Thus, by the regularity theory known for such an equation (e.g. [27, p. 66]), we findv ∈C_loc^α (Ω), which is to sayI_1−su∈C_loc^α (Ω).

5 Integral Functionals of the Fractional Gradient In this section, we consider the variational problem

u∈Hinfg^s,p(Ω)

ˆ

R^df(x, u, D^su)dx.

Under suitable hypothesis, we establish the existence of minimizers, while with further assumptions we show that these minimizers satisfy corresponding Euler- Lagrange equations.

We begin by proving the lower semicontinuity result for strong-weak conver- gence stated in Theorem 1.10.

Proof. Supposeun →ustrongly inL^q(R^d),vn →vweakly inL^p(R^d;R^d). Now we may assume that

lim inf

n→∞

ˆ

R^df(x, u_n, v_n) dx <+∞