Regularity and rigidity theorems for a class of anisotropic nonlocal operators
aa r X i v : . [ m a t h . A P ] S e p REGULARITY AND RIGIDITY THEOREMSFOR A CLASS OF ANISOTROPIC NONLOCAL OPERATORS
ALBERTO FARINA AND ENRICO VALDINOCI
Abstract.
We consider here operators which are sum of (possibly) fractional derivatives, with (possiblydifferent) order. The main constructive assumption is that the operator is of order 2 in one variable. Byconstructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutionsin such direction with respect to the oscillation of the nonlinearity in the same direction.As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity isindependent of a coordinate direction, then so is any global solution (provided that the solution does notgrow too much at infinity). A Liouville type result then follows as a byproduct. Introduction
Recently a good deal of research has been performed about nonlocal operators of fractional type, alsoin consideration of their probabilistic interpretation of L´evy processes. In this framework, it is naturalto consider the superposition of different nonlocal operators in different directions, possibly with different(fractional) orders, in relation with the nonlocal diffusive equations in anisotropic media.A first attempt to systematically study these anisotropic fractional operators was given in [6, 8, 7]. Inparticular, the regularity theory of these anisotropic operators is perhaps harder than expected, it stillpresents several open questions and some lack of regularity occurs in concrete examples (for instance,solutions of rather simple equations with smooth data and smooth domains may fail in this case to besmooth, see Theorem 1.2 in [8]). Roughly speaking, the lack of regularity may be caused by the combinationof the nonlocal properties of the operator and the anisotropic structure of the operator. Namely, first thenonlocal feature may cause the solution to be only H¨older continuous at the boundary; then the anisotropicstructure may relate the solution in the interior to values at (or close to) the boundary, and the nonlocaleffect can somehow “propagate” the boundary singularity towards the interior, making a smooth interiorregularity theory false in this case (see [8] for more details about it).The goal of this paper is to provide a very simple approach to a Lipschitz-type regularity theory for afamily of anisotropic integro-differential operators, obtained by the superposition of different operators indifferent coordinate directions, and possibly with different order of differentiation.The main structural assumption that we take is that there is one “special” coordinate (say the last one)in which the operator is local and of second order. In this framework, we will control the derivative of thesolution in this variable by uniform and universal quantities, depending on the data of the problem.More precisely, the mathematical framework in which we work is the following. We denote by { e , . . . , e n } the Euclidean base of R n . Given a point x ∈ R n , we use the notation x = ( x , . . . , x n ) = x e + · · · + x n e n , with x i ∈ R .We divide the variables of R n into m subgroups of variables, that is we consider m ∈ N and N , . . . , N m ∈ N , with N + · · · + N m − = n −
1. For i ∈ { , . . . , m } , we use the notation N ′ i := N + · · · + N i , and we Mathematics Subject Classification.
Key words and phrases.
Nonlocal anisotropic integro-differential equations, regularity results.The authors have been supported by the ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces andLayers for Odd Nonlinearities”. take into account the set of coordinates X := ( x , . . . , x N ) ∈ R N X := ( x N +1 , . . . , x N ′ ) ∈ R N ... X i := ( x N ′ i − +1 , . . . , x N ′ i ) ∈ R N i ... X m − := ( x N ′ m − +1 , . . . , x N ′ m − ) ∈ R N m − and X m := x n . (1)Given i ∈ { , . . . , m − } and s i ∈ (0 ,
1] we will consider the (possibly fractional) s i -Laplacian in the i thset of coordinates X i . For this scope, given y = ( y , . . . , y N i ) ∈ R N i it is useful to consider the incrementinduced by y with respect to the i th set of coordinates in R n , that is one defines(2) y ( i ) := y e N ′ i − +1 + · · · + y N i e N ′ i ∈ R n . With this notation, one can define the N i -dimensional (possibly fractional) s i -Laplacian in the i th set ofcoordinates X i , for a (smooth) function u : R n → R by(3) ( − ∆ X i ) s i u ( x ) := − ∂ x N ′ i − u ( x ) − . . . − ∂ x N ′ i u ( x ) if s i = 1 ,c N i ,s i Z R Ni u ( x ) − u ( x + y ( i ) ) − u ( x − y ( i ) ) | y ( i ) | N i +2 s i dy ( i ) if s i ∈ (0 , , The quantity c N i ,s i in (3) is just a positive normalization constant, whose explicit value for N ∈ N and s ∈ (0 ,
1) is taken to be(4) c N,s := 2 s − Γ( s + N ) π N | Γ( − s ) | , where Γ is the Euler’s Gamma Function. We refer to [5, 9, 4] and to the references therein for furthermotivations about fractional operators.In this paper we consider a pseudo-differential operator, which is the sum of (possibly) fractional Lapla-cians in the different coordinate directions X i , with i ∈ { , . . . , m − } , plus a local second derivative inthe direction x n . The operators involved may have different orders and they may be multiplied by possiblydifferent coefficients: that is, given a , . . . , a m − > a >
0, we define L := m − X i =1 a i ( − ∆ X i ) s i − a∂ x n = m X i =1 a i ( − ∆ X i ) s i , (5)where in the latter identity we used the convention that a m := a , s , . . . , s m − ∈ (0 ,
1] and s m := 1.Given the operator in (5), we stress that a very important structural difference with respect to theclassical local case is that fractional objects are in general not reduced to the sum of their directionalcomponents We wrote the value of c N,s as in (4) to be consistent with the literature, see e.g. notation of [3]. Of course, such valuecan be equivalently written in other forms, according to the different tastes. The explicit value of the normalization constantin (4) plays no major role in this paper, but it is useful for consistency properties as s i → That is, if x = ( x , . . . , x n ) ∈ R n the following formulas are false, unless s = 1:( − ∂ x ) s + · · · + ( − ∂ x n ) s = ( − ∆ x ) s and ( − ∆ ( x ,...,x N ) ) s + ( − ∆ ( x N +1 ,...,x N + K ) ) s = ( − ∆ ( x ,...,x N + K ) ) s . NISOTROPIC NONLOCAL OPERATORS 3
The main result that we prove in this paper is a Lipschitz regularity theory in the last coordinate variablethat extends the one of [1] (which was obtained in the classical setting of local operators). To this goal, wedenote by B Nr the open ball of R N centered at the origin and with radius R . Also, given d , . . . , d m > d := ( d , . . . , d m ) and Q d := B N d × · · · × B N m − d m − × ( − d m , d m ) = m Y i =1 B N i d i , where in the latter identity we used the convention that N m := 1.Also, given κ > Q d,κ the dilation of factor κ in the last coordinate (leaving the othersfixed), that is Q d,κ := B N d × · · · × B N m − d m − × ( − κd m , κd m ) . Of course, by construction Q d, = Q d . In accordance with the constant fixed in (3), it is also convenientto introduce the following notation for a suitable universal quantity, for any i ∈ { , . . . , m } :(6) η i := Γ (cid:0) N i (cid:1) s i Γ( s i + 1) Γ (cid:0) s i + N i (cid:1) . With this notation, we have the following result:
Theorem 1.1.
Let f : Q d, → R and u : R n → R be a solution of Lu = f in Q d, . Then, for any t ∈ ( − d m , d m ) , (7) | u ( te n ) − u ( − te n ) || t | d m S a + ˜ C d m k u k L ∞ ( R n ) min i ∈{ ,...,m } ( η i d s i i ) , where S := sup ( x,t ) ∈ Q d × (0 ,d m ) | f ( x + te n ) − f ( x − te n ) | , and ˜ C := 2( a + · · · + a m ) a + 1 . Higher regularity results (for different types of nonlocal anisotropic operators) have been obtained in [6, 8](indeed, general anisotropic operators can be considered in [6, 8], but only the kernel with the samehomogeneity were taken into account). Some advantages are offered by Theorem 1.1 with respect to theother results available in the literature. First of all, Theorem 1.1 comprises the case of operators of differentorders (e.g. the s i can be all different and both local and nonlocal operators can be superposed). Moreover,Theorem 1.1 may select the “local” coordinate direction independently on the others, in order to take intoaccount the behavior of the nonlinearity in this single coordinate and detect its effect on the oscillationof the solution (notice in particular the term S appearing in Theorem 1.1, which only depends on theoscillation of f in the last coordinate direction). As a matter of fact, the diffusive operators in the othervariables can also degenerate (indeed a i may vanish for some i ∈ { , . . . , m − } ).In addition, all the constants appearing in Theorem 1.1 can be computed explicitly without effort and theproof is rather simple and it makes use only of one explicit barrier (the barrier will be given in formula (18)and, as a matter of fact, this argument may be seen as the fractional counterpart of the regularity theorydeveloped by [1] in the local framework).As a technical remark, we point out that, for simplicity, the notion of solution in Theorem 1.1 is takenin the classical sense, i.e. the function u will be implicitly assumed to be smooth enough to compute theoperator L pointwise (in this sense, formula (7) reads as an “a priori estimate”). Nevertheless, the sameargument that we present goes through, for instance, by applying the operator to smooth functions thattouch the solution from above/below, that is one can assume simply that the solution in Theorem 1.1 istaken in the viscosity sense (in this case, formula (7) reads as an “improvement of regularity”). We observe that η i = 1 / (2 N i ) if s i = 1, since Γ (cid:0) N i (cid:1) = N i Γ (cid:0) N i (cid:1) . ALBERTO FARINA AND ENRICO VALDINOCI
We point out that, as a simple consequence of Theorem 1.1, we obtain an interior Lipschitz estimate inthe last variable:
Corollary 1.2.
Let u : R n → R be a solution of Lu = f in B . Then (8) k ∂ x n u k L ∞ ( B / ) C (cid:0) k f k L ∞ ( B ) + k u k L ∞ ( R n ) (cid:1) , for some C > , depending on a , . . . , a m , s , . . . , s m − , and N , . . . , N m − . As a matter of fact, when s = · · · = s m = 1, Corollary 1.2 reduces to the classical Lipschitz regularitytheory as presented in [1].We observe that the regularity results obtained in this paper can be also combined efficiently with otherresults available in the literature, possibly leading to higher regularity results. To make a simple exampleof this feature, we give the following result: Corollary 1.3.
Let s ∈ (0 , , a , . . . , a m − , a > and (9) L ∗ := m − X i =1 a i ( − ∆ X i ) s − a∂ x n . Let f ∈ L ∞ ( R n ) be Lipschitz continuous in B with respect to the variable x n . Let u : R n → R be a solutionof L ∗ u = f in B n − × R . Then k u k C γ ( B / ) C (cid:0) k f k L ∞ ( R n ) + k ∂ x n f k L ∞ ( B ) + k u k L ∞ ( R n ) (cid:1) , where γ := (cid:26) s if s < / , − ǫ if s > / for some C > , depending on a , . . . , a m , s , and N , . . . , N m − (with the caveat that when s > / , onecan choose ǫ arbitrarily in (0 , and C will also depend on ǫ ). Another interesting consequence of Theorem 1.1, is also the following rigidity result, valid when all thefractional exponents are larger than 1 / Theorem 1.4.
Let f : R n → R . Assume that σ := 2 min { s , . . . , s n } − > . Let u : R n → R be a solution of Lu = f in the whole of R n . Assume that f does not depend on the n thcoordinate and that (10) k u k L ∞ ( B R ) = o ( R σ ) as R → + ∞ .Then u does not depend on the n th coordinate. Remark 1.5.
A simple, but interesting, consequence of Theorem 1.4 is that if (10) holds and f is identicallyzero, then L ∗ u = 0 in R n − ,where we used the notation in (9). Therefore, if L ∗ enjoys a Liouville property, then u is necessarilyconstant.This feature holds, in particular, when L ∗ = ( − ∂ x ) s + · · · + ( − ∂ x n − ) s , see Theorem 2.1 in [6]. Remark 1.6.
The observation in Remark 1.5 also says that, if (10) is satisfied and L ∗ enjoys a Liouvilleproperty, then the problem Lu = f possesses a unique solution, up to an additive constant. As customary, the notation in (10) simply means thatlim R → + ∞ k u k L ∞ ( B R ) R σ = 0 . NISOTROPIC NONLOCAL OPERATORS 5
The rest of the paper is organized as follows. The proof of Theorem 1.1, based on the barrier methodof [1], is contained in Section 2. Then, in Section 3, we combine our results with those of [6] and we proveCorollary 1.3. The proof of Theorem 1.4, which combines our result with a cutoff argument, is containedin Section 4. 2.
Proof of Theorem 1.1
A useful explicit barrier.
We recall here a useful barrier. Here and in what follows we use thestandard “positive part” notation for any t ∈ R , i.e. t + := max { t, } . We will also exploit the notation in (1) and (6).
Lemma 2.1.
Let s i ∈ (0 , and d i > . For any x = ( X , . . . , X m − , x n ) ∈ R n let Φ d i ( z ) := η i ( d i − | X i | ) s i + . Then, for any x ∈ R n with X i ∈ B N i d i , we have that ( − ∆ X i ) s i Φ d i ( x ) = 1 . Proof.
The result is obvious for s i = 1 (recall the footnote on page 3), hence we suppose s i ∈ (0 , d i := η − i Φ d i ( z ) = ( d i − | X i | ) s i + . By scaling variables x ∗ := xd i , X ∗ ,i := X i d i and ζ ∗ := ζd i , we obtain that( − ∆) s i Ψ d i ( x ) = c N i ,s i Z R Ni d i − | X i | ) s i + − ( d i − | X i + ζ | ) s i + − ( d i − | X i − ζ | ) s i + | ζ | N i +2 s i dζ = c N i ,s i d s i i d N i i Z R Ni − | X ∗ ,i | ) s i + − (1 − | X ∗ ,i + ζ ∗ | ) s i + − (1 − | X ∗ ,i − ζ ∗ | ) s i + d N i +2 s i i | X ∗ ,i | N i +2 s i dζ ∗ = ( − ∆ X i ) s i Ψ ( x ∗ )= 2 s i Γ( s i + 1) Γ (cid:0) s i + N i (cid:1) Γ (cid:0) N i (cid:1) , see for instance Table 3 of [3] for the last identity (here, we used the notation Ψ to denote Ψ d i when d i =1). (cid:3) Completion of the proof of Theorem 1.1.
For any t ∈ R , we define (11) u ± ( x, t ) := u ( x ± t + e n ) = u ( x , . . . , x n − , x n ± t + ) . Similarly, we define f ± ( x, t ) := f ( x ± t + e n ). Let also v ( x, t ) := u + ( x, t ) − u − ( x, t ) and g ( x, t ) := f + ( x, t ) − f − ( x, t ) . We fix ν ∈ (0 , a ) (to be taken as close to a as we wish in what fallows). Recalling (5), we introduce theoperator L ∗ := L + ν∂ x n − ν∂ t = m X i =1 a i ( − ∆ X i ) s i − ν ( − ∂ x n ) − ν∂ t = m − X i =1 a i ( − ∆ X i ) s i − ( a − ν ) ∂ x n − ν∂ t . (12) As a technical remark, we point out that the assumption that the operator is “local” in the last coordinate is used at thispoint, since if t > u ± ( x, t ) = u ( x ± te n ), and so, if we differentiate with respect to t in the domain { t ∈ (0 , d m ) } ,we have that ∂ t u ± ( x, t ) = ± ∂ x n u ( x ± te n ). ALBERTO FARINA AND ENRICO VALDINOCI
Notice that L ∗ is an operator with one variable more than L (namely, the new variable t ∈ R ). We claimthat(13) L ∗ v = g for any ( x, t ) ∈ Q d × (0 , d m ). To check (13), we first notice that if ( x, t ) ∈ Q d × (0 , d m ) then x ± te n ∈ Q d,n, ,and we know that Lu = f in the latter set. Also, since the (fractional) Laplacian is translation invariant,(14) ( − ∆ X i ) s i u ± = (cid:16) ( − ∆ X i ) s i u (cid:17) ± for any i ∈ { , . . . , m } and any ( x, t ) ∈ Q d × (0 , d m ) (notice that the variable t plays the role of a fixedparameter here). Moreover(15) ∂ t u ± = (cid:16) ∂ x n u (cid:17) ± for any ( x, t ) ∈ Q d × (0 , d m ). In turn, we see that (12), (14) and (15) imply that L ∗ u ± = ( Lu ) ± and thus, by linearity, L ∗ v = L ∗ ( u + − u − ) = ( Lu ) + − ( Lu ) − = f + − f − = g, which establishes (13). Now we set c o := m X i =1 η i d s i i + d m m − X i =1 d s i i + d m and A := m X i =1 a i . (16)Let also A := A A + k g k L ∞ ( Q d × (0 ,d m )) + ( a − ν ) , where A := k v k L ∞ ( R n +1 ) min i ∈{ ,...,m } ( η i d s i i ) . (17)We consider the barrierΦ( x, t ) := A ν Φ ( t ) + A Φ ( x, t )with Φ ( t ) := t + ( d m − t ) + ( x, t ) = Φ ( X , . . . , X m − , X m , t ) := c o − m X i =1 η i ( d i − | X i | ) s i + − ( d m − t ) + . (18)Notice that Φ > > c o − m X i =1 η i d s i i − d m . Consequently,(19) Φ > ( x, t ) = c o − n X i =1 Ψ d i ( x i ) − Ψ d m ( t ) . NISOTROPIC NONLOCAL OPERATORS 7
Therefore, making use of Lemma 2.1, we conclude that, for any ( x, t ) ∈ Q d × (0 , d m ), L ∗ Φ( x, t ) = m X i =1 a i ( − ∆ X i ) s i Φ( x, t ) + ν∂ x n Φ( x, t ) − ν∂ t Φ( x, t )= A m X i =1 a i A ( − ∆ X i ) s i Φ ( x, t ) + A ν∂ x n Φ ( x, t ) − ν∂ t Φ( x, t )= − A m X i =1 a i − A ν + ν (cid:18) A ν + A (cid:19) = − A m X i =1 a i + A . That is, recalling (16) and (17), L ∗ Φ( x, t ) = − A A + A = k g k L ∞ ( Q d × (0 ,d m )) + ( a − ν ) > ± g ( x, t ) + ( a − ν ) = ± L ∗ v ( x, t ) + ( a − ν ) , (20)for any ( x, t ) ∈ Q d × (0 , d m ), where we used (13) in the last step.Now we claim that(21) Φ( x, t ) ± v ( x, t ) > x, t ) ∈ R n +1 \ (cid:0) Q d × (0 , d m ) (cid:1) .To check this, we take ( x, t ) outside Q d × (0 , d m ), and we distinguish three cases: either t
0, or t > d m ,or x ∈ R n \ Q d .First, when t
0, we have that t + = 0, so we use (11) to see that u + ( x,
0) = u ( x ) = u − ( x,
0) andso v ( x,
0) = 0. Then ± v ( x,
0) = 0 Φ( x,
0) in this case, thanks to (19) and this establishes (21)when t t > d m . In this case ( d m − t ) + = 0, hence, by (18),Φ( x, t ) > A Φ ( x, t )= A " c o − m X i =1 η i ( d i − | X i | ) s i + > A " c o − m X i =1 η i d s i i = A d m > k v k L ∞ ( R n +1 ) > ± v ( x, t ) , as desired. It remains to consider the case x ∈ R n \ Q d . Under this circumstance, we have that thereexists i o ∈ { , . . . , m } such that | X i o | > d i o . Accordingly m X i =1 ( d i − | X i | ) s i + = X i mi = io ( d i − | X i | ) s i + X i mi = io d s i i , ALBERTO FARINA AND ENRICO VALDINOCI and so Φ( x, t ) > A Φ ( x, t ) > A c o − X i mi = io η i d s i i − d m = A η i o d s io i o > k v k L ∞ ( R n +1 ) > ± v ( x, t ) , which completes the proof of (21).Now we show that the inequality in (21) propagates inside Q d × (0 , d m ), namely that(22) Φ( x, t ) ± v ( x, t ) > x, t ) ∈ R n +1 .The proof of (22) is mostly Maximum Principle. The details are as follows. Suppose, by contradiction,that (22) were false. Then we set h := Φ ± v . Notice that, since u is assumed to be continuous, so is h ,due to (11), and then (21) would imply that min Q d × (0 ,d m ) h =: µ < . Let ¯ p := (¯ x, ¯ t ) attaining the minimum of h , that is( −∞ , ∋ µ = h (¯ p ) h ( ξ ) , for any ξ ∈ R n +1 . By (21), we have that ¯ p lies in Q d × (0 , d m ), hence, by (20),(23) L ∗ h (¯ p ) > a − ν > . On the other hand, recalling the notation in (2), for any i ∈ { , . . . , m − } and any y ∈ R N i , we have that2 h (¯ p ) − h (¯ p + y ( i ) ) − h (¯ p − y ( i ) ) , due to the minimality of ¯ p . Similarly ( − ∂ x j ) h (¯ p ) j ∈ { , . . . , n } , as well as ( − ∂ t ) h (¯ p ) − ∆ X i ) s i (¯ p )
0, for any i ∈ { , . . . , m } . Consequently, by (12), we infer that L ∗ h (¯ p )
0. Thelatter inequality is in contradiction with (23) and thus we have proved (22).By choosing the sign in (22), we deduce that(24) | v ( x, t ) | Φ( x, t ) for any ( x, t ) ∈ R n +1 .Moreover, recalling (18) and (16), for any t ∈ (0 , d m ),Φ (0 , t ) = c o − m X i =1 η i d s i i − ( d m − t ) + d m − ( d m − t )2= t . In addition, Φ ( t ) d m t + , therefore, by (18), for any t ∈ (0 , d m ), Φ(0 , t ) A d m tν + A t . NISOTROPIC NONLOCAL OPERATORS 9
This and (24) imply that, for any t ∈ (0 , d m ), | u ( te n ) − u ( − te n ) || t | = | u + (0 , t ) − u − (0 , t ) | t = | v (0 , t ) | t Φ(0 , t ) t A d m ν + A t (cid:2) A A + k g k L ∞ ( Q d × (0 ,d m )) + ( a − ν ) (cid:3) d m ν + A t . Now we observe that the first term in the above inequality remains unchanged if we replace t with − t , andtherefore the inequality is valid for any t ∈ ( − d m , d m ). Furthermore, we can now take ν as close to a aswe wish (recall that A and A do not depend on ν ), hence we obtain that, for any t ∈ ( − d m , d m ), | u ( te n ) − u ( − te n ) || t | s n (cid:2) A A + k g k L ∞ ( Q d × (0 ,d m )) (cid:3) d m a + A t (cid:2) A A + k g k L ∞ ( Q d × (0 ,d m )) (cid:3) d m a + A d m k g k L ∞ ( Q d × (0 ,d m )) d m a + A d m (cid:18) A a + 12 (cid:19) = k g k L ∞ ( Q d × (0 ,d m )) d m a + k v k L ∞ ( R n +1 ) d m min i ∈{ ,...,m } ( η i d s i i ) (cid:18) a + · · · + a m a + 12 (cid:19) . This completes the proof of Theorem 1.1.3.
Proof of Corollary 1.3
The proof combines Corollary 1.2 here with Theorem 1.1(a) in [6]. To this goal, fixed t ∈ (cid:2) − , (cid:3) (to be taken arbitrarily small in the sequel) we define(25) u ♯ ( x ) := u ( x + te n ) − u ( x ) t and f ♯ ( x ) := f ( x + te n ) − f ( x ) t . By formula (8) in Corollary 1.2, we already know that(26) k u ♯ k L ∞ ( R n ) k ∂ x n u k L ∞ ( R n ) C (cid:0) k f k L ∞ ( R n ) + k u k L ∞ ( R n ) (cid:1) . Also, we point out that L ∗ u ♯ = f ♯ in B / , and so, using again Corollary 1.2, k ∂ x n u ♯ k L ∞ ( B / ) C (cid:0) k f ♯ k L ∞ ( B / ) + k u ♯ k L ∞ ( R n ) (cid:1) . This, combined with (26), gives thatsup x ∈ B / (cid:12)(cid:12)(cid:12)(cid:12) ∂ x n u ( x + te n ) − ∂ x n u ( x ) t (cid:12)(cid:12)(cid:12)(cid:12) = k ∂ x n u ♯ k L ∞ ( B / ) C (cid:0) k ∂ x n f k L ∞ ( B ) + k f k L ∞ ( R n ) + k u k L ∞ ( R n ) (cid:1) . Hence, taking t to the limit,(27) sup x ∈ B / | ∂ x n u ( x ) | C (cid:0) k ∂ x n f k L ∞ ( B ) + k f k L ∞ ( R n ) + k u k L ∞ ( R n ) (cid:1) . Now, given i ∈ { , . . . , m − } we consider the sphere S N i − in the Euclidean space R N i (of course, if N i = 1,then S N i − reduces to two points).We also set S n − := { ( x , . . . , x n − ) s.t. x + · · · + x n − = 1 } and we observe that each S N i − is naturallyimmersed into S n − (in the same way as R N i − is immersed into R n − ). We denote by H i the ( N i − S N i − (if N i = 1, we replaceit by the Dirac’s delta on the two points given by S N i − ). Then we consider the measure µ := m − X i =1 a i c N i ,s i H i . We fix ˜ x n ∈ (cid:2) − , (cid:3) and we set˜ u ( x , . . . , x n − ) := u ( x , . . . , x n − , ˜ x n )and ˜ f ( x , . . . , x n − ) := f ( x , . . . , x n − , ˜ x n ) + a∂ x n u ( x , . . . , x n − , ˜ x n ) . We use (3) and polar coordinates on R N i to see that, for any ˜ x = ( x , . . . , x n − ) ∈ B n − / ,˜ L ˜ u (˜ x ) := Z S n − (cid:20)Z R (cid:16) ˜ u (˜ x + θr ) + ˜ u (˜ x − θr ) − u (˜ x ) (cid:17) dr | r | s (cid:21) dµ ( θ )= m − X i =1 a i c N i ,s i Z S Ni − (cid:20)Z R (cid:16) ˜ u (˜ x + θr ) + ˜ u (˜ x − θr ) − u (˜ x ) (cid:17) drr s (cid:21) d H i ( θ )= m − X i =1 a i c N i ,s i Z S Ni − (cid:20)Z + ∞ (cid:16) ˜ u (˜ x + θr ) + ˜ u (˜ x − θr ) − u (˜ x ) (cid:17) drr s (cid:21) d H i ( θ )= m − X i =1 a i c N i ,s i Z R Ni ˜ u (˜ x + y ( i ) ) + ˜ u (˜ x − y ( i ) ) − u (˜ x ) | y ( i ) | N i +2 s dy ( i ) = m − X i =1 a i ( − ∆ X i ) s i ˜ u (˜ x )= L ∗ ˜ u (˜ x ) + a∂ x n u (˜ x, ˜ x n )= f (˜ x, ˜ x n ) + a∂ x n u (˜ x, ˜ x n )= ˜ f (˜ x ) . Notice that, with this setting, the operator ˜ L satisfies formula (1.1) in [6].Furthermore, we have that(28) inf ν ∈ S n − Z S n − | ν · θ | dµ ( θ ) > λ, for some λ >
0. To prove it, we observe that if ν = ( ν , . . . , ν n − ) ∈ S n − , we have that | ν j | > ( n − − / ,for at least one j ∈ { , . . . , n − } . Up to relabeling variables, we assume that j = 1, and thus Z S n − | ν · θ | s dµ ( θ ) > a c N ,s Z S N − | ν θ | s dµ ( θ ) > a c N ,s n − s Z S N − | θ | s dµ ( θ ) , which proves (28).In addition, µ ( S n − ) m − X i =1 a i c N i ,s i H N i − ( S N i − ) < + ∞ . From this and (28), we conclude that condition (1.2) in [6] is satisfied. Accordingly, we can exploitTheorem 1.1(a) in [6] and conclude that k ˜ u k C γ ( B n − / ) C (cid:0) k ˜ u k L ∞ ( R n ) + k ˜ f k L ∞ ( B n − / ) (cid:1) C (cid:0) k u k L ∞ ( R n ) + k f k L ∞ ( B ) + k ∂ x n u k L ∞ ( B / ) (cid:1) . NISOTROPIC NONLOCAL OPERATORS 11
This and (27) imply that k ˜ u k C γ ( B n − / ) C (cid:0) k ∂ x n f k L ∞ ( B ) + k f k L ∞ ( R n ) + k u k L ∞ ( R n ) (cid:1) . This gives the desired regularity in the set of variables ( x , . . . , x n − ). The regularity in the last variablefollows from (27) and so the proof of Corollary 1.3 is complete.4. Proof of Theorem 1.4
A cutoff argument.
The purpose of this section is to localize the estimate of Theorem 1.1 by usinga cutoff function. As customary in the fractional problems, regularity estimates cannot be completelylocalized, due to nonlocal effect, nevertheless our objective is to give quantitative bounds on the contri-bution “coming from infinity”. For this scope, we use the notation s min := min { s , . . . , s n } and s max :=max { s , . . . , s n } (a similar notation will also be exploited in the sequel for a min := min { a , . . . , a m } and a max := max { a , . . . , a m } ). Lemma 4.1.
Let R > . If w vanishes identically in ( − R, R ) n , then k Lw k L ∞ (( − R,R ) n ) C o Z + ∞ R k w k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ, where (29) C o := 2 n X i =1 a i c N i ,s i H N i − ( S N i − ) . Proof.
Let x ∈ ( − R, R ). We claim that(30) | ( − ∆ X i ) s i w ( x ) | c N i ,s i H N i − ( S N i − ) Z + ∞ R k w k L ∞ ( B ρ \ B ρ/ ) ρ s i dρ for each i ∈ { , . . . , m } . To prove this, we notice that if s i = 1 then the fact that w vanishes identically ina neighborhood of x implies that − ∆ X i w ( x ) = 0, and so (30) is obvious in this case. Thus, we can supposethat s i ∈ (0 , y ( i ) ∈ [ − R, R ] N i then x + y ( i ) ∈ ( − R, R ) n and so w ( x + y ( i ) ) = 0.From this, it follows that(31) ( − ∆ X i ) s i w ( x ) = c N i ,s i Z R Ni ∩{| y ( i ) | > R } − w ( x + y ( i ) ) − w ( x − y ( i ) ) | y ( i ) | N i +2 s i dy ( i ) . Also, if | y ( i ) | > R then | y ( i ) | > | x | , thus | x ± y ( i ) | > | y ( i ) | − | x | > | y ( i ) | | x ± y ( i ) | | x | + | y ( i ) | < | y ( i ) | , and so | w ( x ± y ( i ) ) | k w k L ∞ ( B | y ( i ) | \ B | y ( i ) | / ) . As a consequence of this and of (31), we obtain | ( − ∆ X i ) s i w ( x ) | c N i ,s i Z R Ni ∩{| y | > R } k w k L ∞ ( B | y | \ B | y | / ) | y | N i +2 s i dy = 2 c s i H N i − ( S N i − ) Z + ∞ R k w k L ∞ ( B ρ \ B ρ/ ) ρ s i dρ, which proves (30). The desired claim then follows recalling (5) and adding up the estimate in (30). (cid:3) Corollary 4.2.
Let R > . There exists η R ∈ C ∞ ( R n ) such that η R = 1 in ( − R, R ) n , η R = 0 in R n \ ( − R, R ) , and (32) k Lu − L ( η R u ) k L ∞ (( − R,R ) n ) C o Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ, (33) with C o as in (29) . Proof.
Let η o ∈ C ∞ ( R , [0 , η o = 1 in ( − ,
1) and η o = 0 outside ( − , η R ( x ) := n Y i =1 η o (cid:16) x i R (cid:17) . Then η R satisfies (32). Also, if we set w := (1 − η R ) u , we have from (32) that w = 0 in ( − R, R ) n .Thus, the estimate in (33) follows by writing u − η R u = w , using the linearity of the operator L andLemma 4.1. (cid:3) By combining Theorem 1.1 and Corollary 4.2, we can obtain a refined estimate in which the “contributionfrom infinity” in the right hand side of (7) is weighted “ring by ring”:
Theorem 4.3.
Let R > and f : B √ nR → R . Let u : R n → R be a solution of Lu = f in B √ nR . Then,for any t ∈ ( − R √ n , R √ n ) , | u ( te n ) − u ( − te n ) || t | C R sup ( x,t ) ∈ B R × (0 ,R ) | f ( x + te n ) − f ( x − te n ) | + R k u k L ∞ ( B R ) R s min + R Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ (cid:19) , (34) where C > here only depends on n , s min , s max , a min and a max .Proof. In this argument, we will take the freedom of renaming constants as we please, line after line, bykeeping the same name C . Using the notation of Corollary 4.2, we define ˜ u := η R u and, for any x ∈ B √ nR ,˜ f ( x ) := L ˜ u ( x ). Let also ˜ g := L ˜ u − Lu . By Corollary 4.2, k ˜ g k L ∞ (( − R,R ) n ) C Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ. By construction ˜ f = f + ˜ g , thereforesup ( x,t ) ∈ ( − R √ n , R √ n ) n × (0 , R √ n ) | ˜ f ( x + te n ) − ˜ f ( x − te n ) | sup ( x,t ) ∈ ( − R √ n , R √ n ) n × (0 , R √ n ) | f ( x + te n ) − f ( x − te n ) | + 2 k ˜ g k L ∞ (( − R √ n , R √ n ) n ) sup ( x,t ) ∈ B R × (0 ,R ) | f ( x + te n ) − f ( x − te n ) | + C Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ. Notice also that ˜ u = u in ( − R, R ) n and ˜ u = 0 outside B R . Thus, by applying Theorem 1.1 (herewith d = · · · = d m = R √ n ) to the function ˜ u , for any t ∈ ( − R √ n , R √ n ) we obtain that | u ( te n ) − u ( − te n ) || t | = | ˜ u ( te n ) − ˜ u ( − te n ) || t | CR sup ( x,t ) ∈ ( − R √ n , R √ n ) n × (0 , R √ n ) | ˜ f ( x + te n ) − ˜ f ( x − te n ) | + CR k ˜ u k L ∞ ( R n ) R s min CR sup ( x,t ) ∈ B R × (0 ,R ) | f ( x + te n ) − f ( x − te n ) | + CR Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ + CR k u k L ∞ ( B R ) R s min , as desired. (cid:3) NISOTROPIC NONLOCAL OPERATORS 13
Completion of the proof of Theorem 1.4.
Using L’Hˆopital’s Rule, we see thatlim R → + ∞ R Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρ = lim R → + ∞ Z + ∞ R k u k L ∞ ( B ρ \ B ρ/ ) ρ s min dρR = lim R → + ∞ k u k L ∞ ( B R \ B R ) (2 R ) s min R − = 0 , and lim R → + ∞ R k u k L ∞ ( B R ) R s min = 0 , thanks to (10). So, we can use Theorem 4.3 and pass formula (34) to the limit as R → + ∞ , and obtainthat | u ( te n ) − u ( − te n ) || t | = 0 , for any fixed t ∈ R . This says that u ( te n ) = u ( − te n ) for any t ∈ R Since the problem is translation invariant, we can apply the argument above in the neighborhood of anypoint, so we obtain that(35) u ( p + te n ) = u ( p − te n )for any p ∈ R n and any t ∈ R Now take any point x ∈ R n and any ρ ∈ R . We take p := x + ρ e n and t := ρ . Notice that p − te n = x and p + te n = x + ρe n , therefore (35) implies that u ( x ) = u ( x + ρe n ), which completes the proof ofTheorem 1.4. References [1] Achi E. Brandt,
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Alberto Farina: LAMFA – CNRS UMR 6140 – Universit´e de Picardie Jules Verne – Facult´e des Sciences– 33, rue Saint-Leu – 80039 Amiens CEDEX 1, France – Email: [email protected]