Strong maximum principles for fractional Laplacians
aa r X i v : . [ m a t h . A P ] F e b Strong maximum principles for fractional Laplacians
Roberta Musina ∗ and Alexander I. Nazarov † Abstract
We prove strong maximum principles for a large class of nonlocal operators of the or-der s ∈ (0 , Keywords:
Fractional Laplace operators, maximum principle.
In this paper we prove strong maximum principles for a large class of fractional Laplacians oforder s ∈ (0 , Dirichlet Laplacian ( − ∆) s u ( x ) = C n,s · P . V . Z R n u ( x ) − u ( y ) | x − y | n +2 s dy ,the Restricted Neumann Laplacian ( − ∆ N Ω ) s R u ( x ) = C n,s · P . V . Z Ω u ( x ) − u ( y ) | x − y | n +2 s dy (also called Regional
Laplacian), and intermediate operators, such as the
Semirestricted Neu-mann Laplacian ( − ∆ N Ω ) s Sr u = χ Ω · ( − ∆) s u + χ Ω c · ( − ∆ N Ω ) s R u . (1.1)Here Ω is a domain in R n , n ≥
1, Ω c = R n \ Ω, χ V is the characteristic function of the set V ⊂ R n , C n,s = s s Γ( n + s ) π n Γ(1 − s ) and ”P.V.” means ”principal value”. ∗ Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit`a di Udine, via delle Scienze, 206 –33100 Udine, Italy. Email: [email protected]. Partially supported by Miur-PRIN 2009WRJ3W7-001. † St.Petersburg Department of Steklov Institute, Fontanka 27, St.Petersburg, 191023, Russia, and St.PetersburgState University, Universitetskii pr. 28, St.Petersburg, 198504, Russia. E-mail: [email protected]. Partiallysupported by RFBR grant 17-01-00678. − ∆) s is the mono-graph [17]. Restricted Neumann Laplacians appear as generators of so-called censored processes;a vaste literature about the operator ( − ∆ N Ω ) s R is available as well, see for instance [9, 10] and[18, 19, 20], where Neumann, Robin and mixed boundary value problems on not necessarilyregular domains Ω are studied. The Semirestricted Neumann Laplacian ( − ∆ N Ω ) s Sr has been pro-posed in [7] to set up an alternative approach to Neumann problems, and can be used to studynon-homogeneous Dirichlet problems for ( − ∆) s , see for instance the survey paper [13].In this paper we propose a unifying approach to handle, in particular, all fractional Laplaciansabove. Let us describe the class of nonlocal operators we are interested in.Consider a domain Ω ⊆ R n and open sets U , U ⊆ R n such that Ω ⊆ U ∩ U . We put Z = (cid:0) U × U (cid:1) ∪ (cid:0) U × U (cid:1) ⊆ R n × R n , so that Ω × Ω ⊆ Z . For s ∈ (0 ,
1) we introduce the space X s (Ω; Z ) = (cid:8) u : U ∪ U → R measurable (cid:12)(cid:12) u ( x )1 + | x | n +2 s ∈ L ( U ∪ U ) , u ∈ H s loc (Ω) (cid:9) . Notice that, in particular, X s (Ω; Z ) contains functions u ∈ L ( U ∪ U ) such that E s ( u ; Z ) := C n,s Z Z Z ( u ( x ) − u ( y )) | x − y | n +2 s dxdy (1.2)is finite. For u ∈ X s (Ω; Z ) we introduce the distribution L sZ u ∈ D ′ (Ω) defined via h L sZ u, ϕ i = C n,s Z Z Z ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | n +2 s dxdy , ϕ ∈ C ∞ (Ω) , see Lemma 2.1.Notice that the operator L sZ might be used in modeling symmetric, (possibly) censored L´evyflights of a particle that can only jump from points x ∈ U to points y ∈ U , and vice versa.Our approach can be easily generalized for a wider class of kernels A ( x,y ) | x − y | n +2 s with A measurable,symmetric, bounded and bounded away from zero.We understand the inequality L sZ u ≥ h L sZ u, ϕ i ≥ , if ϕ ∈ C ∞ (Ω) , ϕ ≥ . In our main result, see Theorem 4.1, we provide a strong maximum principle for solutionsto L sZ u ≥ L sZ = ( − ∆) s , Ω bounded andsmooth, n ≥ u ∈ H s ( R n ) and u ≥ R n \ Ω. We cite also [5, Theorem 1.2] for a relatedresult involving the fractional Dirichlet p -Laplacian.The paper is organized as follows. In Section 2 we prove some auxiliary statements. Sec-tion 3 is devoted to Caccioppoli type estimates and to De Giorgi-type maximum estimates for(sub)solutions. In Section 4 we state and prove Theorem 4.1, and formulate correspondingresults for the Dirichlet, Restricted and Semirestricted Neumann Laplacians.In the Appendix we collect some more strong maximum principles for nonlocal Laplacians.First, we formulate a strong maximum principle for ( − ∆) s that is essentially contained in theremarkable paper [15] by Silvestre, who extended the classical theory of superhamonic functionsto the case of fractional Laplacian. Then we discuss strong maximum principles for spectralfractional Laplacians. The Spectral Dirichlet Laplacian ( − ∆ Ω ) s Sp (also called the Navier Lapla-cian ) is widely studied. Notice that for Ω = R n we have ( − ∆ Ω ) s Sp = ( − ∆) s , for other Ω theseoperators differ, see [12] for some integral and pointwise inequalities between them. The SpectralNeumann Laplacian ( − ∆ N Ω ) s Sp is less investigated; we limit ourselves to cite [1, 3, 8] and referencestherein. Notation.
Here we recall some basic notions taken from [17]. For Z ⊆ R n × R n and u measurable, let E s ( u ; Z ) be the quadratic form in (1.2). We put H s ( R n ) = (cid:8) u ∈ L ( R n ) | E s ( u ; R n × R n ) < ∞ (cid:9) , that is an Hilbert space with respect to the norm k u k H s ( R n ) := E s ( u ; R n × R n ) + k u k L ( R n ) . For any domain G ⊂ R n , we introduce the following closed subspace of H s ( R n ): e H s ( G ) = (cid:8) u ∈ H s ( R n ) | u = 0 on R n \ G (cid:9) , and its dual space e H s ( G ) ′ .We write u ∈ H s loc (Ω) if for any G ⋐ Ω, the function u is the restriction to G of some v ∈ H s ( R n ),and we put k u k H s ( G ) := inf (cid:8) k v k H s ( R n ) | v = u on G (cid:9) . It is well known that u ∈ H s loc (Ω) if and only if ηu ∈ e H s (Ω) for any η ∈ C ∞ (Ω), see for instance [17,Subsection 4.4.2].We adopt the following standard notation: B r ( x ) is the Euclidean ball of radius r centered at x , and B r = B r (0); ± = max {± u, } ; sup G u and inf G u stand for essential supremum/infimum of the measurable function u on the measurable set G ;Through the paper, all constants depending only on n and s are denoted by c . To indicate that aconstant depends on other quantities we list them in parentheses: c ( . . . ). For any function ϕ on R n we put Ψ ϕ ( x, y ) = ( ϕ ( x ) − ϕ ( y )) | x − y | n +2 s . (2.1) Lemma 2.1
Let u ∈ X s (Ω; Z ) . Then L sZ u is a well defined distribution in Ω . Moreover, forany Lipschitz domain G ⋐ Ω , we have L sZ u ∈ e H s ( G ) ′ and Z Z G × G | u ( x ) u ( y ) | Ψ ϕ ( x, y ) dxdy < ∞ for any ϕ ∈ C ∞ ( G ) . Proof.
Let ϕ ∈ C ∞ (Ω). In order to have that L sZ u is well defined we need to show that g ( x, y ) := ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | n +2 s ∈ L ( Z ) . Take two Lipschitz domains G, e G , such that supp( ϕ ) ⊂ G ⋐ e G ⋐ Ω. From u ∈ H s ( e G ), we have Z Z e G × e G | g ( x, y ) | dxdy ≤ k u k H s ( e G ) k ϕ k H s ( R n ) ≤ c ( G ) k u k H s ( e G ) k ϕ k e H s ( G ) . Next, since ϕ vanishes outside G , and since (cid:2) Z \ ( e G × e G ) (cid:3) \ ( G c × G c ) = h G × (cid:0) ( U ∪ U ) \ e G (cid:1)i ∪ h(cid:0) ( U ∪ U ) \ e G (cid:1) × G i , (2.2)it is enough to prove that g ∈ L ( G × (cid:0) ( U ∪ U ) \ e G )). We have Z Z G × (( U ∪ U ) \ e G ) | g ( x, y ) | dxdy ≤ Z G | ϕ ( x ) | (cid:20) Z ( U ∪ U ) \ e G | u ( x ) | + | u ( y ) || x − y | n +2 s dy (cid:21) dx ≤ c (dist( G, ∂ e G )) (cid:16) k ϕ k L ( G ) k u k L ( G ) + k ϕ k L ( G ) Z U ∪ U | u ( y ) | | y | n +2 s dy (cid:17) ≤ c ( G, e G, u ) k ϕ k e H s ( G ) .
4e proved that L sZ u ∈ D ′ (Ω) and actually L sZ u ∈ e H s ( G ) ′ , by the density of C ∞ ( G ) in e H s ( G ).Further, take again ϕ ∈ C ∞ ( G ) and notice that Ψ ϕ ( x, · ) ∈ L ( R n ) for any x ∈ R n , becauseΨ ϕ ( x, y ) ≤ c ( ϕ ) (cid:16) χ {| x − y | < } | x − y | n − − s ) + χ {| x − y | > } | x − y | n +2 s (cid:17) . Actually Z R n Ψ ϕ ( x, y ) dy ≤ c ( ϕ ), and by the Cauchy-Bunyakovsky-Schwarz inequality we infer Z Z G × G | u ( x ) u ( y ) | Ψ ϕ dxdy ≤ Z Z G × G | u ( x ) | Ψ ϕ dxdy ≤ c ( ϕ ) Z G | u ( x ) | dx ≤ c ( u, ϕ, G ) < ∞ . The lemma is proved. (cid:3)
Next, for any domain G ⊆ U ∩ U we introduce the relative killing measure M ZG ∈ L ∞ loc ( G ), M ZG ( x ) = C n,s Z ( U ∪ U ) \ G dy | x − y | n +2 s , x ∈ G .
When U ∪ U = R n , that happens for instance in the Dirichlet and in the Semirestricted cases,see Section 4, the weight M ZG coincides with so-called killing measure of the set G : M G ( x ) := M R n × R n G ( x ) = C n,s Z R n \ G dy | x − y | n +2 s . In the Restricted case we have U ∪ U = Ω and M ZG = M Ω × Ω G ; if G ⊂ Ω then M Ω × Ω G is thedifference between the killing measures of the sets G and Ω. Lemma 2.2
Let G ⊆ U ∩ U be a Lipschitz domain. If u ∈ e H s ( G ) , then E s ( u ; Z ) = E s ( u ; G × G ) + Z G M ZG ( x ) | u ( x ) | dx and in particular u is square integrable on G with respect to the measure M ZG ( x ) dx . Proof.
Trivially E s ( u ; Z ) < ∞ , as u ∈ e H s ( G ) ֒ → H s ( R n ). Since u vanishes on G c , using (2.2)with e G = G we have E s ( u ; Z ) = E s ( u ; G × G ) + 2 E s ( u ; G × (cid:2) ( U ∪ U ) \ G (cid:3) )= E s ( u ; G × G ) + C n,s Z G | u ( x ) | (cid:16) Z ( U ∪ U ) \ G dy | x − y | n +2 s (cid:17) dx, and the lemma is proved. (cid:3) u ∈ X s (Ω; Z ), that will be involved in the crucial Caccioppoli-type inequality in the next section.For u ∈ X s (Ω; Z ) and for any domain G ⊆ Ω we use Lemma 2.1 to introduce the distribution h (( − ∆ N ( U ∪ U ) \ G ) s R u ) , ϕ i := C n,s Z Z G × [( U ∪ U ) \ G ] ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | n +2 s dxdy, ϕ ∈ C ∞ ( G ) , that is the restriction on G of the Regional Laplacian of u relative to the set ( U ∪ U ) \ G . Lemma 2.3
Let G ⊆ Ω be a domain, u ∈ X s (Ω; Z ) . Then for any ϕ ∈ C ∞ ( G ) Z G | u ( x ) || ϕ ( x ) | (cid:16) Z ( U ∪ U ) \ G | u ( x ) − u ( y ) || x − y | n +2 s dy (cid:17) dx < ∞ . (2.3) In particular, u · ( − ∆ N ( U ∪ U ) \ G ) s R u ∈ L ( G ) . Proof.
Similarly as in the proof of Lemma 2.1, we estimate the integral in (2.3) by Z G | u ( x ) || ϕ ( x ) | (cid:20) Z ( U ∪ U ) \ G | u ( x ) | + | u ( y ) || x − y | n +2 s dy (cid:21) dx ≤ c (dist(supp( ϕ ) , ∂G )) k ϕ k L ∞ ( G ) (cid:16) k u k L (supp( ϕ )) + k u k L (supp( ϕ )) Z U ∪ U | u ( y ) | | y | n +2 s dy (cid:17) < ∞ , and the lemma follows. (cid:3) Lemma 2.4 If u ∈ X s (Ω; Z ) , then u ± ∈ X s (Ω; Z ) ; moreover for any G ⋐ Ω we have E ( u ± ; G × G ) < E ( u ; G × G ) , unless u has constant sign on G . Proof.
We compute( u ( x ) − u ( y )) − ( u + ( x ) − u + ( y )) = ( u − ( x ) − u − ( y )) + 2 (cid:16) u + ( x ) u − ( y ) + u − ( x ) u + ( y ) (cid:17) ≥ . Thus E ( u + ; G × G ) ≤ E ( u ; G × G ) < ∞ for any G ⋐ Ω. Therefore u + ∈ H s loc (Ω), and u + ∈ X s (Ω; Z ) follows.Next, assume that E ( u ; G × G ) = E ( u + ; G × G ) on some domain G ⋐ Ω. Then( u − ( x ) − u − ( y )) + 2 (cid:16) u + ( x ) u − ( y ) + u − ( x ) u + ( y ) (cid:17) = 0for a.e. ( x, y ) ∈ G × G . We infer that u − is constant a.e. on G . If u − = 0 then u ≥ G ; if u − = 0 we get u + = 0, that is, u ≤ G . The proof for the ”minus” sign follows by replacing u by − u . (cid:3) emark 2.5 If u ∈ L ( U ∪ U ) and E s ( u ; Z ) is finite, then E s ( | u | ; Z ) < E s ( u ; Z ) , unless u has constant sign on U ∪ U . The proof runs with no changes. Our proof of Theorem 4.1 requires the construction of a suitable barrier function. The nextLemma slightly generalizes a result by Ros-Oton and Serra [14].
Lemma 2.6
Let B R ( x ) ⊂ Ω . For any r ∈ (0 , R ) there exists a constant c = c ( R/r ) > and afunction Φ ∈ H s ( R n ) satisfying L sZ Φ ≤ in B R ( x ) \ B r ( x ) ;Φ ≡ in B r ( x ) , Φ ≡ in R n \ B R ( x ) , Φ( x ) ≥ c ( R − | x | ) s in B R ( x ) . (2.4) Proof.
Without loss of generality we can assume x = 0. Lemma 3.2 in [14], see also [11,Lemma 2.2], provides the existence of Φ ∈ e H s ( B R ) satisfying (2.4) and ( − ∆) s Φ ≤ B R \ B r .To conclude we claim that the distribution ( − ∆) s Φ − L sZ Φ is nonnegative in Ω. Indeed, take η ∈ C ∞ (Ω). Since both Φ and η vanish on R n \ Ω, we have h ( − ∆) s Φ − L sZ Φ , η i = C n,s Z Z R n \ Z (Φ( x ) − Φ( y ))( η ( x ) − η ( y )) | x − y | n +2 s dxdy = C n,s Z Ω Φ( x ) η ( x ) (cid:16) Z R n \ ( U ∪ U ) dy | x − y | n +2 s dy (cid:17) dx, and the claim follows. In particular, L sZ Φ ≤ ( − ∆) s Φ ≤ B R \ B r , and we are done. (cid:3) Remark 2.7
It is worth to note that if Z ⊂ Z ′ then for any nonnegative Φ ∈ e H s (Ω) theinequality L sZ Φ ≤ L sZ ′ Φ holds in Ω . The proof runs without changes. We conclude this preliminary section by the following remark. We fix an exponent ¯ p > p = 4 (for instance) if n = 1 ≤ s , and ¯ p = 2 ∗ s = nn − s if n > s . Take anyradius r ∈ (cid:0) , (cid:3) . The Sobolev embedding theorem implies E s ( u ; R n × R n ) ≥ c (cid:16) Z B r | u | ¯ p dx (cid:17) / ¯ p for any u ∈ e H s ( B r ).Now let ρ ∈ (cid:0) , r (cid:1) . Since for u ∈ e H s ( B ρ ) one has E s ( u ; R n × R n ) = E s ( u ; B r × B r ) + C n,s Z B r | u ( x ) | (cid:16) Z R n \ B r dy | x − y | n +2 s (cid:17) dx,
7e plainly infer that E s ( u ; B r × B r ) + 1( r − ρ ) s Z B r | u | dx ≥ c (cid:16) Z B ρ | u | ¯ p dx (cid:17) p for any u ∈ e H s ( B ρ ). (2.5) L sZ -subharmonic functions First, we prove a Caccioppoli-type inequality. We use again the notation introduced in (2.1).
Lemma 3.1
Let G ⊆ Ω be a Lipschitz domain, w ∈ X s (Ω; Z ) and ϕ ∈ C ∞ ( G ) . Then E s ( ϕw + ; G × G ) ≤ h L sZ w, ϕ w + i (3.1)+ C n,s Z Z G × G w + ( x ) w + ( y )Ψ ϕ ( x, y ) dxdy − Z G w + ϕ ( − ∆ N ( U ∪ U ) \ G ) s R w + dx . Proof.
Note that all quantities in (3.1) are finite by Lemmata 2.1, 2.3 and 2.4. We compute( w ( x ) − w ( y ))(( ϕ w + )( x ) − ( ϕ w + )( y )) − (( ϕw + )( x ) − ( ϕw + )( y )) = − w + ( x ) w + ( y )( ϕ ( x ) − ϕ ( y )) + (cid:0) ϕ ( y ) w − ( x ) w + ( y ) + ϕ ( x ) w + ( x ) w − ( y ) (cid:1) ≥ − w + ( x ) w + ( y )( ϕ ( x ) − ϕ ( y )) to infer h L sZ w, ϕ w + i = C n,s Z Z Z ( w ( x ) − w ( y ))(( ϕ w + )( x ) − ( ϕ w + )( y )) | x − y | n +2 s dxdy ≥ E s ( ϕw + ; Z ) − C n,s Z Z Z w + ( x ) w + ( y )Ψ ϕ dxdy = E s ( ϕw + , G × G ) + Z G M ZG ( x ) | w + ϕ | dx − C n,s Z Z Z w + ( x ) w + ( y )Ψ ϕ dxdy by Lemma 2.2. We compute Z G M ZG ( x ) | w + ϕ | dx = C n,s Z G w + ( x ) | ϕ ( x ) | (cid:16) Z ( U ∪ U ) \ G ( w + ( x ) − w + ( y ) + w + ( y )) | x − y | n +2 s dy (cid:17) dx = Z G w + | ϕ | ( − ∆ N ( U ∪ U ) \ G ) s R w + dx + C n,s Z G w + ( x ) | ϕ ( x ) | (cid:16) Z ( U ∪ U ) \ G w + ( y ) | x − y | n +2 s dy (cid:17) dx . ϕ ≡ G c × G c we have, by (2.2) with e G = G , C n,s Z Z Z w + ( x ) w + ( y )Ψ ϕ dxdy = C n,s Z Z G × G w + ( x ) w + ( y )Ψ ϕ dxdy + C n,s Z G w + ( x ) (cid:16) Z ( U ∪ U ) \ G w + ( y )Ψ ϕ dy (cid:17) dx = C n,s Z Z G × G w + ( x ) w + ( y )Ψ ϕ dxdy + C n,s Z G w + ( x ) ϕ ( x ) (cid:16) Z ( U ∪ U ) \ G w + ( y ) | x − y | n +2 s dy (cid:17) dx , and the Lemma follows. (cid:3) Remark 3.2
Inequality (3.1) was essentially proved in [6, Theorem 1.4], in a weaker form butin a non-Hilbertian setting and for more general kernels.
The next De Giorgi-type result is obtained by suitably modifying the argument for [6, The-orem 1.1].
Lemma 3.3
For any u ∈ X s (Ω; Z ) such that L sZ u ≤ in Ω and for every ball B r ( x ) ⊆ Ω ,one has sup B r ( x ) u ≤ (cid:16) b cr n Z B r ( x ) | u + ( x ) | dx (cid:17) + r s Z ( U ∪ U ) \ B r ( x ) | u + ( x ) || x − x | n +2 s dx , (3.2) where b c > depends only on n and s . In particular, u is locally bounded from above in Ω . Proof.
First of all let us recall that for U ∪ U = R n the last term in (3.2) is called nonlocal tail .For Z = R n × R n we call this term relative nonlocal tail and denote it by Tail Z ( u + ; x , r ).By rescaling we can assume without loss of generality that r = 1 and x = 0. We introducea parameter ˜ k > k ≥ Tail Z ( u + ; 0 ,
1) (3.3)(its value will be chosen later). For any integer j ≥ r j = 1 + 2 − j , k j = ˜ k (1 − − j ) , B j = B r j ;˜ r j = r j + r j +1 , ˜ k j = k j + k j +1 , e B j = B ˜ r j ; w j = ( u − k j ) + , e w j = ( u − ˜ k j ) + , α j = (cid:0) Z B j | w j | dx (cid:1) . r j ց , r j +1 < ˜ r j < r j ; k j ր ˜ k , k j < ˜ k j < k j +1 ; e w j ≤ w j , e w j ≤ w j ˜ k j − k j = 2 j +2 ˜ k w j ; (3.4) α = Z B | u + | dx , α j → Z B | ( u − ˜ k ) + | dx as j → ∞ . (3.5)In addition, we have w j +1 (cid:16) ˜ k j +2 (cid:17) ¯ p − = w j +1 ( k j +1 − ˜ k j ) ¯ p − ≤ e w ¯ pj , (3.6)where the exponent ¯ p > j ≥ ϕ j satisfying ϕ j ∈ C ∞ ( e B j ) , ≤ ϕ j ≤ , ϕ ≡ B j +1 , k∇ ϕ k ∞ ≤ j +3 . Since e w j ϕ j ∈ e H s ( B j ) and 1 < ˜ r j < r j <
2, by (2.5) with ρ = ˜ r j and r = r j , we have c (cid:16) Z B j | e w j ϕ j | ¯ p (cid:17) p ≤ E s ( e w j ϕ j ; B j × B j ) + 2 s ( j +2) Z B j | e w j ϕ j | dx. (3.7)Notice that h L sZ ( u − ˜ k j ) , e w j ϕ j i ≤
0, since L sZ ( u − ˜ k j ) = L sZ u ≤ B j and e w j ϕ j ∈ e H s ( B j )is nonnegative. Using Lemma 3.1 with w = u − ˜ k j , we infer E s ( e w j ϕ j ; B j × B j ) ≤ c Z Z B j × B j e w j ( x ) e w j ( y )Ψ ϕ j ( x, y ) dxdy − Z B j e w j ϕ j ( − ∆ N ( U ∪ U ) \ B j ) s R e w j dx, so that (cid:16) Z B j | e w j ϕ j | ¯ p dx (cid:17) p ≤ c (cid:16) J − J + ˜ k sj (cid:16) α j ˜ k (cid:17) (cid:17) (3.8)by (3.7), where J = Z Z B j × B j e w j ( x ) e w j ( y )Ψ ϕ j ( x, y ) dxdy , J = Z B j e w j ϕ j ( − ∆ N ( U ∪ U ) \ B j ) s R e w j ) dx. We estimate from below the left-hand side of (3.8) via (3.6): Z B j | e w j ϕ j | ¯ p dx ≥ Z B j +1 | e w j | ¯ p dx ≥ c (cid:16) ˜ k j (cid:17) ¯ p − Z B j +1 | w j +1 | dx = c ˜ k ¯ p j (2 − ¯ p ) (cid:16) α j +1 ˜ k (cid:17) . (3.9)10e estimate J by usingΨ ϕ j ( x, y ) ≤ k∇ ϕ j k ∞ | x − y | − ( n +2 s − ≤ c j | x − y | − ( n +2 s − and the Cauchy-Bunyakovsky-Schwarz inequality, to obtain J ≤ c j Z Z B j × B j e w j ( x ) | x − y | n +2 s − e w j ( y ) | x − y | n +2 s − dxdy ≤ c j Z Z B j × B j | e w j ( x ) | | x − y | n +2 s − dxdy = c j Z B j | e w j ( x ) | (cid:16) Z B j dy | x − y | n +2 s − (cid:17) dx ≤ cr − sj j α j ≤ c ˜ k j (cid:16) α j ˜ k (cid:17) . (3.10)We handle J as follows. For x ∈ supp( ϕ j ) ⊂ e B j and y ∈ Ω \ B j we have | y || x − y | ≤ | x || x − y | ≤ r j r j − ˜ r j ≤ c j . Hence, using also (3.4) we can estimate e w j ( x ) | ϕ j ( x ) | e w j ( y ) − e w j ( x ) | x − y | n +2 s ≤ c ˜ k | w j ( x ) | j ( n +2 s +1) w j ( y ) | y | n +2 s , so that − J = Z B j e w j ( x ) | ϕ j ( x ) | (cid:16) Z ( U ∪ U ) \ B j e w j ( y ) − e w j ( x ) | x − y | n +2 s dy (cid:17) dx ≤ c ˜ k j ( n +2 s +1) (cid:16) Z ( U ∪ U ) \ B j w j ( y ) | y | n +2 s dy (cid:17) Z B j | w j | dx ≤ c ˜ k j ( n +2 s +1) Tail Z ( u + ; 0 , (cid:16) α j ˜ k (cid:17) because B j ⊃ B and w j ≤ u + . Comparing with (3.8), (3.9) and (3.10) we arrive at2 − ¯ p )¯ p j (cid:16) α j +1 ˜ k (cid:17) p ≤ c j ( n +2 s +1) (cid:16) k − Tail Z ( u + ; 0 , (cid:17)(cid:16) α j ˜ k (cid:17) . Taking (3.3) into account, we can conclude that α j +1 ˜ k ≤ ( b c β η − β ) η j (cid:16) α j ˜ k (cid:17) β +1 , (3.11)where β = ¯ p − > , η = 2 ¯ p ( n +2 s +1)+ β >
1. Now we choose the free parameter ˜ k , namely˜ k = Tail Z ( u + ; 0 ,
1) + b c α = Tail Z ( u + ; 0 ,
1) + (cid:16)b c Z B | u + | dx (cid:17) , k guarantees that b c α j ˜ k ≤ η − jβ (3.12)for j = 0. Using induction and (3.11) one easily gets that (3.12) holds for any j ≥
0. Thus α j → u − ˜ k ) + ≡ B by (3.5). The proof is complete. (cid:3) We are in position to state and prove a strong maximum principle for the nonlocal operator L sZ ,that is the main result of the present paper. Theorem 4.1
Let u be a nonconstant measurable function on U ∪ U such that u ∈ H s loc (Ω) , Z U ∪ U | u ( x ) | | x | n +2 s dx < ∞ , L sZ u ≥ in Ω .Then u is lower semicontinuous on Ω , locally bounded from below on Ω and u ( x ) > inf U ∪ U u for every x ∈ Ω . Proof.
First, local boundedness from below follows from Lemma 3.3.To check the first claim it suffices to show that u has a representative that is lower semicon-tinuous on any fixed domain G ⋐ Ω. From C n,s Z G (cid:16) Z G ( u ( x ) − u ( y )) | x − y | n +2 s dy (cid:17) dx < ∞ we infer that Z G ( u ( x ) − u ( y )) | y − x | n +2 s dy < ∞ (4.1)for a.e. x ∈ G . Let G be the set of Lebesgue points x ∈ G for u that satisfy (4.1). We canassume that u ( x ) = lim inf x → x u ( x ) for any x ∈ G \ G , because G \ G has null Lebesgue measure.Our next goal is to show that u ( x ) ≤ lim inf x → x u ( x ) for any x ∈ G . We use Lemma 3.3 with u replaced by u ( x ) − u to getinf B r ( x ) u ≥ u ( x ) − Tail Z (( u ( x ) − u ) + ; x , r ) − (cid:16) b cr n Z B r ( x ) | ( u ( x ) − u ) + | dx (cid:17) (4.2)12or any r > Z (( u ( x ) − u ) + ; x , r ) = r s Z ( U ∪ U ) \ B r ( x ) | u ( x ) − u ( x ) || x − x | n +2 s dx ≤ r s Z ( U ∪ U ) \ G | u ( x ) | + | u ( x ) || x − x | n +2 s dx + r s Z G \ B r ( x ) | u ( x ) − u ( x ) || x − x | n +2 s dx =: P r + Q r . We readily obtain P r ≤ c (dist( x , ∂G )) r s Z ( U ∪ U ) | u ( x ) | + | u ( x ) | | x | n +2 s dx → r →
0. Next we use the Cauchy-Bunyakovsky-Schwarz inequality to estimate Q r ≤ r s (cid:16) Z G ( u ( x ) − u ( x )) | x − x | n +2 s dx (cid:17) (cid:16) Z R n \ B r ( x ) dx | x − x | n +2 s (cid:17) = cr s (cid:16) Z G ( u ( x ) − u ( x )) | x − x | n +2 s dx (cid:17) . Since (4.1) is satisfied at x = x , we have that Q r →
0. Thus Tail Z (( u ( x ) − u ) + ; x , r ) → r →
0. Further, the last term in (4.2) goes to zero as r → x is a Lebesgue point for u . Thus lim inf x → x u ( x ) ≥ u ( x ), and the first statement is proved.Next, assume by contradiction that u is bounded from below andΩ + := { x ∈ Ω | u ( x ) > m := inf U ∪ U u } is strictly contained in Ω. Since u is lower semicontinuous on Ω, the set Ω + is open and has anonempty boundary in Ω.Fix a point ξ ∈ Ω ∩ ∂ Ω + , so that u ( ξ ) = m . Using again the lower-semicontinuity of u , we canfind R > r > x ∈ Ω + , such that ξ ∈ B R ( x ) ⋐ Ω and u ( x ) ≥ ( u ( x ) + m ) > m for every x ∈ B r ( x ). We can assume that x = 0 to simplify notations. Thus we have thefollowing situation: ξ ∈ B R ⊂ Ω , u ( ξ ) = m , inf B r u ( x ) ≥ m + δ (4.3)for some δ >
0. Let Φ be the function defined in Lemma 2.6. We claim that u ≥ m + δ Φ > m in B R \ B r , that gives a contradiction with (4.3).Indeed, define v = u − δ Φ, so that v = u − δ ≥ m in B r , v = u ≥ m in ( U ∪ U ) \ B R .13ur goal is to show that v ≥ m also on B R \ B r .Clearly v ∈ X s (Ω; Z ) as u, Φ ∈ X s (Ω; Z ). By Lemma 2.4 this implies v m ± := ( v − m ) ± ∈ X s (Ω; Z ). Next, notice that v m − = 0 out of B R \ B r . Therefore v m − ∈ e H s ( B R \ B r ), and usingLemma 2.1 we obtain h L sZ v, v m − i = h L sZ u, v m − i − δ h L sZ Φ , v m − i ≥ . (4.4)However, h L sZ v, v m − i = C n,s Z Z Z (( v ( x ) − m ) − ( v ( y ) − m ))( v m − ( x ) − v m − ( y )) | x − y | n +2 s dxdy = − C n,s Z Z Z v m + ( x ) v m − ( y ) + v m + ( y ) v m − ( x ) | x − y | n +2 s dxdy − E s ( v m − ; Z ) ≤ −E s ( v m − ; B R × B R ) , so that (4.4) implies E s ( v m − ; B R × B R ) = 0, that together with v m − ∈ e H s ( B R \ B r ) gives theconclusion. (cid:3) By choosing U = U = R n we have Z = R n × R n . It is well known that L s R n × R n u = ( − ∆) s u pointwise on R n if u ∈ C ( R n ). Thanks to Lemma 2.1, we can say that L s R n × R n u = ( − ∆) s u in e H s ( G ) ′ , for every u ∈ X s (Ω; R n × R n ) and for any Lipschitz domain G ⋐ Ω. From Theorem 4.1we immediately infer the next result.
Corollary 4.2 (Dirichlet Laplacian)
Let Ω ⊆ R n be a domain, and let u be a nonconstantmeasurable function on R n such that u ∈ H s loc (Ω) , Z R n | u ( x ) | | x | n +2 s dx < ∞ , ( − ∆) s u ≥ in Ω .Then u is lower semicontinuous on Ω , locally bounded from below on Ω and u ( x ) > inf R n u forevery x ∈ Ω . The Restricted Laplacian is obtained by choosing U = U = Ω. In fact, L s Ω × Ω u = ( − ∆ N Ω ) s R u on Ω if u ∈ C (Ω), and L s Ω × Ω u = ( − ∆ N Ω ) s R u in e H s ( G ) ′ , for every u ∈ X s (Ω; R n × R n ) and everyLipschitz domain G ⋐ Ω. From Theorem 4.1 we infer the next result.
Corollary 4.3 (Restricted Laplacian)
Let Ω ⊂ R n be a domain, and let u be a nonconstantmeasurable function on Ω such that u ∈ H s loc (Ω) , Z Ω | u ( x ) | | x | n +2 s dx < ∞ , ( − ∆ N Ω ) s R u ≥ in Ω . hen u is lower semicontinuous on Ω , locally bounded from below on Ω and u ( x ) > inf Ω u forevery x ∈ Ω . Next, we choose U = Ω , U = R n , so that Z = R n \ (Ω c ) . By [7, Lemma 3] we have that L s R n \ (Ω c ) u = ( − ∆ N Ω ) s Sr u if u ∈ C ( R n ) ∩ L ∞ ( R n ), compare with (1.1). From the computationsthere and thanks to Lemma 2.1 we can identify the distributions L s R n \ (Ω c ) u and ( − ∆ N Ω ) s Sr u forfunctions u ∈ X s (Ω; R n \ (Ω c ) ) (see also [7, Definition 3.6]). Theorem 4.1 immediately impliesthe next corollary, see also [2, Theorem 1.1] for a related result. Corollary 4.4 (Semirestricted Laplacian)
Let Ω ⊂ R n be a domain, and let u be a noncon-stant measurable function on R n such that u ∈ H s loc (Ω) , Z R n | u ( x ) | | x | n +2 s dx < ∞ , ( − ∆ N Ω ) s Sr u ≥ in Ω .Then u is lower semicontinuous on Ω , locally bounded from below on Ω and u ( x ) > inf R n u forevery x ∈ Ω . We conclude by recalling that in the local case s = 1, the strong maximum principle statesthat every nonconstant superharmonic function u on Ω satisfies u ( x ) > inf Ω u for every x ∈ Ω.Notice that in the non local, Neumann Restricted case we reached the same conclusion. Incontrast, in the Dirichlet and in the Semirestricted cases a similar result can not hold, see theexample in the next remark.
Remark 4.5
Take any bounded domain Ω ∈ R n and two nonnegative functions u, ψ ∈ C ∞ ( R n ) such that ≤ u ≤ , u ≡ on Ω , supp ψ ⊂ Ω . For any x ∈ Ω we have ( − ∆) s u ( x ) = ( − ∆ N Ω ) s Sr u ( x ) = C n,s · P . V . Z R n − u ( y ) | x − y | n +2 s dy > . Since ( − ∆) s u, ( − ∆) s ψ are smooth functions, we have that ( − ∆) s ( u − εψ ) ≥ in Ω , for somesmall ε > . Then u − εψ satisfies the assumptions in Corollaries 4.2 and 4.4, but inf Ω ( u − εψ ) is achieved in Ω . Clearly, inf R n ( u − εψ ) = 0 is not achieved in Ω . Appendix
We start with a proposition in fact proved in [15]. It gives the same conclusion as in Corollary4.2 under weaker summability assumptions on u . Notice however that n > s is needed (this is15 restriction only if n = 1), and that Silvestre’s construction cannot be easily extended to moregeneral operators such as the Restricted and Semirestricted ones. Proposition A.1
Assume n > s and let u be a nonconstant measurable function on R n suchthat u ( x )1+ | x | n +2 s ∈ L ( R n ) and ( − ∆) s u ≥ in the distributional sense on Ω , that is, h ( − ∆) s u, ϕ i = Z R n u ( − ∆) s ϕ dx ≥ for any ϕ ∈ C ∞ ( R n ) , ϕ ≥ .Then u is lower semicontinuous on Ω and u ( x ) > inf R n u for every x ∈ Ω . Proof.
First, notice that ( − ∆) s u is a well defined distribution, as (1 + | x | n +2 s ) ( − ∆) s ϕ isa bounded function on R n , for any ϕ ∈ C ∞ ( R n ). Proposition 2.2.6 in [15] gives the lowersemicontinuity of u in Ω and the relations u ( x ) ≥ Z R n u ( x ) γ sr ( x − x ) dx > −∞ for any ball B r ( x ) ⊂ Ω, where γ sr is certain continuous and positive function on R n . If u is unbounded frombelow we are done; otherwise, we can assume inf R n u = 0. Suppose that there exists x ∈ Ω suchthat u ( x ) = 0. Take a ball B r ( x ) ⊂ Ω. Then 0 ≥ Z R n u ( x ) γ sr ( x − x ) dx ≥
0, that immediatelyimplies u ≡ R n , a contradiction. (cid:3) Now we recall that the Spectral Dirichlet/Neumann fractional Laplacian is the s -th powerof standard Dirichlet/Neumann Laplacian in Ω in the sense of spectral theory.A strong maximum principle for the Spectral Dirichlet Laplacian follows from [4, Lemma2.6] and reads as follows. Proposition A.2
Let Ω ⊂ R n be a bounded domain, and let a function u ∈ e H s (Ω) be suchthat ( − ∆ Ω ) s Sp u ≥ in Ω the sense of distributions. Then either u ≡ or inf K u > for arbitrarycompact set K ⊂ Ω . A strong maximum principle for the Spectral Neumann Laplacian can be obtained from theresults in [4].
Theorem A.3
Let Ω ⊂ R n be a bounded Lipschitz domain, and let a function u ∈ H s (Ω) be such that ( − ∆ N Ω ) s Sp u ≥ in a subdomain G ⊂ Ω in the sense of distributions. Then either u ≡ const or inf K u > inf Ω u for arbitrary compact set K ⊂ G . Proof.
It is well known, see [16], [3] and [1] for a general setting, that for any u ∈ H s (Ω) theboundary value problem − div( y − s ∇ w ) = 0 in Ω × R + ; w (cid:12)(cid:12) y =0 = u ; ∂ n w (cid:12)(cid:12) x ∈ ∂ Ω = 0 , (A.1)16as a unique weak solution w Ns ( x, y ), and( − ∆ N Ω ) s Sp u ( x ) = − s − Γ( s )Γ(1 − s ) · lim y → + y − s ∂ y w Ns ( x, y )(the limit is understood in the sense of distributions).Without loss of generality we can assume that inf Ω u = 0. Then by the maximum principlefor (A.1) we have w ≥
0. By [4, Lemma 2.6] the statement follows. (cid:3)
Acknowledgments . This paper was completed while the second author was visiting SISSA(Trieste), in January 2017. The authors wish to thank SISSA for the hospitality.
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