CHENCHEN MOU AND YINGFEI YI

Transcription

1 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN CHENCHEN MOU AND YINGFEI YI Abstract. In this paper, we study interior regularity properties for the regional fractional Laplacian operator. We obtain the integer order differentiability of the regional fractional Laplacian, which solves a conjecture of Guan and Ma [9]. We further extend the integer order differentiability to the fractional order of the regional fractional Laplacian. Schauder estimates for the regional fractional Laplacian are also provided.. Introduction This paper is devoted to the study of interior regularity for the regional fractional Laplacian operators. Given real numbers 0 < s <, ɛ > 0, and an open set, denote (.) s u(y),ɛ = A(n, s) dy, Bɛ c(x) x y n+s where A(n, s) = s s Γ( n+s ) π n Γ( s ), B ɛ (x) is the open ɛ-ball in centered at x, and u dx (+ x ) n+s ), i.e., L (, is defined as (+ x ) n+s dx <. (.) s = lim ɛ 0 s,ɛ, The regional s-fractional Laplacian s on u L (, dx ), ( + x ) n+s provided that the limit exists. The regional s-fractional Laplacian can be also defined on the closure of by talking in place of in the above. We note that, if x, then s = s. When is a bounded Lipschitz open set, the regional s-fractional Laplacian s is in fact the Feller generator of the so-called reflected symmetric s-stable process (X t ) t 0 on, i.e., a Hunt process associated with the regular Dirichlet form (E, F) on L (, dx): E(u, v) = A(n, s) ( u(y))(v(x) v(y)) x y n+s dxdy, { F = u L ( u(y)) } ( ), x y n+s dxdy <. It is first shown in [] that if 0 < s, then the censored s-stable process in is essentially the reflected s-stable process (X t ) t 0, and if < s <, then the censored s-stable process in is identified as a proper subprocess of (X t ) t 0 killed upon leaving. Later, it is 000 Mathematics Subject Classification. Primary 35J05; Secondary 3B0, 60G5, 60J75. Key words and phrases. Interior regularity, Regional fractional Laplacian, Differentiability, Schauder estimates, Reflected symmetric s-stable process. This research was partially supported by NSF grant DMS090. The second author was also supported by a NSERC discovery grant, a scholarship from Jilin University, and a faculty development fund from the University of Alberta.

2 C. MOU AND Y. YI shown in [3] that (X t ) t 0 can be refined to be a Feller process starting from each point of which admits a Hölder continuous transition density function. In [9], not only does the Feller generator of (X t ) t 0 on shown to be the regional s-fractional Laplacian, but also a semi-martingale decomposition of (X t ) t 0 is obtained by studying the differentiability of the regional fractional Laplacian and its integration by parts property. For other studies on regional fractional Laplacians, we refer the reader to [7] for a more general integration by parts formula of the regional fractional and fractional-like Laplacian, and to [8] for some boundary Harnack inequalities for the regional fractional Laplacian on C,β (), s < β. If =, the regional fractional Laplacian s becomes the usual fractional Laplacian ( ) s defined via Fourier transform: F(( ) s u)(ξ) = ξ s F(u)(ξ) (see [4]). If we furtherly let s tend to, then the fractional Laplacian ( ) s becomes the classical Laplacian, and it is clear that u C α for some integer α > implies that u C α. In the case that u C α for some α > s with α s not being an integer, one also has ( ) s u C αs ([, Proposition.7]). A natural problem then is whether the regional fractional Laplacian shares similar regularity properties as that of the classical and fractional Laplacian. This problem is first investigated in [9] in which the following results are proved. Theorem. Let be an open set in and u L dx (, ) for some 0 < s <. Then (+ x ) n+s the following holds. a) ([9, Proposition 8.3]) If u C,α () for some α > s when 0 < s < or u C,α () for some α > s when s <, then s u C,0 (). b) ([9, Theorem 8.]) In the case n =, if r is a non-negative integer such that u C r,α () for some some α > s when 0 < s < or u C r+,α () for some α > s when s <, then s u C r,0 (). In the above and also through the rest of the paper, for each non-negative integer r and 0 α, we denote by C r,α () (C r,α ( )) the subspace of C r,0 () (C r,0 ( )) consisting functions whose rth partial derivatives are locally (uniformly) α-hölder continuous in. It is conjectured in [9] that part b) of the above theorem should hold for higher dimensions as well. In this paper, we give an affirmative answer to this conjecture. More precisely, we will prove the following result. Theorem A. Let be an open set and u L (, (+ x ) n+s ) for some 0 < s <. If r is a non-negative integer such that u C r,α () for some α > s or u C r+,α () for some α with + α > s α, then s u C r,0 (). Unlike the fractional Laplacian, the differential operator and the regional fractional Laplacian are not exchangeable in order. To overcome this difficulty in the proof of Theorem A, we derive a class of integral identities (see Lemma.) and use them to conclude that all possible singular terms of D r ( s,ɛ u) as ɛ 0 + are in fact non-singular. Making further estimates, we are able to extend the integer order differentiability result in Theorem A to a fractional order. Throughout the paper, for each j = (j, j j n ) N n, we denote j = j + j + + j n and j u = j u ( x ) j ( x ) j ( x. For any u n) jn Cr,α (), where r

3 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 3 is a non-negative integer and 0 α, define { supx, j =r j, if α = 0; [u] r,α; = sup j j u(y) x,y,x y, j =r xy, if α > 0. α Then we have the follows result analogous to [, Proposition.7] in the case of regional fractional Laplacian. Theorem B. Let be an open set and u L (, (+ x ) n+s ) for some 0 < s <. Then the following holds. (i) If u C r,α () for some positive number α with s < α and a non-negative integer r, then s C r,αs (), and moreover, [ s u] r,αs; C[u] r,α;. (ii) If u C r,α () for some positive number α with α < s < + α and a positive integer r, then s C r,+αs (), and moreover, [ s u] r,+αs; C[u] r,α;. Schauder estimate is well-known for the classical Laplacian (see [6]) as well as for the fractional Laplacian (see [, 5, 0, ]). We refer the reader to [, 5] for interior and boundary regularity theory for more general fractional operators. Using Schauder estimates for the fractional Laplacian, we are able to show a similar Schauder estimate holds for the regional fractional Laplacian. For each u C r,α ( ), denote u C r,α ( ) = { r j=0 [u] j,0,, if α = 0; u C r,0 ( ) + [u] r,α;, if α > 0. For simplicity, through the rest of the paper, we use the notation C α () (C α ( )), where α > 0, to denote the space C r,α () (C r,α ( )), where r is the largest integer smaller than α and α = α r. We note that if α is an integer r, then C α () = C α, () C α,0 () (C α ( ) = C α, ( ) C α,0 ( )). For a given open set in with, let d x = dist(x, ) and δ = {x ; d x < δ}. We denote Cc () as the space of C functions with compact support in, S as the Schwartz space of rapidly decreasing C function in, and Λ ( ) as the Zygmund space of all bounded functions on such that [u] := u(x + h) + u(x h) Λ ( ) <. h sup x,x+h,xh We equip the space Λ ( ) with the norm u := u Λ ( ) L ( ) +[u] Λ ( ). It is easy to see that the regional fractional Laplacian s is well-defined for functions u C () L (). We can then extend the definition of the regional fractional Laplacian to the space L (, dx For any u L (, ), we define (+ x ) n+s (.3) < s u, ϕ > := u(y) s ϕ(y)dy, for any ϕ Cc (). dx (+ x ) n+s ).

4 4 C. MOU AND Y. YI In Theorems C and D below, the definition of s is understood in the sense of (.3). Theorem C. Let 0 < α and 0 < s <. If, for some w C α ( ), u L () solves the equation s u = w in, then u C α+s ( δ ) C( u L () + w C α ( ) ), where δ > 0 is sufficiently small and C is a constant depending only on n, s, δ and α. Remark.. The notions of C α+s ( δ ) and C α ( ) we defined earlier have unified different cases for α, α+s being or not being natural numbers in the statement of Theorem C. We note that C ( δ ) = C 0, ( δ ) C,0 ( δ ), C ( ) = C 0, ( ) C,0 ( ), and C ( δ ) = C, ( δ ) C,0 ( δ ). Theorem D. Let 0 < s <. Suppose that, for some w L (), u L () solves the equation s u = w in. Then, for any sufficiently small δ > 0, there exists a constant C > 0 depending only on n, s, and δ such that the following holds: (i) If s, then (ii) If s =, then u C s ( δ ) C( u L () + w L ()). u Λ ( δ ) C( u L () + w L ()). The rest of sections are devoted to the proof of these results. In Section, we will give a class of integral identities and use them to obtain the integer order differentiability of the regional fractional Laplacian s. Both Theorem A and B will be proved in this section. In Section 3, Schauder estimate for the regional fractional Laplacian will be given and Theorem C and D will be proved.. Differentiability In this section, we will study the differentiability of the regional fractional Laplacian in an open set and prove Theorems A, B. The proof will be based on some integral identities in... Integral identities. Given n N, z = (z, z,, z n ) and k = (k, k,, k n ) N n, we denote by z k the monomial n i= zk i i. Also, for each j =,,, n, we let e j to denote the jth standard coordinate vector in. Lemma.. Consider an annulus domain R δɛ (0) := {z : ɛ < z < δ}, where 0 < ɛ < δ. Then for any i, j, m N and k = (k, k,, k n ) N n, we have z k+e i k i + z m dz = z k+e j k j + z m dz. R δɛ (0) R δɛ (0) Proof. The lemma clearly holds if i = j. Let i j, and without loss of generality, we assume that i = n and j = n. Denote z = (z, z,, z n ), k = (k, k,, k n ), and consider polar coordinate (r, θ) in the (z n, z n )-plane. Then under the new coordinate

5 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 5 R δɛ (0) becomes R δɛ (0) (0, π) = {( z, r) R + : ɛ < z + r < δ } (0, π), and integration by parts yields that z k zn R δɛ (0) z m dz π z k = r k n+k n+3 d zdr cos kn+ θsin R δɛ (0) ( z + r ) m kn θdθ 0 π z k = (k n + ) r k n+k n+3 d zdr cos k R δɛ (0) ( z + r ) m n θsin kn+ θdθ 0 π z k k n r k n+k n+3 d zdr cos kn+ θsin ( z + r ) m kn θdθ R δɛ (0) = (k n + ) R δɛ (0) from which the lemma follows. z k z n z m dz k n R δɛ (0) 0 z k zn z m dz, Lemma.. ([9, Lemma 8.]) Let be an open subset of and u L (). Suppose that u is continuous in an open neighborhood U of x 0 = (x,, x n ) and dist(x 0, U) > ɛ > 0. Then the function f(x) = Bɛ c (x) u(y)dy is differentiable at x 0 and f x i (x 0 ) = B ɛ(x 0 ) u(y) x i y i x y m(dy), where m(dy) is the n dimensional surface Lebesgue measure... Proof of Theorems A. We first prepare some technical lemmas concerning derivatives of s,ɛ. Lemma.3. Let be an open set in and 0 < s <. Suppose that u L (, (+ x ) n+s ) C,0 (). Then for any x, δ < d x = dist(x, ), and 0 < ɛ < δ, we have (.) s A(n, s) x,ɛ i n j= = ( u x j (y) u x j (x))(y j x j )(x i y i ) x y n+s+ dy [u(y) + n j= u x (n ) j (x)(x j y j )](x i y i ) x y n+s+ dy u x i (x) (u(y) )(x i y i ) dy + x y n+s x y n+s+ m(dy) B c δ (x) (n + s) where := B δ (x) B c ɛ(x). B δ (x) B c δ (x) (u(y) )(x i y i ) y x n+s+ dy, Proof. The proof follows from that of [9, Proposition 8.3].

6 6 C. MOU AND Y. YI Lemma.4. Let be an open set in and 0 < s <. Suppose that u L (, (+ x ) n+s ) C r+,0 () for some positive integer r. For any x, ɛ > 0, δ < d x = dist(x, ), and l = (l, l, l n ), k = (k, k,, k n ) N n, if k i = 0 for some i n, then x i = (n + p m ) l u(y) j =m A j j! (y x)j l+j x y n+s+p (x y) k dy l u(y) j = A j j! (y x)j l+j x y n+s+p+ (x y) k+e i dy [ l+j u(y) j =m A j j! (y x)j l+j +j x y n+s+p+ (y x) j ] (x y) k+e i dy, j! where A j = j!j! j n! for each j = (j, j,, j n ) N and m, p N are such that l + m = r and m + k = p. Proof. Since k i = 0, we have by Lemma. that I : = l u(y) j =m A j j! (y x)j l+j x i x y n+s+p (x y) k dy = I + I + I 3, where I := m! j =m A j l+j+e i (y x) j (x y) k x y n+s+p dy, I := l u(y) j =m A j j! (n + s + p) l+j x y n+s+p+ (x y) k+e i dy, I 3 := B ɛ(x) B δ (x) l u(y) j =m A j j! (y x)j l+j x y n+s+p+ (x y) k+e i m(dy) l u(y) j =m A j j! (y x)j l+j x y n+s+p+ (x y) k+e i m(dy). Using the identity m + k = p, integration by parts yields δ I = (n + s + p) r nsp dr r n+pm (x y) k+e i = ɛ B (x) [ j =m l A j u(x + r(y x)) j! (y x)j l+j r j ] m(dy) δ r n+pm{ ɛ B (x) } (x y) k+e i m(dy) = I 3 (n + p m ) j =m [ l u(x + r(y x)) dr (n+s+p) δ ɛ B (x) r ms (x y) k+e i A j j! (y x)j l+j r j ]

7 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 7 [ j =m l A j u(x + r(y x)) j! (y x)j l+j r j ] m(dy)dr δ r ms (x y) k+e i ɛ B (x) [ j =m A l+j u(x + r(y x))(y x) j j j! (y +j x)j l+j +j r j ] m(dy)dr l u(y) j =m A j j! = I 3 (n + p m ) (y x)j l+j x y n+s+p+ (x y) k+e i dy [ l+j u(y) j =m A j j! (y x)j l+j +j x y n+s+p+ (y x) j ] (x y) k+e i dy. Therefore, where I () 3 = (n + p m ) I () 3 = n + p I (3) 3 = Observe that I + I () 3 = t N,t m n + p = t N,t m C t m m! j = [ j =m,j i =t t N,t m C t+ I + I 3 = I () 3 + I() 3 + I(3) 3, A j l+j l u(y) j = A j j! (y x)j l+j x y n+s+p+ (x y) k+e i dy, (y x) j (x y) k+e i x y n+s+p+ dy, l+j u(y) j =m A j j! (y x)j l+j +j x y n+s+p+ (y x) j ] (x y) k+e i dy. C t+ j =m,j i =t Rδɛ(x) (y x) j (x y) k x y n+s+p+ n + p t N,t m C t+ l+j+e i j =,j i =t+ Rδɛ(x) (y x) j (x y) k x y n+s+p+ (x y) j dy l+j (y x) j (x y) k+e i x y n+s+p+ dy (t + ) l+j+e i (x y) j dy j =,j i =t+ where j = (j, j,, j n ). An application of Lemma. yields I + I () 3 l+j (y x) j (x y) k+e i x y n+s+p+ dy,

8 8 C. MOU AND Y. YI = t N,t m C t+ j =m,j i =t (y x) j (x y) k+e i x y n+s+p+ dy = 0. ] [n + p ( k + n + m) l+j+e i Thus, I = I () 3 + I(3) 3 and the lemma is proved. Lemma.5. Let be an open set in and 0 < s <. Suppose that u L (, (+ x ) n+s ) C r+,0 () for some positive integer r. For any x, ɛ > 0, δ < d x = dist(x, ), and l = (l, l, l n ), k = (k, k,, k n ) N n, if k i 0 for some i n, then x i = k i l u(y) j =m A j j! (y x)j l+j x y n+s+p (x y) k dy l u(y) j = A j j! (y x)j l+j x y n+s+p (x y) ke i dy (n + p m ) l u(y) j = A j j! (y x)j l+j x y n+s+p+ (x y) k+e i dy [ l+j u(y) j =m A j j! (y x)j l+j +j x y n+s+p+ (y x) j ] (x y) k+e i dy, where m, p N are such that l + m = r and m + k = p. Proof. Since k i 0, we have by Lemma. that I = I + Ī + I + I 3, where I and I i, i =,, 3, are as in the proof of Lemma.4 and l u(y) j =m A j j! Ī = k (y x)j l+j i x y n+s+p (x y) ke i dy. By the proof of Lemma.4, we have I + I 3 = I () 3 + I() 3 + I(3) 3. where I (i) 3, i =,, 3, are defined in the proof of Lemma.4. Write Ī = Ī () + Ī (), where Note that Ī () = ki Ī () = k i I + Ī () + I () 3 l u(y) j = A j j! (y x)j l+j x y n+s+p (x y) ke i dy, j = A j l+j (y x) j (x y) ke i x y n+s+p dy.

9 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 9 = m! j =m + k i n + p A j l+j+e i (y x) j (x y) k x y n+s+p+ j = j = A j l+j A j l+j First let k i be an even number. Then I + Ī () + I () 3 = Cm t m! t N,t m + k i n + p = t N,t m j =m,j i =t t N,t m C t+ l+j+e i j =,j i =t+ (y x) j (x y) ke i x y n+s+p+ t N,t m C t+ C t+ j =m,j i =t Rδɛ(x) (y x) j (x y) k x y n+s+p+ + t N,t m C t+ j =,j i =t+ j =m,j i =t It follows from Lemma. that = I + Ī () + I () 3 C t+ t N,t m (x y) j dy (y x) j (x y) ke i x y n+s+p+ (y x) j (x y) k+e i x y n+s+p+ dy. Rδɛ(x) (y x) j (x y) k x y n+s+p+ l+j (x y) j dy (x y) j dy (x y) j dy l+j (y x) j (x y) k+e i x y n+s+p+ dy (t + k i + ) l+j+e i (x y) j dy (n + p) l+j+e i j =m,j i =t (y x) j (x y) k+e i x y n+s+p+ dy = 0. Now let k i be an odd number. Then I + Ī () + I () 3 = t N,t m C t+ m m! j =m,j i =t+ (y x) j (x y) k+e i x y n+s+p+ dy. ] [(n + p) ( k + n + m) l+j+e i l+j+e i (y x) j (x y) k x y n+s+p+ (x y) j dy

10 0 C. MOU AND Y. YI = + k i n + p k i (n + p) t N,t m + k i n + p t N,t C t j =,j i =t (y x) j (x y) ke i x y n+s+p+ t N,t j =,j i =0 j =,j i =0 C t+ m m! C t j =,j i =t A j l+j A j l+j j =m,j i =t+ (y x) j (x y) k x y n+s+p+ t N,t m C t+ (y x) j (x y) ke i x y n+s+p+ t N,t m C t+ l+j (x y) j dy l+j (y x) j (x y) k+e i x y n+s+p+ dy (y x) j (x y) ke i x y n+s+p+ l+j+e i (x y) j dy j =,j i =t+ It again follows from Lemma. that = I + Ī () + I () 3 t N,t m + t N,t m j =,j i =0 (x y) j dy j =,j i =t+ (y x) j (x y) k+e i x y n+s+p+ dy C t+ (y x) j (x y) k+e i x y n+s+p+ dy l+j (x y) j dy l+j (y x) j (x y) k+e i x y n+s+p+ dy. ] A j [(m + n + k (n + p)) l+j j =m,j i =t+ (y x) j (x y) k x y n+s+p+ C t+ j =m,j i =t+ (y x) j (x y) k+e i x y n+s+p+ dy (t + + k i ) l+j+e i (x y) j dy (n + p) l+j+e i

11 = INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN t N,t m C t+ j =m,j i =t+ (y x) j (x y) k+e i x y n+s+p+ dy = 0. ] [(n + p) ( k + n + m) l+j+e i Thus, for any k i 0, I = Ī () + I () 3 + I(3) 3 and the lemma is proved. Lemma.6. Let be an open set in and 0 < s <. Suppose that u L (, (+ x ) n+s ) C r,0 () for some positive integer r. Then for any x, ɛ > 0, ˆr = (r, r,, r n ) N n with ˆr = r, δ < d x = dist(x, ), all ɛ-dependent terms of ˆr s,ɛ have the form (.) I ɛ,δ l,k,m,p (x) = l u(y) j =m A j j! (y x)j l+j x y n+s+p (x y) k dy, where l = (l, l, l n ), k = (k, k,, k n ) N n, m, p N are such that l + m = r and m + k = p. Proof. We will prove the lemma by induction. In the case of r =, we observe that the only ɛ-dependent terms on the right hand side of (.) are its first two terms. They clearly have the form (.) with the first term corresponding to l =, m = 0, k = and p =, and the second term corresponding to l = 0, m =, k = and p =. Now suppose that (.) is satisfied when r = q, where q is a fixed positive integer. We want to show that it is also satisfied when r = q +, i.e., for any l, k N with l + m = q and m + k = p, all ɛ-dependent terms of have the form (.3) I = x i l u(y) j =m A j j! (y x)j l+j x y n+s+p (x y) k dy l u(y) j =m A j j! (y x)j l +j (x y) k dy, x y n+s+p where l + m = q + and m + k = p. If k i = 0, then, by Lemma.4, we have I = I () 3 + I(3) 3, where I() 3, I(3) 3 are as in the proof of Lemma.4 which clearly have the form (.3) with l + m = l + m + = q + and m + k = m + k + = (p + ). If k i 0, then, by Lemma.5, we have I = Ī () + I () 3 + I(3) 3, where Ī () is as in the proof of Lemma.5 which is clearly of the form (.3) with l + m = l + m + = q + and m + k = m + k = p. Proof of Theorem A. For any ɛ > 0, x, ˆr = (r, r,, r n ) N n with ˆr = r, and δ < d x = dist(x, ), we have by Lemma.6 that all ɛ-dependent terms of ˆr s,ɛ have the form (.). In the case α > s, we note that any integral of the form (.) is bounded above in absolute value by a constant times δ ɛ ραs dρ which is convergent as ɛ 0. It follows that

12 C. MOU AND Y. YI ˆr s,ɛ u converges uniformly on any compact subset of as ɛ 0. Thus, ˆr s u C(), i.e., s u C r,0 (). In the case +α > s α, we again consider an integral I ɛ,δ l,k,m,p (x) of the form (.) for some l, k N n, m, p N satisfying l + m = r and m + k = p. Since m + + k = p + is an odd number, we have, for any j N n with j = m +, that (y x) j (x y) k x y n+s+p dy = 0. Hence I ɛ,δ l,k,m,p (x) can be re-written as l u(y) j = A j j! (y x)j l+j x y n+s+p (x y) k dy which is bounded above in absolute value by a constant times δ ɛ ραs dρ that is convergent as ɛ 0. It follows again that ˆr s,ɛ u converges uniformly on any compact subset of as ɛ 0. Thus, ˆr s C(), i.e., s u C r,0 ()..3. Proof of Theorem B. Let u L (, (+ x ) n+s ) C r,α () for some positive integer r and some real numbers 0 < s <, 0 < α. For each ɛ > 0 sufficiently small, x, and any ˆr N n with ˆr r, we write ˆr s,ɛ = I ɛ (x) + I (x), where I ɛ (x) denotes the ɛ-dependent term of ˆr s,ɛ and I (x) denotes the remaining term. Lemma.7. Let u, s, r, α be as in the above. If either α > s or + α > s α, then ˆr s = I 0 (x) + I (x), if ˆr r, where I 0 (x) = lim ɛ 0 I ɛ (x) which consists of terms of the form (.4) I δ l,k,m,p (x) = B δ (x) l u(y) j =m A j j! (y x)j l+j x y n+s+p (x y) k dy, for any δ < d x = dist(x, ) and some l = (l, l, l n ), k = (k, k,, k n ) N n, m, p N with l + m = ˆr and m + k = p. Proof. It follows immediately from Lemma.6 and the proof of Theorem A. Lemma.8. Let u, s, r, α be as in the above. Then the following holds. a) If α > s and ˆr = r, then there exists a constant C > 0 such that I (x) I (y) C[u] r,α. x y αs, x, y, x y. b) If + α > s α and ˆr = r, then there exists a constant C > 0 such that I (x) I (y) C[u] r,α; x y +αs, x, y, x y.

13 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 3 Proof. The function I can be derived simply by taking higher order derivatives of the right hand side of (.) and identifying all ɛ-independent terms of the derivatives. As these terms involves only regular integrals, the lemma follows from straightforward estimates. Proof of Theorem B. Let x, y and take δ < d x,y = min{d x, d y }. For given l = (l, l, l n ), k = (k, k,, k n ) N n, m, p N, consider where Il,k,m,p δ is as in (.4). It is clear that where J := J := J = I δ l,k,m,p (x) Iδ l,k,m,p (y), B η(0) R δη (0) J = J + J, [ l u(x + z) j =m z n+s+p A j j! zj l+j l u(y + z) j =m A j j! zj l+j u(y) ] z n+s+p z k dz, [ l u(x + z) j =m A j j! zj l+j z n+s+p l u(y + z) j =m A j j! zj l+j u(y) ] z n+s+p z k dz, and η = x y < δ. (i) In this case, we let l +m = r and m+ k = p in J, J. On one hand, since l +m = r and u C r,α (), there exists a constant C > 0 such that j =m l u(x + z) l A j ( u(y + z) l+j l+j u(y) ) z j C [u] r,α; z m+α, z B η (0). j! Using the fact m + k = p, it follows that C [u] r,α; z p+α J dz B η(0) z n+s+p C [u] r,α; η αs = C [u] r,α; x y αs for some constant C > 0. On the other hand, we also have j =m l u(x + z) l A j ( u(y + z) l+j l+j u(y) ) z j j! m C 3 [u] r,α; [ x y m+αi z i + i=0 m z m+αi x y i ], z Bη(0), c where C 3 > 0 is a constant. It follows that C 3 [u] r,α; ( m i=0 J x y m+αi z i + m i= z m+αi x y i ) dz B c η(0) i= z n+s+m

14 4 C. MOU AND Y. YI m C[u] r,α; ( η ims x y m+αi + i=0 for some constant C 4 > 0. Hence (.5) J J + J (C + C 4 )[u] r,α. x y αs. m η αsi x y i ) C 4 [u] r,α; x y αs Let I 0 be as in Lemma.7. Then Lemma.7 together with (.5) imply that i= I 0 (x) I 0 (y) C 5 [u] r,α. x y αs for some constant C 5 > 0. With this estimate, the proof is now complete by Lemma.7 and Lemma.8 a). (ii) In this case, we let l + m = r and m + k = p in J, J. Since m + + k = p + is an odd number, we have, for any j N n with j = m +, any w, and any ρ < d w that z j+k dy = 0. z n+s+p It follows that J = B η(0) B ρ(w) [ l u(x + z) j = z n+s+p A j j! zj l+j l u(y + z) j = A j j! zj l+j u(y) ] z n+s+p z k dz, J = R δη (0) [ l u(x + z) j = z n+s+p A j j! zj l+j l u(y + z) j = A j j! zj l+j u(y) ] z n+s+p z k dz. The rest of the proof is similar to that of (i). We only note that using facts l + m = r and u C r,α (), the estimate of J follows from the inequality j = l u(x + z) l A j ( u(y + z) l+j l+j u(y) ) z j C 5 [u] r,α; z +α, j! z B η (0), where C 5 > 0 is a constant, while, using facts m + k = p and s < + α, the estimate of J follows from the inequality j = l u(x + z) l A j ( u(y + z) l+j l+j u(y) ) z j j! C 6 [u] r,α; ( x y +αi z i + z +αi x y i ), z Bη(0), c i=0 where C 6 > 0 is a constant. i=

15 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 5 3. Schauder estimates for the regional fractional Laplacian In this section, we will show the Schauder estimates for the regional fractional Laplacian using those for the fractional Laplacian. Recall that the fractional Laplacian ( ) s is welldefined in S, the Schwartz space of rapidly decreasing C functions in, and we can then extend its definition to the space L ( dx, ) by (+ x ) n+s (3.) < ( ) s u, ϕ >= u(y)( ) s ϕ(y)dy, ϕ S, for any u L (, dx (+ x ) n+s ). 3.. Schauder estimates for the fractional Laplacian. In the following Lemma 3. and 3., the definition of ( ) s is understood in the sense of (3.). We refer the reader to [] for a more general definition of the fractional Laplacian. Lemma 3.. Let 0 < α and 0 < s <. If, for some w C α ( ), u L ( ) solves the equation ( ) s u = w in, then for any δ > 0 sufficiently small there exists a constant C > 0 depending only on n, s, δ and α such that u C α+s ( δ ) C( u L ( ) + w C α ( ) ). Proof. The proof follows from that of [, Proposition.8]. Lemma 3.. Let 0 < s <. Suppose that, for some w L (), u L ( ) solves the equation ( ) s u = w in. Then, for any sufficiently small δ > 0, there exists a constant C > 0 depending only on n, s and δ such that the following holds: (i) If s, then (ii) If s =, then u C s ( δ ) C( u L ( ) + w L ()). u Λ ( δ ) C( u L ( ) + w L ()). Proof. We first use the argument in the proof of [, Proposition.8]. By covering and rescaling arguments, we only need to consider the case δ = B (0) and = B (0). Let (0). η Cc (R) be such that range(η) [0, ], supp(η) B (0), and η(x) = for any x B 3 4 Denote u 0 (x) := A(n, s) η(y)w(y) x y ns dy = ( ) s ηw(x). Then ( ) s u 0 = w = ( ) s u in B 3 (0). It follows that u u 0 C ( B (0)) and 4 u u 0 C ( B (0)) C u u 0 L ( ) C( u L ( ) + w L (B (0))), where C > 0 is a constant depending only on n. We note that C ( B (0)) = C, ( B (0)) C,0 ( B (0)). The lemma now follows from [, Proposition 5.].

16 6 C. MOU AND Y. YI 3.. Proof of Theorems C and D. Let 0 < s <, w L (), and u L () solves the equation s u = w in. Also let ū L ( ) be such that ū u in and ū 0 outside of. Then for any ϕ S, ( s ),ɛ ū(x) + A(n, s)ū(x) x y n+s dy ϕ(x)dx = A(n, s) ( = A(n, s) ( = A(n, s) \B ɛ(x) \B ɛ(x) \B ɛ(x) = A(n, s) ū(x) c \B ɛ(x) ū(x) ū(y) dyϕ(x)dx x y n+s \B ɛ(x) ū(x)ϕ(x) dydx x y n+s ū(x)ϕ(x) dydx x y n+s In particular, for any ϕ Cc (), we have s,ɛ ϕ(x)dx + A(n, s) = ( s,ɛ ū(x) + A(n, s)ū(x) = A(n, s) ū(x) \B ɛ(x) ϕ(x) ϕ(y) dydx. x y n+s supp(ϕ) c \B ɛ(x) ϕ(x) ϕ(y) dydx. x y n+s \B ɛ(y) \B ɛ(x) ϕ(x) c \B ɛ(x) ) x y n+s dy ϕ(x)dx Letting ɛ 0 in the above, we easily obtain that, for any x, (3.) ( ) s ū(x) = w(x) + A(n, s) dy. c x y n+s ū(y)ϕ(x) ) x y n+s dxdy ū(x)ϕ(y) ) x y n+s dydx dydx x y n+s Proof of Theorems C, D. Let δ > 0 be sufficiently small. We have by Lemma 3. and (3.) that there exists a constant C depending on n, s, and δ such that u C s ( δ ) C( ū L ( ) + w + A(n, s)u c y n+s dy L ( δ )) when s. Similarly, C( u L () + w L ()) u Λ ( δ ) C( u L () + w L ()). when s =. This proves Theorem D. Using bootstrap arguments, Lemma 3. and (3.) for both cases of α, α + s being or not being natural numbers, we have, similarly to the above, that u C α+s ( δ ) C( u L () + w C α ( ) ), where C > 0 is a constant depending on n, s, δ and α. This proves Theorem C. Acknowledgement We would like to thank the referee for pointing out references [, 5] to us and for valuable comments which lead to a significant improvement of the paper.

17 INTERIOR REGULARITY FOR REGIONAL FRACTIONAL LAPLACIAN 7 References [] K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields 7 (003), [] L. Caffarelli and P. Stinga, Fractional elliptic equations: Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire (04), to appear. [3] Z. Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process Appl. 08 (003), 7 6. [4] E. Di Nezza, G. Palatucci and E. Valdinoci, Hithchhiker s guide to the fractional Sobolev spaces, Bull. Sci. Math. 36 (0), [5] G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, arxiv: (04). [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 00. [7] Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys. 66 (006), [8] Q. Y. Guan, Boundary Harnack inequality for regional fractional Laplcian, arxiv: (007). [9] Q. Y. Guan and Z. M. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Related Fields 34 (006), [0] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 0 (04), [] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (007), 67. C. Mou: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3033, USA address: cmou3@math.gatech.edu Y. Yi: Department of Mathematical & Statistical Sci, University of Alberta, Edmonton Alberta, Canada T6G G, School of Mathematics, Jilin University, Changchun, 300, PRC, and School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3033, USA address: yingfei@ualberta.ca,yi@math.gatech.edu