Singularly Perturbed Time-Periodic Eigenvalue Problems With Spectral Fractional Laplace Operators

Suppose that (H1) and (H2) are satisfied. Let ( − L B ) s | X 1 : D [ ( − L B ) s ] ∩ X 1 → X be given by ( − L B ) s | X 1 u = ( − L B ) s u , u ∈ D [ ( − L B ) s ] ∩ X 1 . Then (i)

( − L B ) s | X 1 = ( − L B | X 1 ) s , where

( − L B | X 1 ) s u = 1 Γ ( − s ) ∫ 0 ∞ e L B | X 1 t u − u t 1 + s d t , u ∈ D [ ( − L B | X 1 ) s ] = D [ ( − L B ) s ] ∩ X 1 .

We denote − L B by A and first verify that D ( ( A | X 1 ) s ) = D ( A s ) ∩ X 1 . Let ϵ > 0 be fixed, then Theorem 5.1.7 of Carracedo and Alix (2001) and Theorem 1.4.2 of Henry (1981) show that D ( A s ) = D ( ( ϵ I + A ) s ) = R ( ( ϵ I + A ) − s ) , where

( ϵ I + A ) − s = sin ( π s ) π ∫ 0 ∞ μ − s ( ( μ + ϵ ) I + A ) − 1 d μ .

( ϵ I + A ) − s = sin ( π s ) π { ∫ 0 ∞ τ − s ( ( τ + ϵ ) I + A ) − 1 P d τ + ∫ 0 ∞ τ − s ( ( τ + ϵ ) I + A ) − 1 ( I − P ) d τ } = sin ( π s ) π { ∫ 0 ∞ τ − s ( ( τ + ϵ ) I + A | X 0 ) − 1 P d τ + ∫ 0 ∞ τ − s ( ( τ + ϵ ) I + A | X 1 ) − 1 ( I − P ) d τ } = sin ( π s ) π { ∫ 0 ∞ τ − s ( τ + ϵ + μ b ) − 1 P d τ } + ( ϵ I + A | X 1 ) − s ( I − P ) = 1 ( ϵ + μ b ) s P + ( ϵ I + A | X 1 ) − s ( I − P ) .

D ( A s ) = X 0 ⊕ D ( ( A | X 1 ) s ) .

Now given any u ∈ D ( A s ) , let P be given by (2.3). Then u = τ φ 0 + ( I − P ) u , where φ 0 ∈ X 0 = ker ( μ d I − A ) and τ ∈ R . Since e − A t ( I − P ) = e − A | X 1 t , it follows that

A s u = 1 Γ ( − s ) { τ ∫ 0 ∞ e − A t φ 0 − φ 0 t 1 + s d t + ∫ 0 ∞ ( e − A t − I ) ( I − P ) u t 1 + s d t } = 1 Γ ( − s ) { τ φ 0 ∫ 0 ∞ e − μ b t − 1 t 1 + s d t + ∫ 0 ∞ ( e − A | X 1 t − I ) ( I − P ) u t 1 + s d t } = τ μ b s φ 0 + ( A | X 1 ) s ( I − P ) u = μ b s P u + ( A | X 1 ) s ( I − P ) u .

A s u = A s | X 1 u = ( A | X 1 ) s ( I − P ) u , u ∈ D ( A s ) ∩ X 1 .

We now turn to (ii). By virtue of Theorem 5.3.1 of Carracedo and Alix (2001), σ ( A s ) = { μ s ∣ μ ∈ σ ( A ) } . Likewise, σ ( ( A | X 1 ) s ) = { μ s ∣ μ ∈ σ ( A | X 1 ) } . By (H1), we see that

μ b s ∈ σ ( A s ) , σ ( A s ) ∖ { μ b s } = σ ( ( A | X 1 ) s ) ⊂ { μ ∈ C ∣ Re μ ≥ κ s } .

( ( A | X 1 ) s ) − 1 := ( A | X 1 ) − s = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e − A | X 1 t d t .

Next let f ∈ X 1 , then the inevitability of μ b s I − ( A | X 1 ) s shows that μ b s I − ( A | X 1 ) s u f = f for a unique u f ∈ X 1 . Hence, X 1 ⊆ R ( μ b s I − A s ) . Moreover, Proposition 5.3.2 of Carracedo and Alix (2001) implies that A s has compact resolvent. Hence, μ b s I − A s is a Fredholm operator of index zero. Consequently,

codim X 1 = dim X 0 = codim R ( μ b s I − A s ) .

Corollary 2.2

Suppose that (H1) and (H2) are fulfilled and μ b = 0 , where μ b is the principal eigenvalue of − L B . Then

( − d L B ) s u = f , f ∈ X

(2.6)

has a solution if and only f ∈ X 1 := R ( L B ) . If f ∈ X 1 , then each solution of (2.6) can be expressed as τ φ 0 + u f , where τ ∈ R , φ 0 ∈ ker ( L B ) with ‖ φ 0 ‖ X = 1 , u f is given by the following equation:

u f = d − s ( − L B | X 1 ) − s f ,

where

( − L B | X 1 ) − s f = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e L B | X 1 t f d t .

(2.7)

In particular, assume that B := ∂ n + b 0 ( x ) I on ∂ Ω , where ∂ n is defined in (2.1), either b 0 ≡ 0 or b 0 ≢ 0 on ∂ Ω . Let κ > 0 be the constant given in (H1). Then the following estimates hold.

Case 1. X = L p ( Ω ) ( 1 < p < ∞ ) . (i)

‖ u f ‖ W 2 α , p ( Ω ) ≤ C d − s ‖ f ‖ L p ( Ω ) for any 0 < α < s . Moreover, if s > p + 1 / 2 p , then u f ∈ W B 2 α , p ( Ω ) for any p + 1 / 2 p < α < s , where C > 0 is a constant depending on s , p , α , κ , L , b 0 , and Ω .

(ii)

If β ∈ ( 0 , p + 1 2 p ) ∖ { 1 2 p , 1 2 } , and s + β < 1 , f ∈ W 2 β , p ( Ω ) , then u f ∈ W 2 s + 2 α , p ( Ω ) for any 0 < α < β , and ‖ u f ‖ W 2 s + 2 α , p ( Ω ) ≤ C d − s ‖ f ‖ W 2 β , p ( Ω ) . Moreover, if s + β > p + 1 / 2 p , then u f ∈ W B 2 s + 2 α , p ( Ω ) for any s + α > p + 1 / 2 p , where C > 0 is a constant depending on s , p , α , β , κ , L , b 0 , and Ω .

(iii)

If p + 1 / 2 p < β , s + β < 1 , and f ∈ W B 2 β , p ( Ω ) , then u f ∈ W B 2 s + 2 α , p ( Ω ) for any p + 1 2 p < α < β , and ‖ u f ‖ W 2 s + 2 α , p ( Ω ) ≤ C d − s ‖ f ‖ W 2 β , p ( Ω ) , where C > 0 is a constant depending on s , p , α , β , κ , L , b 0 , and Ω .

(iv)

If f ∈ L ∞ ( Ω ) , then u f ∈ W 2 α , p ( Ω ) for any 0 < α < s and p > 1 , and ‖ u f ‖ W 2 α , p ( Ω ) ≤ C d − s | f | L ∞ ( Ω ) for some C > 0 depending on s , p , α , κ , L , b 0 , and Ω . Moreover, if s > 1 2 , then u f ∈ W B 2 α , p ( Ω ) for any 0 < α < s and p > 1.

Case 2. X = C ( Ω ¯ ) . (v)

If f ∈ C ( Ω ¯ ) , then u f ∈ C 2 s ( Ω ¯ ) as long as s ≠ 1 2 , and ‖ u f ‖ C 2 s ( Ω ¯ ) ≤ C d − s ‖ f ‖ C ( Ω ¯ ) . Moreover, u f ∈ C B 2 s ( Ω ¯ ) if s > 1 2 . In addition, u f ∈ C B 1 ( Ω ¯ ) and ‖ u f ‖ C 1 ( Ω ¯ ) ≤ C d − 1 2 ‖ f ‖ C ( Ω ¯ ) when s = 1 2 , where each C is a positive constant depending on s , κ , L , b 0 , and Ω .

(vi)

If 0 < α < 1 2 , s + α < 1 , and f ∈ C 2 α ( Ω ¯ ) , then u f ∈ C 2 s + 2 α ( Ω ¯ ) provided that s + α ≠ 1 2 , and ‖ u f ‖ C 2 s + 2 α ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 2 α ( Ω ¯ ) . Moreover, u f ∈ C B 2 s + 2 α ( Ω ¯ ) if 1 2 < s + α < 1. In case that s + α = 1 2 , then u f ∈ C B 1 ( Ω ¯ ) and ‖ u f ‖ C 1 ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 2 α ( Ω ¯ ) , where each C is a positive constant depending on s , α , κ , L , b 0 , and Ω .

(vii)

If 1 2 < α , s + α < 1 , and f ∈ C B 2 α ( Ω ¯ ) , then u f ∈ C B 2 s + 2 α ( Ω ¯ ) and ‖ u f ‖ C 2 s + 2 α ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 2 α ( Ω ¯ ) , where the constant C > 0 depends on s , α , κ , L , b 0 , and Ω . Furthermore, suppose that f ∈ C B 1 ( Ω ¯ ) and s < 1 2 , then u f ∈ C B 2 s + 1 ( Ω ¯ ) , and ‖ u f ‖ C 2 s + 1 ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 1 ( Ω ¯ ) , where the constant C > 0 depends on s , κ , L , b 0 , and Ω .

Proof.

We first establish the solvability of (2.6). Let again A = − L B throughout the proof. If (2.6) has a solution, then f ∈ R ( A s ) . Thus, Proposition 2.1 shows that f ∈ X 1 . Conversely, if f ∈ X 1 , in view of Proposition 2.1, the inevitability of ( A | X 1 ) s implies that A s u = d − s f has a unique solution u f in X 1 , which is given by (2.7). That is, ( d A ) s u f = f . Furthermore, given any solution u of (2.6), note that u = P u + ( I − P ) u , where P is defined in (2.3) and P u = τ φ 0 for some τ ∈ R . It follows from Proposition 2.1 that P u ∈ ker ( A s ) and

( d A ) s u = ( d A | X 1 ) s ( I − P ) u = f .

This implies that ( I − P ) u = u f . Clearly, ( d A ) s ( τ φ 0 + u f ) = f for any τ ∈ R . Hence solutions of (2.6) differ only by a constant multiple of φ 0 .

We now employ interpolation theory built on the K -method and the trace method to prove (i)–(vi), part of the theory characterizes various Sobolev spaces and spaces of Hölder continuous and bounded functions on Ω ¯ by an interpolation couple of X and D ( − A ) , where X = L p ( Ω ) with 1 < p < ∞ or X = C ( Ω ¯ ) (a detailed presentation of this theory can be found in Chapters 1 and 2 of Lunardi, 1995). In fact, given any 0 < θ < 1 , set

D − A ( θ , p ) = { u ∈ X ∣ w ( t ) = t 1 − θ − 1 p ‖ A e − A t u ‖ X ∈ L p ( 0 , 1 ) , [ u ] D − A ( θ , p ) = | w | L p ( 0 , 1 ) } ,

where 1 < p < ∞ . Also let

D − A ( θ , ∞ ) = { u ∈ X ∣ w ( t ) = t 1 − θ ‖ A e − A t u ‖ X ∈ L ∞ ( 0 , 1 ) , [ u ] D − A ( θ , ∞ ) = | w | ∞ } .

Set 1 < χ ≤ ∞ ; that is, either χ = p with 1 < p < ∞ or χ = ∞ , D − A ( θ , χ ) is an interpolation couple (or an intermediate space between D ( − A ) and X ). Now if X = L p ( Ω ) with 1 < p < ∞ . Then by Theorem 4.16 of Daners and Medina (1992), for any α ∈ ( 0 , 1 ) ∖ { 1 2 p , 1 2 , p + 1 2 p } we have

D − A ( α , p ) ≐ { W 2 α , p ( Ω ) , α ∈ ( 0 , p + 1 2 p ) ∖ { 1 2 p , 1 2 } , W B 2 α , p ( Ω ) , p + 1 2 p < α < 1 ,

where ≐ means that two spaces are the same up to equivalent norms. Hence, there exist two positive constants C i ( i = 1 , 2 ) such that

‖ ⋅ ‖ W 2 α , p ( Ω ) ≤ C 1 ‖ ⋅ ‖ D − A ( α , p ) and ‖ ⋅ ‖ D − A ( α , p ) ≤ C 2 ‖ ⋅ ‖ W 2 α , p ( Ω ) ,

where C i ( i = 1 , 2 ) depend on L , b 0 , α , and p . Therefore, to show (i)–(vi), we only need to make estimates for ‖ u f ‖ D − A ( α , p ) . To this end, let again { e − A t } t ≥ 0 be the analytic semigroup generated by − A in X . Since e − A | X 1 t = e − A t ( I − P ) , it follows from (H1) and Proposition 2.3.3 of Lunardi (1995) that

‖ e − A | X 1 t ‖ L ( X , D − A ( α , χ ) ) ≤ c 0 t − α e − ω t , ‖ e − A | X 1 t ‖ L ( D − A ( β , χ ) , D − A ( s + α , χ ) ) ≤ c 1 t β − s − α e − ω t ,

(2.8)

where 0 < α < 1 , 0 < β ≤ s + α < 1 , 0 < ω < κ , c 0 , and c 1 are two positive constants. A careful look at the proofs of Propositions 1.2.6 and 2.2.9 of Lunardi (1995) shows that c 0 depends on κ − ω and Ω only, and c 1 depends on s , β , α , κ − ω , and Ω when χ = ∞ . While if 1 < χ = p < ∞ , then c i ( i = 0 , 1 ) also depend on p . Since u f = ( A | X 1 ) − s d − s f , u f ∈ D ( A | X 1 ) s = D ( A s ) ∩ X 1 . Thus, (4.11) and (4.12) of Daners and Medina (1992) show that u f ∈ D − A ( α , χ ) for any 0 < α < s . Using the trivial fact that W 2 α 1 , p ( Ω ) ⊂ W 2 α 2 , p ( Ω ) for α 1 < α 2 , if s ≤ p + 1 2 p , then u f ∈ W 2 α , p ( Ω ) for any 0 < α < s . Likewise u f ∈ W B 2 α , p ( Ω ) for all 0 < α < s provided that s > p + 1 / 2 p as there exists p + 1 / 2 p < α < s . In either case, by means of (2.7) and (2.8), chose α ∈ ( 0 , s ) ∖ { 1 2 p , 1 2 , p + 1 2 p } , we find that

‖ u f ‖ W 2 α , p ( Ω ) ≤ d − s C 1 Γ ( s ) ∫ 0 ∞ t s − 1 ‖ e − A | X 1 t f ‖ D − A ( α , p ) d t ≤ c 0 C 1 d − s ‖ f ‖ X Γ ( s ) ∫ 0 ∞ t s − α − 1 e − ω t d t = c 0 C 1 Γ ( s − α ) ω s − α Γ ( s ) d − s ‖ f ‖ L p ( Ω ) .

This confirms (i).

Now suppose that

f ∈ { W 2 β , p ( Ω ) ∩ X 1 if β ∈ ( 0 , 1 2 p ) ∪ ( 1 2 p , 1 2 ) ∪ ( 1 2 , p + 1 2 p ) and 0 < s < 1 − β , W B 2 β , p ( Ω ) ∩ X 1 if p + 1 2 p < β and 0 < s < 1 − β .

Then f ∈ D − A ( β , p ) ∩ X 1 . Subsequently, it follows from Remark A.1 and (4.12) of Daners and Medina (1992) that f ∈ D ( A | X 1 ) α for any 0 < α < β . Hence the additivity formula of Carracedo and Alix (2001) implies that u f ∈ D [ ( A | X 1 ) s + α ] ⊂ D − A ( s + α , p ) . In other words, u f ∈ W 2 s + 2 α , p ( Ω ) for any 0 < α < β . Moreover, u f ∈ W B 2 s + 2 α , p ( Ω ) if s + β > p + 1 / 2 p as there exists 0 < α < β such that s + α > p + 1 / 2 p . In addition, for fixed 0 < s < 1 and 0 < β < 1 − s satisfying the assumptions given above, chose any α < β such that s + α ∈ ( β , s + β ) ∖ { 1 2 p , 1 2 , p + 1 2 p } , then similar calculation yields that

‖ u f ‖ W 2 s + 2 α , p ( Ω ) ≤ d − s C 1 Γ ( s ) ∫ 0 ∞ t s − 1 ‖ e − A | X 1 t f ‖ D − A ( s + α , p ) d t ≤ c 1 C 1 d − s ‖ f ‖ D − A ( β , p ) Γ ( s ) ∫ 0 ∞ t s − 1 + β − s − α e − ω t d t ≤ c 1 C 1 C 2 Γ ( β − α ) ω β − α Γ ( s ) d − s ‖ f ‖ W 2 β , p ( Ω ) .

This verifies (ii) and (iii).

To show (iv), as f ∈ L ∞ ( Ω ) ⊂ L p ( Ω ) for any p > 1 , it follows from (i) that u f ∈ W 2 α , p ( Ω ) for any 0 < α < s . If s > 1 2 , given any p > 1 , there exists p ′ > p such that s > p ′ + 1 / 2 p ′ . Hence, the second part of (iv) follows again from (i).

Next given that X = C ( Ω ¯ ) , it follows from Theorems 3.1.30 and 3.1.31 of Lunardi (1995) that

D − A ( θ , ∞ ) ≐ { C 2 θ ( Ω ¯ ) , 0 < θ < 1 2 , C B 1 ( Ω ¯ ) , θ = 1 2 , C B 2 θ ( Ω ¯ ) , 1 2 < θ < 1.

(2.9)

We now proceed to estimate ‖ u f ‖ D − A ( θ , ∞ ) . If f ∈ X 1 then u f = d − s ( A | X 1 ) − s f ∈ D ( A ) s ⊂ D − A ( s , ∞ ) . By virtue of (H1) and Proposition 2.3.3 of Lunardi (1995), it is easy to see that

‖ u f ‖ X ≤ ‖ f ‖ X d s Γ ( s ) ∫ 0 ∞ t s − 1 ‖ e − A | X 1 t ‖ L ( X ) d t ≤ C ‖ f ‖ X d s Γ ( s ) ∫ 0 ∞ t s − 1 e − ω t d t = C ( d ω ) − s ‖ f ‖ X ,

(2.10)

for any 0 < ω < κ , where C > 0 is a constant depending on κ − ω and Ω . Furthermore, it holds that

‖ − A | X 1 e − A | X 1 t ‖ L ( X ) = ‖ A e − A t ( I − P ) ‖ L ( X ) ≤ C t − 1 e − ω t , t > 0.

(2.11)

Hence, for any 0 < τ ≤ 1 , − A | X 1 e − A | X 1 τ is a bounded operator. As u f ∈ X 1 , it follows that

− A e − A τ u f = − A | X 1 e − A | X 1 τ u f = d − s Γ ( s ) ∫ 0 ∞ t s − 1 ( − A | X 1 ) e − A | X 1 ( t + τ ) f d t .

Consequently,

‖ A e − A τ u f ‖ X ≤ C d − s ‖ f ‖ X Γ ( s ) ∫ 0 ∞ t s − 1 ( τ + t ) − 1 d t = C d − s ‖ f ‖ X Γ ( s ) ∫ 0 ∞ t s − 1 τ − 1 ( 1 + t τ ) − 1 d t = C d − s τ s − 1 ‖ f ‖ X Γ ( s ) ∫ 0 ∞ ζ s − 1 ( 1 + ζ ) − 1 d ζ = C π d − s τ s − 1 ‖ f ‖ X sin [ ( 1 − s ) π ] Γ ( s ) .

Therefore,

[ u f ] D − A ( s , ∞ ) = sup 0 < τ ≤ 1 ‖ τ 1 − s A e − A τ u f ‖ X ≤ C d − s ‖ f ‖ X .

This along with (2.10) yield

‖ u f ‖ D − A ( s , ∞ ) ≤ C d − s ‖ f ‖ X ,

for some C > 0 depending on s , κ , and Ω . This confirms (v).

We now turn to (vi) and (vii). Given any f ∈ D − A ( α , ∞ ) ∩ X 1 , it follows from the same reasoning that u f ∈ D − A ( s + θ , ∞ ) for any 0 ≤ θ < α . To complete the proof, we only need to show u f ∈ D − A ( s + α , ∞ ) . To this end, first observe that ‖ A e − A t u ‖ X = ‖ A | X 1 e − A | X 1 t u ‖ X ≤ t α − 1 [ u ] D − A ( α , ∞ ) for 0 < t ≤ 1 if u ∈ D − A ( α , ∞ ) . This along with (2.11) imply that

‖ A e − A t ( I − P ) ‖ L ( D − A ( α , ∞ ) , X ) ≤ C t α − 1 e − ω t , t > 0 ,

(2.12)

for some C > 0 depending on κ − ω , ω , and Ω . Thus, given that 0 < s + α < 1 , similar computation yields

‖ A e − A τ u f ‖ X ≤ C d − s ‖ f ‖ D − A ( α , ∞ ) Γ ( s ) ∫ 0 ∞ t s − 1 ( τ + t ) α − 1 d t = C d − s ‖ f ‖ D − A ( α , ∞ ) Γ ( s ) ∫ 0 ∞ t s − 1 τ α − 1 ( 1 + t τ ) α − 1 d t = C d − s τ s + α − 1 ‖ f ‖ D − A ( α , ∞ ) Γ ( s ) ∫ 0 ∞ ζ s − 1 ( 1 + ζ ) α − 1 d ζ = C Γ ( 1 − s − α ) d − s τ s + α − 1 ‖ f ‖ D − A ( α , ∞ ) Γ ( 1 − α ) .

Consequently,

[ u f ] D − A ( s + α , ∞ ) = sup 0 < τ ≤ 1 ‖ τ 1 − s − α A e − A τ u f ‖ X ≤ C d − s ‖ f ‖ D − A ( α , ∞ ) ,

which together with (2.10) shows that

‖ u f ‖ D − A ( s + α , ∞ ) ≤ C d − s ‖ f ‖ D − A ( α , ∞ )

for some C > 0 depending on s , α , κ , and Ω . In view of (2.9), we readily infer (vi) and (vii). The proof is completed. □

Corollary 2.3

Assume that B := I on ∂ Ω . Suppose that (H1) and (H2) are fulfilled and μ b = 0 , where μ b is the principal eigenvalue of − L B . Then

( − d L B ) s u = f , f ∈ X

(2.13)

has a solution if and only f ∈ X 1 := R ( L B ) . If f ∈ X 1 , then each solution of (2.6) can be expressed as τ φ 0 + u f , where τ ∈ R , φ 0 ∈ ker ( L B ) with ‖ φ 0 ‖ X = 1 , u f is given by the following equation:

u f = d − s ( − L B | X 1 ) − s f ,

where

( − L B | X 1 ) − s f = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e L B | X 1 t f d t .

In particular, the following estimates hold: (i)

‖ u f ‖ W 2 α , p ( Ω ) ≤ C d − s ‖ f ‖ L p ( Ω ) for any 0 ≤ α < s . In addition, if s > 1 2 p , then u f ∈ W 0 2 α , p ( Ω ) for any 1 2 p < α < s , where C > 0 is a constant depending on s , p , α , κ , and Ω .

(ii)

If 0 < β < 1 2 p , s + β < 1 , and f ∈ W 2 β , p ( Ω ) , then u f ∈ W 2 s + 2 α , p ( Ω ) for any 0 < α < β , and ‖ u f ‖ W 2 s + 2 α ( Ω ) ≤ C d − s ‖ f ‖ W 2 β , p ( Ω ) . Moreover, u f ∈ W 0 2 s + 2 α ( Ω ) for any 1 2 p < s + α if s + β > 1 2 p , where C > 0 is a constant depending on s , p , α , β , κ , L , and Ω .

(iii)

If β ∈ ( 1 2 p , 1 ) ∖ { 1 2 , p + 1 2 p } , s + β < 1 , and f ∈ W 0 2 β , p ( Ω ) , then u f ∈ W 0 2 s + 2 α ( Ω ) for any 0 < α < β , and ‖ u f ‖ W 2 s + 2 α ( Ω ) ≤ C d − s ‖ f ‖ W 2 β , p ( Ω ) , where C > 0 is a constant depending on s , p , α , β , κ , L , and Ω .

(iv)

If f ∈ L ∞ ( Ω ) , then u f ∈ C 0 2 α ( Ω ¯ ) and ‖ u f ‖ C 2 α ( Ω ¯ ) ≤ C d − s | f | L ∞ ( Ω ) for any 0 ≤ α < s , where C > 0 is a constant depending on s , α , κ , L , and Ω . If f ∈ C ( Ω ¯ ) , then u f ∈ C 0 2 s ( Ω ¯ ) as long as s ≠ 1 2 , and ‖ u f ‖ C 2 s ( Ω ¯ ) ≤ C d − s ‖ f ‖ C ( Ω ¯ ) . In case that s = 1 2 , then u f ∈ C 0 1 ( Ω ¯ ) , and ‖ u f ‖ C 1 ( Ω ¯ ) ≤ C d − s ‖ f ‖ C ( Ω ¯ ) , where the constants C > 0 depend on s , κ , L , and Ω .

(v)

If f ∈ C 0 2 α ( Ω ¯ ) with 0 ≤ α < 1 , α ≠ 1 2 , and α < 1 − s , then u f ∈ C 0 2 s + 2 α ( Ω ¯ ) given that s + α ≠ 1 2 , and ‖ u f ‖ C 2 s + 2 α ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 2 α ( Ω ¯ ) , whereas u f ∈ C 0 1 ( Ω ¯ ) if s + α = 1 2 , and ‖ u f ‖ C 1 ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 2 α ( Ω ¯ ) , where the constants C > 0 depend on s , α , κ , L , and Ω . In case that f ∈ C 0 1 ( Ω ¯ ) and s < 1 2 , then u f ∈ C 0 2 s + 1 ( Ω ¯ ) and ‖ u f ‖ C 2 s + 1 ( Ω ¯ ) ≤ C d − s ‖ f ‖ C 1 ( Ω ¯ ) , where C > 0 is a constant depending on s , κ , L , and Ω .

Proof.

For the Dirichlet boundary operator, given that X = L p ( Ω ) , as shown in Theorem 4.16 of Daners and Medina (1992), D − A ( α , p ) ≐ W 2 α , p ( Ω ) if 0 < α < 1 2 p . Here A = − L B . While D − A ( α , p ) ≐ W 0 2 α , p ( Ω ) provided that α ∈ ( 1 2 p , 1 ) ∖ { 1 2 , p + 1 2 p } . Hence, (i), (ii), and (iii) are confirmed with the same arguments given in Corollary 2.2. As for (iv), obviously f ∈ L ∞ ( Ω ) ⊂ L p ( Ω ) for any p > 0. Thus, (i) implies u f ∈ D − A ( α , p ) for any α < s and p > 1 ; namely, u f ∈ W 0 2 α , p ( Ω ) . Then the desired conclusion follows from Sobolev embedding theorem. Now assume that f ∈ C ( Ω ¯ ) , to show the rest of (iv), note that − A generates an analytic semigroup in C ( Ω ¯ ) , and Theorem 3.1.29 of Lunardi (1995) shows that

D − A ( θ , ∞ ) ≐ { C 0 2 θ ( Ω ¯ ) , 0 < θ < 1 , θ ≠ 1 2 , C 0 1 ( Ω ¯ ) , θ = 1 2 .

(2.14)

Since (2.10) and (2.11) continue to hold, we readily infer that u f ∈ D − A ( s , ∞ ) , and the rest of (iv) follows from (2.14). Finally, in view of (2.14) again, (v) is an immediate result of the same calculations made in Corollary 2.2 for (v)–(vii). The proof is completed. □

Corollary 2.4

Assume that (H1) and (H2) are satisfied and μ b > 0 , where μ b is the principal eigenvalue of − L B . Then (2.6) has a unique solution u f given by the following equation:

u f = d − s ( − L B ) − s f ,

where

( − L B ) − s f = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e L B t f d t .

In particular, all estimates presented in Corollaries 2.2 and 2.3 remain valid, and all constants C ′ s depend on s , μ b , α , β , L , b 0 , and Ω accordingly.

Proof.

Let A = − L B . Since μ b s > 0 , A s is invertible and its inverse is given above. Also, (2.8) continues to hold for which e − A | X 1 t is replaced by e − A t , and ω satisfies 0 < ω < μ b . Thus, as a result of the same calculations, all estimates given in Corollaries 2.2 and 2.3 are still valid for the case that μ b > 0. The proof is completed. □

Remark 2.5

Corollaries 2.2–2.4 extend some related results established by Caffarelli and Stinga (2016) and Grubb (2016). Note that (H1) and (H2) neither require L B being self-adjoint nor assume a 0 > 0 on Ω ¯ , where a 0 is given in (2.1). Oblique derivative boundary condition is also covered.

2.3. Estimates of Fractional Operators