Boundary roblems for fractional Lalacians and other mu-transmission oerators Gerd Grubb Coenhagen University Geometry and Analysis Seminar June 20, 2014
Introduction Consider P a equal to ( ) a or to A a for some symmetric, strongly ellitic second-order oerator A on R n with real C -coefficients, 0 < a < 1. The Dirichlet-to-Neumann oerator is rincially of this tye, with a = 1 2. Assume for simlicity (P a u, u) 0 for all u C0 (Rn ). For Ω oen R n, the Dirichlet realization P a,dir is defined by a variational construction, as the Friedrichs extension of P a,c 0 (Ω) in L 2 (Ω). Assume Ω smooth bounded. We shall use the notation H s (R n ) = {u S (R n ) F 1 ( ξ s û) L (R n )}, H s (Ω) = {u H s (R n ) su u Ω}, H s (Ω) = r + H s (R n ). Here ξ = (1 + ξ 2 ) 1 2 ; r + restricts to Ω, e + extends by zero on Ω. (The notation with H and H stems from Hörmander s books 63 and 85.) The oerator r + P a mas continuously from H 2 a a (Ω) to H2 (Ω), and P a,dir is the restriction with domain D(P a,dir ) = {u H a 2(Ω) r + P a u L 2 (Ω)}. The oerator is selfadjoint 0 with comact resolvent; for P a = ( ) a the oerator is known to have a ositive lower bound.
Q1. What is the domain? It is relatively easy to show that for a < 1 2, D(P a,dir ) = H 2 2a(Ω). But for a 1 2? Q2. What is the sectrum? Is there a Weyl-tye asymtotic formula for the eigenvalues, λ k C k n/2a? The answer yes is known for P a = ( ) a, but for general A? Q3. Are eigenfunctions smooth in some sense? To my knowledge, very little is known. We shall deal with all three questions, and some more. Consider the roblem r + P a u = f in Ω, su u Ω, called the homogeneous Dirichlet roblem. Unique solvability in L (Ω) is known for P a = ( ) a. But the results on the regularity of the solutions have been sarse. Vishik, Eskin, Shamir 1960 s: u H 1 2 +a ε 2 (Ω) if 1 2 < a < 1. Some analysis of the behavior of solutions at Ω when data are C, Eskin 81, Bennish 93, Chkadua and Duduchava 01. Q4. When f H s (Ω) for some s 0, 1 < <, how regular is u? Q5. Same question in Hölder saces C t, t 0 (continuously differentiable functions when t is integer).
Recent activity: Ros-Oton and Serra (arxiv 2012) showed for ( ) a by otential theoretic and integral oerator methods, when Ω is C 1,1, that f L (Ω) = u d a C α (Ω), for a small α > 0. Here d(x) = dist(x, Ω). Moreover, u C a (Ω) C 2a (Ω). Lifted to at most α 1 when f is more smooth. They stated that they did not know of other regularity results for ( ) a in the literature. P a is a seudodifferential oerator (ψdo), why not ψdo methods? There is a calculus initiated by Boutet de Monvel 1971 for ψdo boundary value roblems. But it does not cover P a. I have been asked about such oerators through the years, but only recently found an answer. It is resented in arxiv 13 as a new systematic theory of seudodifferential boundary roblems covering the oerators P a. A consquence is: f L (Ω) = u d a C a ( ε) (Ω), (1) f C t (Ω) = u d a C a+t ( ε) (Ω), all t > 0; (2) ( ε) is included when a = 1 2 in (1), a + t or 2a + t N in (2). This theory will be the subject of the talk.
1. Pseudodifferential oerators Pseudodifferential oerators (ψdo s) were introduced in the 1960 s as a generalization of singular integral oerators (Calderòn, Zygmund, Seeley, Kohn, Nirenberg, Hörmander, Giraud, Mikhlin,....) They systematize the use of the Fourier transform Fu = û(ξ) = R n e ix ξ u(x) dx: Pu = α m a α(x)d α u = F 1( (x, ξ)û(ξ) ) = OP()u, where (x, ξ) = α m a α(x)ξ α, the symbol. This extends to more general functions (x, ξ) as symbols. In the classical theory, symbols are taken olyhomogeneous: (x, ξ) j N 0 j (x, ξ), where j (x, tξ) = t m j j (x, ξ) for ξ 1, t 1 (here the order m C). The ellitic case is when the rincial symbol 0 is invertible; then Q = OP( 1 0 ) is a good aroximation to an inverse of P. The theory extends to manifolds by use of local coordinates.
An examle of a ψdo is ( ) a = OP( ξ 2a ), of order 2a, but also variable coefficients are allowed. E.g. A a, and much more general oerators. When Ω is a smooth oen subset of R n, there is a need to consider boundary value roblems P + u = f on Ω, Tu = ϕ on Ω; with P + = r + Pe +, the truncation of P to Ω, and T a trace oerator. Boutet de Monvel in 1971 introduced a calculus treating this when P is of integer order and has the transmission roerty: P + mas C (Ω) into C (Ω). Solution oerators for the roblem are tyically of the form ( Q+ + G K ), where Q P 1 and G is an auxiliary oerator called a singular Green oerator, and K is a Poisson oerator (going from Ω to Ω). But there are many interesting ψdo s not having the transmission roerty, e.g. ( ) 1 2 does not have it, although of order 1.
2. The µ-transmission roerty Definition 1. For Re µ > 1, E µ (Ω) consists of the functions u of the form { d(x) µ v(x) for x Ω, with v C (Ω), u(x) = 0 for x Ω; where d(x) is > 0 on Ω, belongs to C (Ω), and is roortional to dist(x, Ω) near Ω. More generally for j N, E µ j is sanned by the distribution derivatives u to order j of functions in E µ. In Hörmander s book 85 Th. 18.2.18, for a classical ψdo P of order m: Theorem 2. r + P mas E µ (Ω) into C (Ω) if and only if the symbol has the µ-transmission roerty for x Ω, with N denoting the interior normal: β x α ξ j (x, N) = e πi(m 2µ j α ) β x α ξ j (x, N), for all indices. It is a, ossibly twisted, arity along the normal to Ω. Simle arity is the case m = 2µ; it holds for ξ 2a with m = 2a, µ = a. Boutet de Monvel s transmission roerty is the case m Z, µ = 0. The oerators are for short said to be of tye µ.
3. Solvability with homogeneous boundary conditions The µ-transmission roerty was actually introduced far earlier by Hörmander in a lecture note from IAS 1965-66, distributed by hotocoying. I received it in 1980, and have only last year studied it in deth. It contains much more, namely a solvability theory in L 2 Sobolev saces for oerators of tye µ, which in addition have a certain factorization roerty of the rincial symbol. Definition 3. P (of order m) has the factorization index µ 0 when, in local coordinates where Ω is relaced by R n + with coordinates (x, x n ), 0 (x, 0, ξ, ξ n ) = (x, ξ, ξ n ) + (x, ξ, ξ n ), with ± homogeneous in ξ of degrees µ 0 res. m µ 0, and ± extending to {Im ξ n 0} analytically in ξ n. Here OP( ± (x, ξ)) on R n reserve suort in R n + res. R n. Examle: For ( ) a on R n we have ξ 2a = ( ξ 2 + ξ 2 n) a = ( ξ iξ n ) a ( ξ + iξ n ) a, so that ± = ( ξ ± iξ n ) a, and the factorization index is a.
The oerators Ξ µ ± = OP(( ξ ± iξ n ) µ ) lay a great role in the theory. Based on the factorization, Vishik and Eskin showed in 64 (extension to L by Shargorodsky 95, 1 < <, 1/ = 1 1/): Theorem 4. When P is ellitic of order m and has the factorization index µ 0, then r + P : H (Ω) s H s Re m (Ω) is a Fredholm oerator for Re µ 0 1/ < s < Re µ 0 + 1/. Note that s runs in a small interval ] Re µ 0 1/, Re µ 0 + 1/[. The roblem is now to find the solution sace for higher s. For this, Hörmander introduced for = 2 a articular sace combining the H and the H definitions: Definition 5. For µ C and s > Re µ 1/, the sace H µ(s) (R n +) is defined by H µ(s) (R n +) = Ξ µ + e + H s Re µ (R n +). Here H µ(s) (R n +) S (R n ), suorted in R n +. Note the jum at x n = 0 in e + H s Re µ (R n +).
Proosition 6. Let s > Re µ 1/. Then + e + : H s Re µ (R n +) H µ(s) (R n +) has the inverse Ξ µ r + Ξ µ + : H µ(s) (R n +) H s Re µ (R n +), and H µ(s) (R n +) is a Banach sace with the norm u µ(s) = r + Ξ µ +u H s Re µ (R n + ). One has that H µ(s) (R n +) H (R s n +), and elements of H µ(s) (R n +) are locally in H s on R n +, but they are not in general H s u to the boundary. The definition generalizes to Ω R n by use of local coordinates. These are Hörmander s µ-saces, very imortant since they turn out to be the correct solution saces. The saces H µ(s) relace the E µ in a Sobolev sace context, in fact one has: Proosition 7. Let Ω be comact, and let s > Re µ 1/. Then E µ (Ω) H µ(s) (Ω) densely, and s Hµ(s) (Ω) = E µ (Ω).
We can now state the basic theorems: Theorem 8. When P is of order m and tye µ, r + P mas H µ(s) (Ω) continuously into H s Re m (Ω) for all s > Re µ 1/. Theorem 9. Let P be ellitic of order m, with factorization index µ 0, and of tye µ 0 (mod 1). Let s > Re µ 0 1/. When u H Re µ0 1/ +ε (Ω), then Moreover, the maing is Fredholm. r + Pu = f H s Re m (Ω) = u H µ0(s) (Ω). r + P : H µ0(s) (Ω) H s Re m (Ω) (4) This answers Q4 in a recise way, with m = 2a, µ 0 = a. The roofs in the old 1965 notes (for = 2) are long and difficult. One of the difficulties is that the Ξ µ ± are not truly ψdo s in n variables, the derivatives of the symbols ( ξ ± iξ n ) µ do not decrease for ξ in the required way.
More recently we have found (G 90) a modified choice of symbol that gives true ψdo s Λ (µ) ± with the same holomorhic extension roerties for Im ξ n 0; they can be used instead of Ξ µ ±, also for 2. This allows a reduction of some of the considerations to cases where the Boutet de Monvel calculus (extended to H s in G 90) can be alied. In fact, when we for Theorem 9 introduce Q = Λ (µ0 m) PΛ ( µ0) +, we get a ψdo of order 0 and tye 0, with factorization index 0; then can be transformed to the equation r + Pu = f, with su u Ω, r + Qv = g, where v = Λ (µ0) + u, g = r + Λ (µ0 m) e + f. Here the Boutet de Monvel calculus alies to Q and rovides good solvability roerties for all s > Re µ 0 1/.
Since s Hµ(s) (Ω) = E µ (Ω), and s Hs Re m (Ω) = C (Ω), one finds as a corollary when s : Corollary 10. Let P be as in Theorem 9 and let u be a function suorted in Ω. If r + Pu C (Ω), then u E µ0 (Ω). Moreover, the maing r + P : E µ0 (Ω) C (Ω) is Fredholm. One can furthermore show that the finite dimensional kernel and cokernel (a comlement of the range) of the maing in Corollary 10 serve as kernel and cokernel also in the maings for finite s in Theorem 9. Note the sharness: The functions in E µ0 have the behavior u(x) = d(x) µ0 v(x) at the boundary with v C (Ω); they are not in C themselves, when µ 0 / N 0! Now, answers to the remaining Q1, Q2, Q3, Q5: Q1. The domain of P a,dir. = H D(P a,dir ) = H a(2a) 2 (Ω) = Λ ( a) + e + H a 2 2a(Ω), if 0 < a < 1 2, 2(Ω) ε>0 H 1 ε 2 (Ω), if a = 1 2, d a H a 2(Ω) + H 2 2a(Ω), if 1 2 < a < 1.
The last line shows the aearance of a factor d a ; the roof uses Poisson oerators from the Boutet de Monvel calculus. (Recall also that D(P a,dir ) H 1 2 +a ε (Ω) when 1 2 < a < 1.) Q2. P 1 a,dir has the structure (Λ( a) + ) + ( Q + + G)(Λ ( a) ) +, where Q is a ψdo and G a singular Green oerator of order 0. This can be used to show that indeed, the eigenvalues λ k of P a,dir satisfy λ k C(P a, Ω) k n/2a, for k. Extends to any bounded oen Ω, by aroximation from smooth cases. Q5. The theory extends, using Johnsen 96, to Besov-Triebel-Lizorkin saces F s,q and B s,q, in articular to the Hölder-Zygmund saces B s,, that coincide with Hölder saces C s for s R + \ N. This leads to: f C t (Ω) = u d a C a+t (Ω), all t 0; with a + t relaced by a + t ε if a + t or 2a + t is integer. Otimal in the non-excetional cases. Q3. Smoothness of eigenfunctions u k? If 0 is an eigenvalue, its eigenfunctions are in E a (Ω). For λ k > 0, one finds by an iterative argument using the regularity theory: u k d a C 2a ( ε) (Ω) C (Ω); ε active if 2a N.
4. Nonhomogeneous boundary conditions For simlicity, consider just P a. The Fredholm ma from Th. 9, Cor. 10, r + P a : H a(s) (Ω) H s 2a (Ω), s > a 1/, (6) r + P a : E a (Ω) C (Ω), reresents the homogeneous Dirichlet roblem for P a. Recall that E a 1 (Ω) = e + {u(x) = d(x) a 1 v(x) v C (Ω)}. Here we have: Theorem 11. The boundary maing E a 1 (Ω) u γ a 1,0 u = γ 0 (d 1 a u) C ( Ω) extends for s > a 1/ to a continuous surjective maing γ a 1,0 : H (a 1)(s) with kernel H a(s) (Ω); in other words, γ a 1,0 : H (a 1)(s) (Ω) B s a+1 1/ ( Ω), (Ω)/H a(s) (Ω) B s a+1 1/ ( Ω). (7) Combination with (6) gives a nonhomogeneous Dirichlet roblem:
Theorem 12. The following mas are Fredholm, for s > a 1/ : {r + P a, γ a 1,0 }: H (a 1)(s) (Ω) H s 2a {r + P, γ a 1,0 }: E a 1 (Ω) C (Ω) C ( Ω). (Ω) B s a+1 1/ ( Ω), What is erhas surising is that this naturally defined nonhomogeneous Dirichlet roblem involves a blow-u at Ω! Namely, even for the smoothest ossible data, the solutions behave like d a 1 at Ω, with a 1 < 0. There is a general version in Hölder saces: Theorem 13. For t 0, the roblem r + P a u = f C t (Ω), γ a 1,0 u = ϕ C a+1+t ( Ω); is Fredholm solvable, with solution u e + d(x) a 1 C a+1+t (Ω) + C 2a+t (Ω), (8) (with t relaced by t ε in (8) if a + t or 2a + t is integer). The comonent e + d(x) a 1 C t+a+1 (Ω) enters nontrivially. Whenever ϕ 0, it creates a term in that sace, which is unbounded at Ω, behaving like d a 1.
In the studies by otential- and integral oerator-methods, this henomenon is called large solutions (blowing u at Ω), e.g. by Abatangelo arxiv 13 working in low-regularity saces. It is rather recisely described by our methods. Further remarks. Our study moreover allows treatments of other boundary conditions (also vector valued). For examle, r + ( ) a u = f with a Neumann condition γ a 1,1 u n (d(x) 1 a u) Ω = ψ can be shown to be define a Fredholm oerator: {r + ( ) a, γ a 1,1 }: H (a 1)(s) (Ω) H s 2a (Ω) B s a 1/ ( Ω), s > a+1/. The current efforts for roblems involving the fractional Lalacian are often concerned with nonlinear equations where it enters, and there is an interest also in generalizations with low regularity of the domain or the coefficients. For roblems where itself enters, one has a old and well-known background theory of boundary value roblems in the smooth case. Such a background theory has been absent in the case of ( ) a, and we can say that the resent results rovide that missing link.
The methods used in the current literature on ( ) a are often integral oerator methods and otential theory. Here is one of the strange formulations used there: Because of the nonlocal nature of ( ) a, when we consider a subset Ω R n, auxiliary conditions may be given as exterior conditions, where the value of the unknown function is rescribed on the comlement of Ω. Then the homogeneous Dirichlet roblem is formulated as: { r + ( ) a U = f on Ω, U = g on R n \ Ω. (8) The nonhomogeneous Dirichlet roblem is then formulated (for more general U) as: r + ( ) a U = f on Ω, U = g on R n \ Ω, (9) d(x) 1 a U = ϕ on Ω. (Abatangelo arxiv November 13.) We can show, when Ω is smooth: Within the framework of our function saces, (8) and (9) can be reduced to roblems with the unknown u suorted in Ω (i.e., g = 0), uniquely solved by our receding theorems.