Propagation of Singularities of Solutions to Semilinear Boundary Value Problems

Let P be a second-order, strictly hyperbolic differential operator on an open region Q C R n (n > 3) with smooth noncharacteristic boundary. Given a solution u G fffoc(n), s > (n + l ) /2 , to Pu = f(x,u), we discuss the propagation of microlocal H singularities in the range s < r < 2s — n/2 in the general case where the Hamilton field of p may be tangent to dT* f2\0 to arbitrarily high finite or infinite order. Introduction. Recently, M. Sablé-Tougeron [10, 11] and A. Alabidi [1] used the calculus of paradifferential operators to obtain, for a quite general class of nonlinear boundary value problems, results describing the reflection of singularities that travel on bicharacteristics transversal to the boundary of a region Q C R n . Using a different approach, similar to the one initiated by Rauch [9] and further developed by Beals and Reed [3] in studying interior propagation, we have obtained results for second-order semilinear problems with Dirichlet conditions in the general case where tangential bicharacteristics as well as gliding rays may carry singularities. Our argument has two main steps. We first prove an analogue of Rauch's Lemma for a class of spaces measuring microlocal H regularity up to the boundary. We also require a precise linear theorem (Theorem 2), a refinement of the results of Melrose and Sjöstrand ([8], see also [6]), which describes the propagation of H regularity along generalized bicharacteristics. This theorem must apply to equations Pu = v for distributions v which lie in our microlocal algebras, but which cannot be assumed to be normally regular. A simple inductive argument combining the above then yields the desired semilinear theorem (Theorem 1). Complete proofs will appear in [4]. Spaces of distributions near dû. Let U be a coordinate chart centered at so G dQ, and let (x±, x') be coordinates in which fi Pi U (= U) = {x\ > 0}. We will use the spaces H°t ,{U) defined in Hörmander [5]. For u e H^l^U) we^define WFr{u) C (T*dU\0) U (T*Ù\0) as follows. If a e T*dU\0, a £ WFru if and only if for some tangential pseudodifferential operator 0(x, D') of order zero, elliptic at (0,a), (j>u G H{oc(U). In this case we write u G H (a). In T*U, WFr coincides with the usual notion of WFr. Note that the definition of WFru is coordinate dependent. However, it is a consequence (see [4]) of the Lemma stated below and the fact that <9fi is noncharacteristic, that for Received by the editors December 10, 1985. 1980 Mathematics Subject Classification. Primary 58G17, 35L70. ©1986 American Mathematical Society 0273-0979/86 $1.00 + $.25 per page 201 202 FLORIN DAVID AND MARK WILLIAMS solutions u G #fo c(n), s > (n + l)/2, to Pu = / (x,u) , WFrw is coordinate independent for s < r < 2s — n/2 + 1/2. It was observed in [10] that the spaces H\°$(U) are algebras stable under f(x,u) if t > ±, t + t' > n/2, and t + 2t' > \. Thus, given u G Hfoc such that Pu = f(x,u), Peetre's Theorem [5, 4.3.1] yields by induction that u G #2s-p+2,-s+p-2(k0 f° all P > 5This suggests considering, for a G T*dU\Q, the following spaces as candidates for microlocal algebras at the boundary: Main results. The following lemma is, we think, the natural analogue of Rauch's Lemma at the boundary. LEMMA. The spaces As^yP(a) defined above are algebras stable under the action of C°° functions f{x,u) provided (n + l)/2 < s < s' < 2s — n/2 — p and pe (1/2,8 -n/2). In the statement of the following theorem we take fi = UJ x RXn, where ou C R n _ 1 is an open region with smooth boundary. H and Q denote the hyperbolic and glancing regions of T*<9fi\0 (see [8 or 6]). THEOREM 1. Let P be a second-order differential operator, noncharacteristic with respect to dU, and strictly hyperbolic with respect to the surfaces xn = c. Suppose that u G Hfoc(Q), s > (n + l)/2, satisfies Pu = f{x,u) in Q. Then for s < r < 2s — n/2, every point 70 £ WFru\WFr(u\dn) is either a characteristic point of P in T*Q, or else contained in M\jQ. An open interval (—T, T) 9 t > i(t) with 7(0) = 70 on a generalized bicharacteristic [S, 6] is contained in WFru. Theorem 1 is proved by an inductive argument using the lemma and the following linear theorem. Note that the assumptions on P in Theorem 1 and the coordinate invariance of WFru for u G H*oc satisfying Pu = f(x,u) (r G [s, 2s — n/2-f \)) allow us to reduce to the case where P and fi are as described in Theorem 2. We denote points in T*R by (xi, x', £1, £'). THEOREM 2. Suppose Q C R£ = {xi > 0} and write dQ = ïïndKf.. Let P = D\x —R(x, D') be a second-order differential operator on O strictly hyperbolic with respect to the planes xn = c. Suppose Pu = ƒ m Q, where for some s' > l,u G ^ 2 , 0 0 ( Ï Ï ) , ƒ G fll?_oo(n), and t i j E ^ ' + H f o ^ O ) : X l > 0, x near dû}. Then every point 70 £ W^s'+i/2?A(^^V ƒ U WFs/+i/2(u|dn)) is ei£/ier a characteristic point of P in T*Q, or else contained in M U Q. An interval (—T,T) 3 t > ^(t) with 7(0) = 70 on a generalized bicharacteristic is contained in WF^+i^u. Sketch of the proofs. We begin with the lemma. Any u G As,s>,p(a) can be extended to an element u G Hfoc(U) such that ü = u\ -f i^, where u\ G Hfóc{Ü) and where u2 G H^Lp,-s+p{U) has the property that (x,D )u2 G #2s-p,oo(R-) for Ö') supported sufficiently near (0,o*). Now given it and v in ^s,s',p(0-), we first choose such extensions ü = u\ -f 1*2, # = v\ + v2We then show that tangential operators of order zero of the form x{D') ° 'l{)) SEMILINEAR BOUNDARY VALUE PROBLEMS 203 where xjj G Co°(R), map üv into H (R) provided they are supported sufficiently near (0, -e of the incoming tangential ray passing over ')*> G iP '+^R/ 1 ) for all A G L° supported sufficiently near (0,<r). (It is obvious that Av G # 5 + 1 (p _ 1 (0 ) ) for such A.) Next choose such an A = 1 near (0, a) and solve the mixed problem in U: Pw' = 0, w\du = M' ~ )\du, w' = 0 in xn < -e/2. Since J 4 V | W G # 5 ' + 1 / 2 , w' G #ioC (£/)The results of Melrose [7] and Taylor [12] imply a $ WFb(u' {v + w')). Hence u' G tf '/(<r), which implies u G H'^~^(a) as desired. In cases of finite-order tangency where the incoming or outgoing generalized bicharacteristic is a gliding ray, and in the case of infinite-order tangency, our proof of Theorem 2 follows the arguments in Hörmander [6, §24.5] closely. Again, an appropriate mixed problem is solved first. The remainder of the proof involves checking that the L pairings in [6, 24.5] remain bounded under the hypotheses of Theorem 2. It is a pleasure to thank Richard Melrose for helpful conversations concerning the linear theorem, and for encouraging us to pursue the case of higher-order tangency. NOTE ADDED IN PROOF. The proof of Theorem 2 has been simplified in [4]. A method similar to that sketched above for a G Qd can be used to handle all points cr G M U Q. The revised argument allows H regularity for r <2s — n/2 + \ to be propagated in Theorem 1.