On the sharp regularity of solutions to hyperbolic boundary value problems

We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to 1/2 + N, as it involves nonlocal compatibility conditions.

These compatibility relations are trivial here due to the solution formula, but are more generally derived considering u (and its derivatives) at the corner x = t = 0, and writing A basic rule of thumb being that any compatibility condition that makes sense should be true.For fractional regularity, not much changes except in the notoriously pathologic case s ≡ 1/2[Z].Indeed even if there is no trace in H 1/2 (R + ), the gluing of two functions in H 1/2 (R + ) is not H 1/2 (R).The simplest way to see this is to consider the map 2 by interpolation.The interpolated space is the famous Lions-Magenes space H 1/2 00 (R + ), and it is different (algebraically and topologically) from H 1/2 (R + ): by interpolation of Hardy's inequality, any function f ∈ H 1/2 00 (R + ) must satisfy x dx < ∞, this is obviously not the case for functions merely in H 1/2 (R + ).
For the regularity of solutions of the BVP, this adds a "global" compatibility condition u ∈ C t H 1/2 ⇔ (u 0 , g) ∈ H 1/2 (R + ) and Our aim here is to prove an analogous result for general hyperbolic boundary value problems.

Settings and results
Let Ω be a smooth open set of R d , we consider first order boundary value problems of the form    (1.1) IBVP The index t in R + t has no meaning except to emphasize the time variable.The A ′ j s are q × q matrices depending smoothly on (x, t), B is a smooth b×q matrix, b is the number of boundary conditions.For data (u 0 , g, f ) ∈ L 2 (Ω) × L 2 (∂Ω × R + t ) × L 2 (R + t × Ω), the well-posedness of such hyperbolic BVP has been obtained in a large variety of settings, that we will only shortly mention.After the pioneering results of Friedrichs Friedrichs [6] for symmetric dissipative systems, Kreiss Kreiss [10] proved the well-posedness of the BVP with zero initial data in the strictly hyperbolic case ( A j ξ j has only real eigenvalues of algebraic multiplicity one) under the now standard Kreiss-Lopatinskii condition on B. In Kreiss's framework, the case of L 2 initial data was then tackled by Rauch Rauch [17].Well-posedness of constantly hyperbolic BVP was later obtained by Métivier Met [15] (zero initial data), the author then proved well-posedness with L 2 initial data Audiard1 [2].A further generalization was obtained by Métivier Metsemigroupe [16] for a new class of hyperbolic operators, larger than the constantly hyperbolic ones.He also gave a new proof, both more general and simpler, of well-posedness with L 2 initial data.For more references and results, in particular for characteristic BVP (that we do not consider here) the reader may refer to the book Benzoni3 [4].
Let n be a normal on ∂Ω, the problem ( IBVP 1.1) is said to be noncharacteristic when A j n j is invertible on ∂Ω.For non characteristic boundary value problems, the main reference on the smoothness of solutions is the classical paper of Rauch and Massey RauchMassey [18], where, under no specific assumption (except of course well-posedness), the authors prove that the solution of ( ) and satisfy natural compatibility conditions that we describe now.For conciseness, when there is no ambiguity we will usually denote We denote A = A j ∂ j and define inductively v j the formal value of (∂ j t u)| t=0 by The first order compatibility condition is Bv 0 | ∂Ω = g| t=0 and the generic compatibility condition of order j is Compatibility at order j : 3) makes sense as soon as (u 0 , g, f ) ∈ (H s ) 3 , s > j − 1/2.If the smoothness of the data is j − 1/2, j ∈ N * , we define a special compatibility condition : when Ω = R d−1 × R + , denote x = (x ′ , y); the condition is Compatibility at order j − 1/2: For general smooth Ω, ( ) is defined similarly through local maps and a partition of unity: near the boundary Ω is diffeomorphic to (a part of) R d−1 × R + thanks to some map Φ, one simply requires ( CCj 1.3) to stand for g(Φ(x ′ , 0), t), (v l • Φ(x ′ , t)) 0≤l≤j−1 .Note that due to Hardy's inequality, the j-th condition implies the condition of order j − 1/2.
we say that data (u 0 , g, f ) ∈ (H s ) 3 satisfy the compatibility conditions at order s when the compatibility conditions are satisfied at order s when that there exists a sequence u n of smooth solutions of ( IBVP 1.1) with data (u 0,n , g n , f n ) that converge to (u 0 , g, f ) in L 2 , and for any T > 0, u − u n C([0,T ],L 2 ) → 0.
Assumptions We need the smoothness of Ω and the well-posedness of ( ⊂ Ω, and ∪ J j=1 U j ⊃ ∂Ω.We do not assume that the U j are bounded sets, but Dϕ j , Dϕ −1 j must be uniformly bounded, and d(Ω \ ∪Im(ϕ j ), ∂Ω) > 0.
3. For data (u 0 , g, f ) ∈ (L 2 ) 3 , there exists a unique strong L 2 solution1 to ( IBVP 1.1) that satisfies the semi-group estimate for γ large enough We use the convention that norms inside the domain are denoted • while norms on the boundary are denoted | • |.
We point out that a consequence of the semi-group estimate is the resolvent estimate: for γ large enough (larger than for ( This is readily obtained by squaring ( semigroupe 1.5) for some fixed γ 0 , multiplication by e −2γt , γ > γ 0 and integration in t.Higher regularity versions of the resolvent and the semi-group estimates are a bit more delicate to state.We define weighted Sobolev spaces H s γ in section notations 2, the weighted resolvent estimate is then The main point of this estimate is that it is sharp with respect to the parameter γ and allows to absorb commutators in a priori estimates.Moreover, it implies the following (simpler to read) estimate We shall not need something as precise for the semi-group estimate: Both estimates should be modified when s = k + 1/2, k ∈ N: it is necessary to add in the right hand side the H 15 for details.This is the (implicit) convention that we use in theorem mainth 1.3, we refer to the proof for more details.An interesting related feature is that the constant in can not be uniform in θ, it blows up as θ → 1/2 and the estimates are actually not true for θ = 1/2.
We can now state more precisely the regularity result of Rauch and Massey: satisfy the compatibility condition up to order k, the solution of The only suboptimal part of the theorem is the regularity assumption on g.This is due to the fact that the theorem is deduced from the homogeneous case g = 0 with a lifting argument.It was already pointed out at the time by the authors that it could be improved (without proof), but quite unfortunately the result that remained in the litterature is the suboptimal one, see for example the reference book Benzoni3 [4], and in somewhat different settings the lecture notes Met2 [14] or the interesting discussion in the introduction of lanigu [9].Our result is that the same property holds with boundary data in H k instead of H k+1/2 , moreover we allow k to be any nonnegative real number rather than an integer.
satisfy the compatibility condition up to order s, the solution of ( The proof when s is an integer is quite similar to the original argument of Rauch and Massey, actually the fact that we handle directly nonzero boundary data leads to some slight simplifications due to the fact that it allows to avoid a reduction to the case where B is constant.The fractional case is essentially an interpolation argument, however it is not trivial due to the presence of the compatibility conditions.For example, in the model case described The litterature on such problems is not very rich.Another related problem is the interpolation of Sobolev spaces with boundary conditions, that are in some sense between H s and H s 0 .This issue appeared quite long ago for elliptic equations on non smooth domains or parabolic problems, see e.g. the last section of Grisvard [7], sections 14-17 of chapter 4 in lionsmagenes2 [13] (where most of the identification problems were left open), or the more recent (and much more involved) book Amann [1], in particular VIII.2.5.Due to the technicity of this last reference (anisotropic Besov spaces are studied), degenerate cases (in our settings s ∈ N + 1/2) are not considered.The Schrödinger equation on a domain and related interpolation problems were also studied by the author in Audiard7 [3], where the natural spaces for the boundary data are Bourgain spaces.
Plan of the article Section 2 is devoted to notations and a brief reminder on interpolation.The proof of theorem mainth 1.3 is then organized in three sections : in section 3 we recall a standard smoothness result for the pure boundary value problem posed for t ∈ R, due to Tartakov.For completeness, we include a sketch of proof that follows an argument of the (unfortunately depleted) book ChaPi [5].Theorem mainth 1.3 in the case s integer is proved in section 4.An important point is a basic lifting lemma which proves also useful for the general case.In section 5, smoothness is first proved for 0 ≤ s ≤ 1 with an interpolation argument, then for any s with a non trivial differentiation argument.

Notations and basic results
notations Basic notations Proofs are often reduced to the case Ω = R d−1 × R + .In such settings, we denote the variable x = (x ′ , y) x ′ ∈ R d−1 .The variables x ′ , t are called tangential, while y is the normal variable.Partial differential operators acting on functions of (x, t) are written as ∂ α , α ∈ N d+1 , by convention α d+1 is the order of differentation in time.A multi-index, or a differential operator, is said to be tangential when Sobolev spaces Ω is assumed to be a smooth open set as in definition page assump1 3. The Sobolev spaces H s (Ω), are defined when s is an integer as When s is not an integer, they are defined by (complex) interpolation, H s = [L 2 , H k ] s/k for any integer k larger than s.This definition does not depend on k.
The Sobolev spaces for functions defined on ∂Ω, Ω × R + t etc are defined in the same standard way.
00 is different algebraically and topologically from H 1/2 .It is a Banach space endowed with the norm where d is the distance to ∂Ω (see lionsmagenes [12]).The only important fact, regularly used in the article, is that if X 0 , X 1 are Banach spaces, an operator T : The weighted Sobolev spaces H s γ are defined as follows : Definition 2.1.When s is a nonnegative integer we define H s γ (Ω × R + t ) as the the set of functions in L 2 such that the following norm is finite When s is not an integer, H s γ is defined by complex interpolation : if k is an integer larger than s, When s is an integer, it is a straightforward consequence of Leibniz formula ∂ j t (e −γt u) = j i (−γ) i e −γt ∂ j−i t u that the H s γ norm is equivalent to e −γt u H s , though with constants that depend on γ, hence the H s γ spaces coincide algebraically and topologically with the set of functions such that e −γt u ∈ H s .
Traces Sobolev spaces on ∂Ω are defined with local maps.The trace operator is an isomorphism: , where ∂ n is the normal derivative on ∂Ω.For functions defined in H s (Ω × R + * ), the trace operator on ∂Ω × R + * and Ω × {0} is more subtle, the map (see lionsmagenes2 [13]).In the case s = 1, and Ω = R d−1 × R + * , surjectivity requires the global compatibility condition (

2.3) compaclassiq
This condition extends to smooth Ω, see the short comment after ( CCj-1.4).Provided such compatibility conditions are added, the trace map is a surjection and has a right inverse, this very well known fact will be proved later in the article in some basic cases where it is needed with more precise estimates.

Regularity for the pure boundary value problem
Consider the boundary value problem When g, f can be smoothly extended by 0 for t < 0, the smoothness of u is well known Tartakoff [19], ChaPi [5].The classical proof is done by first studying the pure boundary value problem posed on t ∈ R, the case t ∈ R + is then deduced by an extension by 0 for t < 0. We give here a minor variation of this argument that directly tackles ( In particular, its belongs to H k (Ω × [0, T ]) for any T > 0.
Proof.The classical plan is to straighten the boundary through local maps, then use a tangential regularization.It is done by induction on k, it suffices to prove the final step where we assume u ∈ H k−1 (R d × R t ) and prove u ∈ H k .

) equivsobo
Friedrich's lemma can be generalized in such settings: for P a first order differential operator with smooth coefficients For details, we refer to ChaPi [5] chapter 2 section 6.We shall use tangential mollifiers ρ ε (x ′ , t) for the functions u j , 1 ≤ j ≤ J, and full mollifiers ρ ε (x, t) for u 0 .Everything in ( eqredresse 3.2) is extended by 0 for t < 0. Note that due to the assumptions on f, g, the extensions of (f j , g j ) are still in H k .We apply ρ ε * to ( eqredresse 3.2) for 1 ≤ j ≤ J: Multiplying by ε −2k−1 1 + (δ/ε) 2 −1 , integrating in ε and using Friedrich's lemma we have

.6) estimtan
The commutator [ψ j , L j ] is the multiplication by a smooth matrix θ j .Due to the special structure of the local maps, ϕ −1 i • ϕ j has the form (ϕ i,j (y ′ ), y d ) hence Thanks to composition rules (in H s,δ , again see For γ large enough, this can be absorbed in (the sum over j of) the left-hand side of ( estimtan 3.6): It seems "moral" that noncharacteristicity should imply the same bound for u j H k−1,δ , however the H k−1,δ norm is a non local norm for functions defined on R d × R t , hence such an assertion is not clear.Instead we first obtain interior estimates with similar, simpler computations

.8) interieur
Decomposing again ψ 0 u = J j=0 ψ 0 ψ j u, and following the same lines that led to ( A simple consequence of the definition of the H s,δ spaces is that for any tangential differential operator D of order 1 and s ≥ 1 and on its support L j is (uniformly) non characteristic, so we may extend it by zero for y d < 0 and use ( transfert 3.10) to deduce

.11) interieur2
Note that the term γ e −γt u 2 is present due to the factor γ in the definition of L j .Thanks to the induction assumption, this lower order term is bounded by g 2 Letting δ → 0 we have u j ∈ L 2 H k , 1 ≤ j ≤ J and u 0 ∈ H k .We conclude that u ∈ H k again thanks to the uniform non characteristicity.

Smoothness of the IBVP: the integer case
We assume in this section that (u 0 , g, f ) ∈ (H k ) 3 satisfy the compatibility conditions ( CCj 1.3) up to order k, and we prove theorem mainth 1.3 in these settings.To prove that u ∈ ∩ k j=0 C j t H k−j , the strategy is to use the regularity for the pure boundary value problem by substracting an approximate solution (actually a Taylor expansion at t = 0) to u.For technical reasons, it is necessary to use much more regular data that satisfy compatibility conditions to higher order.The construction of such data requires the following lifting lemma that is also used in the next section.
releve Lemma 4.1.For m ∈ N, there exists a lifting map R m : H s (∂Ω) → H m+s+1/2 (∂Ω × R t ), continuous for any s > 0 such that Proof.Up to the use of local maps, the problem is reduced to ∂Ω = R d−1 , and to construct a lifting valued in We use the Fourier transform on R d−1 × R t and denote ξ the dual variable of x ′ , τ the dual variable of t, and λ is a large parameter to fix later: The trace relations ( trace 4.1) are obvious from the second formula.The H m+s+1/2 norm is easily bounded With the same computation It is therefore sufficient to choose λ large enough to ensure the smallness of R m L 2 →H r .
compahaute Lemma 4.2 (Construction of smooth compatible data).Let k ≥ 0, (u 0 , g, f ) ∈ (H k ) 3 satisfying the compatibility conditions up to order k.For any m > k, there exists (u 0,n , g n , f n ) ∈ (H ∞ ) 3 satisfying the compatibility conditions up to order m, and such that Proof.By density of smooth functions, there exists a sequence (u 0,n , g n , f n ) ∈ (H ∞ ) 3 converging to (u 0 , g, f ) in (H k ) 3 .We denote v j,n the corresponding functions in ( taylor 1.2).For j ≥ 1 the "compatibility error" is defined as Due to the compatiblity conditions and continuity of traces we have As a consequence, given a lifting operator R j−1 as in lemma releve 4.1, R j−1 ε j,n H k → n 0. For k < j ≤ m, ε j,n is not small in any Sobolev space, nevertheless from lemma releve 4.1 there exists a lifting R j−1,n such that R j−1,n ε j,n H k ≤ 1/n.We then define This choice ensures that compatibility conditions are satisfied by (u 0,n , g n , f n ) up to order m and g n − g H k → 0. 2) for smooth data (u 0,n , g n , f n ).We define the approximate solution

Proof of theorem
We solve then By construction, the data (0, g n − Bu app,n , f n − Lu app,n ) are smooth and it is easily seen that Hence according to proposition regBVP 3.1, the solution w n belongs to H k+2 , this implies by Sobolev embedding w n ∈ ∩ k+1 j=0 C j t H k+1−j .Therefore u n := w n + u app,n is also in ∩ k+1 j=0 C j t H k+1−j , and it is a solution of ( IBVP 1.1) with data (u 0,n , g n , f n ).Using a differentiation argument similar to the proof of proposition regBVP 3.1, but much simpler since no regularization is needed, we see that u n satisfies semigreg 1.9: as well as ( resolvreg 1.7).The same estimates, applied to u p − u q , (p, q) ∈ N 2 , shows that (u n ) is a Cauchy sequence in ∩ k j=0 C j t H k−j , but since (u n ) converges (in L 2 ) to the solution u of ( IBVP 1.1) with data (u 0 , g, f ), this ensures that u ∈ ∩ k j=0 C j t H k−j .The estimate ( resolvreg 1.7) is then an elementary differentiation argument : tangential regularity is obtained directly by differentiation (which is now legal) and use of the L 2 estimate, while normal regularity uses the non characteristicity.

Regularity for positive s
For ease of presentation, we only detail the case Ω = R d−1 × R + .The general case can be obtained by using a partition of unity as in the previous section.In this section, we follow the (non standard) convention that Under such settings, we can assume that A d is invertible and A −1 d is uniformly bounded.Furthermore since B : R p → R b has maximal rang b, there exists a smooth basis of Ker B (as a smooth vector bundle over the contractible space We remind that compatibility conditions of order s = k + θ, k ∈ N * , 0 < θ < 1 are defined as follows: 1.If θ < 1/2, then compatibility conditions ( CCj 1.3) up to order k are satisfied.
3. If θ = 1/2, compatibility conditions up to order k are satisfied and The case 0 < s < 1 From the previous section, the map (u 0 , g, f ) → u solution of ( Let us define for 0 ≤ θ ≤ 1 : the compatibility condition of order θ is satisfied , (note that compatibility conditions of order less than 3/2 do not involve f ).Both the semi-group estimate ( semigreg 1.9) and the resolvent estimate ( resolvreg 1.7) follow from an interpolation argument if we can prove that X θ = [X 0 , X 1 ] θ .
(5.1) interpX More precisely, since the resolvent estimate implies for s = 0, 1 the interpolation identity ( interpX 5.1) implies (5.2) resolvsupers (a better estimate would require to use weighted X θ spaces, a course that we chose not to follow).
Proof of ( interpX 5.1) We extend B on Ω × R + t as B(x ′ , y, t) = B(x ′ , t), and consider the map u 0 → Bu 0 := u 0 .It is an isomorphism (H s (Ω)) p → (H s (Ω)) p , and the compatibility condition can be rewritten This transformation "diagonalizes" ( interpX 5.1), and we are reduced to determine where u 0 and g are now scalar functions.Of course, it is well-known that [L 2 , H 1 ] θ = H θ , so the first case is immediate.In the second case, surprisingly, we were not able to find results in the litterature except in the simplest case θ < 1/2, which is in lionsmagenes2 [13] section 14.

.3) inclusion
On the other hand, from Lions-Peetre reiteration theorem, for any 0 < s, θ < 1 If we have for some On the other hand, the case θ ≤ s is contained in lemma interpfacile 5.1.For any 0 < r < 1 we define the map It is easily seen that Tr is continuous H 3/2 → Y 1 and H 1/2+s → Y s for 0 < s < 1/2.As it is well known that [H s+1/2 , H 3/2 ] θ = H 1/2+θ+(1−θ)s , we deduce by interpolation Tr : We observe now that the lifting R from lemma relevecoin 5.2 is a right inverse for Tr: for fixed 0 < s < 1/2 and any 0 < θ < 1, we have Tr which was the required converse inclusion.
The case s > 1 We denote s = k + θ, 0 ≤ θ < 1.According to the integer case, we already have u ∈ ∩C k−j H j .For any tangential multi-index α of order k (that is, (5.4) derBVP where L α , L ′ α are differential operators of respective order α, α − 1. Regularity will again be obtained by regularization of the data, we distinguish three cases: The case 0 < θ < 1/2 With the same argument as in the integer case (note that the condition θ < 1/2 allows to use lemma releve 4.1), there exists regularized data (u 0,n , g n , f n ) ∈ (H k+1 ) 3 , converging to (u 0 , g, f ) that satisfy the compatibility conditions up to order k + 1.The corresponding solution u n belongs to ∩ k+1 0 C j t H k+1−j so that we may apply the resolvent estimate ( resolvreg 1.7) to ∂ α u n with s = θ, combined with basic trace estimates and the commutator estimate Due to the boundary being non characteristic, we deduce as for the integer case (note that the fractional regularity gained here includes conormal regularity) for γ large enough only depending on s With the resolvent estimate available, the semi group estimate is now an immediate consequence of the case 0 < s < 1 applied to ( derBVP 5.4): Once more, normal regularity is then obtained thanks to the boundary being non characteristic.Letting n → ∞, we deduce that e −γt u is in H s (R + × Ω) ∩ (∩ k j=0 C j (R + , H s−j (Ω)) and satisfies the semi group estimate and the resolvent estimate.
The case 1/2 < θ < 1 This can be done with exactly the same argument.Actually, the construction of regularized data (u 0,n , g n , f n ) ∈ (H k+1 ) 3 that satisfy compatibility conditions up to order k + 1 and converging to (u 0 , g, f ) in (H s ) 3 is even simpler.Indeed (u 0 , g, f ) satisfy compatibility conditions up to order k + 1, hence any regularization of (u 0 , g, f ) satisfies −→ n 0, and it suffices to modify g n as g n − δ n where δ n is a function in H k+1 (∂Ω × R + t ) that satisfies for 1 The case θ = 1/2 When s = k + 1/2, the compatibility conditions are satisfied in particular up to order k.From the previous study, we have e −γt u ∈ (∩ k j=0 C j t H k+θ−j ) ∩ H k+θ for any θ < 1/2, with the estimate e −γt u (∩ k j=0 C j t H j+θ−j ) + e −γt u H k+θ ≤ C(θ) (u 0 , g, f ) (H s ) 3 .
Of course this is not enough to conclude, but the estimate can be sharpened: apply estimate ( Recall that the compatibility conditions at order j are are and at order k + 1/2 As a consequence, for any j ≤ k + 1 and any

regBVP Proposition 3 . 1 .
Let k ∈ N. If the extension of f and g by 0 for t < 0 belongs to H k , then for γ large enough the solution of ( bvp 3.1)

mainth 1 . 3 (
integer case) We follow the notations of lemma compahaute 4.2; v j,n are smooth functions defined by ( taylor 1.