Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

doi:10.1155/2010/524736

Research Article

Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

Alessandro Morando and Paolo Secchi

Dipartimento di Matematica, Facolt`a di Ingegneria, Universit`a di Brescia, Via Valotti, 9, 25133 Brescia, Italy

Correspondence should be addressed to Paolo Secchi,paolo.secchi@ing.unibs.it Received 9 June 2010; Accepted 30 August 2010

Academic Editor: Alberto Cabada

Copyrightq2010 A. Morando and P. Secchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a uniqueL²-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinski˘ıcondition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.

1. Introduction and Main Results

Forn≥2, letRⁿdenote then-dimensional positive half-space

Rⁿ: x

x1, x

, x1>0, x: x2, . . . , xn∈Rⁿ⁻¹

. 1.1

The boundary ofRⁿwill be systematically identified withRⁿ⁻¹_x .

We are interested in the following stationary boundary value problemBVP:

γLB

uF, inRⁿ, 1.2

MuG, onRⁿ⁻¹, 1.3

whereLis the first-order linear partial differential operator

Lⁿ

j1

Aj∂j; 1.4

for eachj 1, . . . , n, the short notation∂j:∂/∂xjis used.

The coefficientsAj j 1, . . . , nofLareN×Nmatrix-valued functions inC^∞₀Rⁿ, the space of restrictions toRⁿof functions ofC₀^∞Rⁿ. In1.2,Bstands for a lower-order term whose form and nature will be specified later; compare toTheorem 1.1andSection 3.2.

The source termF, as well as the unknownu, is aR^N-valued function ofx; we may assume that they are both supported in the unitary positive half-ballB:{x x1, x:x1≥ 0, |x|<1}.

The BVP has characteristic boundary of constant multiplicity 1≤ r < Nin the following sense; the coefficientA1of the normal derivative inLdisplays the blockwise structure

A1x

⎛

⎝A^I,I₁ A^I,II₁ A^II,I₁ A^II,II₁

⎞

⎠, 1.5

whereA^I,I₁ ,A^I,II₁ ,A^II,I₁ ,A^II,II₁ are, respectively,r×r,r×N−r,N−r×r,N−r×N−r submatrices, such that

A^I,II_1|x100, A^II,I_1|x100, A^II,II_1|x100, 1.6 andA^I,I₁ is invertible overB. According to the representation above, we split the unknown uasu u^I, u^II;u^I ∈R^r andu^II ∈R^N−r are said to be, respectively, the noncharacteristic and the characteristic components ofu.

Concerning the boundary condition1.3,Mis assumed to be the matrixId 0, where Id denotes the identity matrix of orderd, 0 is the zero matrix of sized×N−d, anddis a given positive integer≤r. The datumGis a givenR^d-valued function ofx x2, . . . , xnand is supported in the unitaryn−1-dimensional ballB0,1:{|x|<1}.

Section 4will be devoted to prove the following regularity result.

Theorem 1.1. Letk, r, sbe fixed nonnegative integer numbers such thats≥r≥0,s >0, and suppose that the coefficientsAj (j 1, . . . , n) of the operatorLin1.4are given inC^∞₀RⁿandA1fulfils conditions1.5,1.6. One assumes that for anyh >0 there exist some constantsC0 C0h>0, γ0γ0h≥1 such that for everyγ ≥γ0, for every operatorBOp^γb, whose symbolbbelongs to Γ⁰and satisfies|b|_0,k ≤h, and for all functionsF ∈ H^s,r_tan,γRⁿ, andG∈H_γ^sRⁿ⁻¹, the corresponding BVP1.2-1.3admits a unique solutionu∈ L²Rⁿ, withu^I_|x

10 ∈L²Rⁿ⁻¹, and the following a priori energy estimate is satisfied:

γu²_L2Rⁿu^I_|x10²

L^2Rn−1≤C0

1

γ^2s1F²_H^s,r

tan,γRⁿ 1 γ^2sG²_Hs

γ^Rn−1

. 1.7

Then, for everym ∈ Nand any matrix-valued functionB ∈ C^∞₀Rⁿ, there exist some constants Cm > 0, γm (with γm ≥ γ_m−1 ≥ 1) such that if γ ≥ γm and we are given arbitrary functions

F ∈ H^sm,rm_tan,γ Rⁿ and G ∈ H_γ^smRⁿ⁻¹, the uniqueL²-solution uof 1.2-1.3 (with dataF, G, and lower-order termB ≡multiplication by B) belongs toH_tan,γ^m Rⁿ,u^I_|x

10∈H_γ^mRⁿ⁻¹and the a priori estimate of orderm

γu²_H^m

tan,γRⁿu^I_|x10²

Hγ^mRn−1≤Cm

1

γ^2s1F²_Hsm, rm

tan,γ Rⁿ 1

γ^2sG²_Hsm γ ^Rn−1

1.8

is satisfied.

The function spaces involved in the statement ofTheorem 1.1, as well as the norms appearing in1.7,1.8, will be described inSection 2. The kind of lower-order operatorB involved in1.2, that is allowed inTheorem 1.1, will be introduced inSection 3.2.

The BVP1.2-1.3, with the aforedescribed structure, naturally arises from the study of a mixed evolution problem for a symmetric or Friedrichs’symmetrizable hyperbolic system, with characteristic boundary. The analysis of the regularity of the stationary problem, presented in this work, plays an important role for the study of the regularity of time-dependent hyperbolic problems, constituting the final goal of our investigation and developed in 1. In view of the well-posedness property that problems1.2-1.3enjoy in the statement ofTheorem 1.1, here we do not need to assume the hyperbolicity of the linear operatorL in 1.4; the only condition required on the structure ofL is that expressed by conditions1.5and1.6. In the hyperbolic problems, the numberdof the scalar boundary conditions prescribed in 1.3 equals the number of positive eigenvalues of A1 on {x1 0} ∩ B the so-called incoming characteristics of problem 1.2-1.3, compare to 1; this valuedremains constant along the boundary, as a combined effect of the hyperbolicity and the fact thatA1|{x10}∩Bhas constant rank.

In 2, the regularity of weak solutions to the characteristic BVP 1.2-1.3 was studied, under the assumption that the problem is strongly L²-well posed, namely, that a unique L²-solution exists for arbitrarily given L²-data and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that theL²-norms of the interior and boundary values of the solution are measured by the L²-norms of the corresponding dataF, G.

The statement ofTheorem 1.1extends the result of 2, Theorem 15, to the case where only a weak well posedness property is assumed on the BVP1.2-1.3. Here, theL²-solvability of1.2-1.3requires an additional regularity of the corresponding dataF, G; the integers represents the minimal amount of regularity, needed for data, in order to estimate theL²- norm of the solutionuin the interior of the domain, and its trace on the boundary, by the energy inequality1.7.

Several problems, appearing in a variety of different physical contexts, such as fluid dynamics and magneto-hydrodynamics, exhibit a finite loss of derivatives with respect to the data, as considered by estimate 1.7 in the statement of Theorem 1.1. This is the case of some problems that do not satisfy the so-called uniform Kreiss-Lopatinski˘ıcondition; see, for example, 3,4. For instance, when the Lopatinski˘ıdeterminant associated to the problem has a simple root in the hyperbolic region, estimating theL²-norm of the solution makes the loss of one tangential derivative with respect to the data; see, for example, 5, 6. In 7, Coulombel and Gu`es show that, in this case, the loss of regularity of order one is optimal;

they also prove that the weak well posedness, with the loss of one derivative, is independent of Lipschitzean lower-order terms, but not independent of bounded lower-order terms. This is a major difference with the strongly well-posed case, where there is no loss of derivatives and

one can treat lower-order terms as source terms in the energy estimates. Also, this yields that the techniques we used in 2, for studying the regularity of stronglyL²-well-posed BVPs, cannot be successfully performed in the case of weakly well-posed problemsseeSection 4 for a better explanation.

The paper is organized as follows. InSection 2we introduce the function spaces to be used in the following and the main related notations. InSection 3we collect some technical tools, and the basic concerned results, that will be useful for the proof of the regularity of BVP 1.2-1.3, given inSection 4.

A final Appendix contains the proof of the most of the technical results used in Section 4.

2. Function Spaces

The purpose of this section is to introduce the main function spaces to be used in the following and collect their basic properties.

Forj1,2, . . . , n, we set

Z1:x1∂1, Zj:∂j, forj ≥2. 2.1

Then, for every multi-indexα α1, . . . , αn∈Nⁿ, the conormal derivativeZ^αis defined by Z^α:Z^α₁¹· · ·Z^αnⁿ; 2.2

we also write∂^α∂^α₁¹· · ·∂^αnⁿ for the usual partial derivative corresponding toα.

Forγ≥1 ands∈R, we set

λ^s,γξ:

γ²|ξ|²s/2

2.3

and, in particular,λ^s,1:λ^s.

The Sobolev space of order s ∈ R in Rⁿ is defined to be the set of all tempered distributions u ∈ SRⁿ such that λ^su ∈ L²Rⁿ, being u the Fourier transform of u; in particular, fors∈N, the Sobolev space of ordersreduces to the set of all functionsu∈L²Rⁿ, for which∂^αu∈L²Rⁿfor allα∈Nⁿwith|α| ≤s.

Throughout the paper, for realγ≥1,H_γ^sRⁿwill denote the Sobolev space of orders, equipped with theγ-depending norm · _s,γ defined by

u²_s,γ : 2π⁻ⁿ

Rⁿλ^2s,γξ|uξ|²dξ, 2.4 whereξ ξ1, . . . , ξnare the dual Fourier variables ofx x1, . . . , xn. The norms defined by2.4, with different values of the parameterγ, are equivalent to each other. Forγ 1 we set for brevity · _s: · _s,1and, accordingly,H^sRⁿ:H₁^sRⁿ.

It is clear that, fors ∈ N, the norm in2.4turns out to be equivalent, uniformly with respect toγ, to the norm · _Hγ^s_Rndefined by

u²_Hγ^s_Rⁿ:

|α|≤s

γ^2s−|α|∂^αu²_L2Rⁿ. 2.5

Another useful remark about the parameter depending norms defined in2.4is provided by the following counterpart of the usual Sobolev imbedding inequality:

u_s,γ ≤γ^s−ru_r,γ, 2.6

for arbitrarys≤randγ ≥1.

InSection 4, the ordinary Sobolev spaces, endowed with the weighted norms above, will be considered inRⁿ⁻¹ interpreted as the boundary of the half-spaceRⁿ; regardless to the different dimension, the same notations and conventions as before will be used there.

Let us introduce now some classes of function spaces of Sobolev type, defined over the half-spaceRⁿ; these spaces will be used to measure the regularity of solutions to characteristic BVPs with sufficiently smooth datacf.Theorem 1.1andSection 4.

Given an integerm ≥ 1, the conormal Sobolev space of ordermis defined as the set of functionsu∈L²Rⁿsuch thatZ^αu∈L²Rⁿ, for all multi-indicesαwith|α| ≤ m. Agreeing with the notations set for the usual Sobolev spaces, forγ ≥ 1, H_tan,γ^m Rⁿ will denote the conormal space of ordermequipped with theγ-depending norm

u²_H^m

tan,γRⁿ:

|α|≤m

γ^2m−|α|Z^αu²_L2Rⁿ, 2.7

and we again writeH_tan^m Rⁿ:H_tan,1^m Rⁿ.

For later use, we need to consider also a class of mixed tangential/conormal spaces, where different orders of tangential and conormal smoothness are allowed. Namely, for every m, r ∈ N, with m ≥ r, we let H^m,r_tanRⁿdenote the space of all functions u ∈ L²Rⁿsuch that Z^αu ∈ L²Rⁿ, whenever |α| ≤ mand 0 ≤ α1 ≤ r: here derivativesZ^α are required belonging toL²up to the orderm, but conormal derivativesnamely, derivatives involving the operatorZ1are allowed only up to the lower orderr, the remainingm−rderivatives being purely tangentiali.e, involving only differentiation with respect to tangential variables x. This space is provided with the expectedγ-depending norm

u²_H^m,r

tan,γRⁿ:

|α|≤m,0≤α1≤r

γ^2m−|α|Z^αu²_L2Rⁿ. 2.8

The notationH^m,r_tan,γRⁿis used, here and below, with the same meaning as for usual and conormal Sobolev spacesaccordingly, one hasH^m,r_tanRⁿ H^m,r_tan,1Rⁿ.

For a given Banach spaceY with norm · _Yand 1 ≤ p ≤ ∞, L^p0,∞;Ywill denote the space of theY-valued measurable functions on0,∞such that_∞

0 ut^p_Ydt <

∞.

It is easy to see that the following identities hold true forH^m,r_tanRⁿ, in the border cases r0 andrm:

H^m,0_tanRⁿ L²

0,∞;H^m Rⁿ⁻¹

, H^m,m_tan Rⁿ H_tan^m Rⁿ. 2.9

Actually all of the previously collected observations and properties ofγ-weighted norms on usual Sobolev spaces can be readily extended to the weighted norms defined on conormal and mixed spaces.

Remark 2.1. The above-considered tangential-conormal spacesH^m,r_tanRⁿcan be viewed as a conormal counterpart, by the action of the mapping introduced below, of corresponding mixed spaces of Sobolev type inRⁿ, studied in H ¨ormander’s 8.

3. Preliminaries and Technical Tools

In this section, we collect several technical tools that will be used in the subsequent analysis cf.Section 4.

We start by recalling the definition of two operatorsand , introduced by Nishitani and Takayama in 9, with the main property of mapping isometrically square integrable resp., essentially boundedfunctions over the half-spaceRⁿ onto square integrableresp., essentially boundedfunctions over the full spaceRⁿ.

The mappings : L²Rⁿ → L²Rⁿ and : L^∞Rⁿ → L^∞Rⁿ are, respectively, defined by

wx:w e^x1, x

e^x1/2, ax a e^x1, x

, ∀x x1, x

∈Rⁿ. 3.1 They are both norm preserving bijections.

It is also useful to notice that the above operators can be extended to the setDRⁿof Schwartz distributions inRⁿ. It is easily seen that bothand are topological isomorphisms of the spaceC^∞₀ Rⁿof test functions in Rⁿ resp.,C^∞Rⁿonto the space C^∞₀ Rⁿof test functions inRⁿresp.,C^∞Rⁿ. Therefore, a standard duality argument leads to defineand

onDRⁿ, by setting for everyϕ∈C^∞₀ Rⁿ

u, ϕ:u, ϕ⁻¹,

u, ϕ

:

u, ϕ 3.2

·,·is used to denote the duality pairing between distributions and test functions either in the half-spaceRⁿor the full spaceRⁿ. In the right-hand sides of3.2,⁻¹is just the inverse operator of, while the operatoris defined by

ϕx 1 x1ϕ

logx1, x

, ∀x1>0, x∈Rⁿ⁻¹, 3.3

for functionsϕ ∈ C^∞₀Rⁿ. The operators⁻¹ andarise by explicitly calculating the formal adjoints ofand , respectively.

Of course, one has thatu, u ∈ DRⁿ; moreover the following relations can be easily verifiedcf. 9:

ψu

ψ u, 3.4

∂j

u

Zju

, j 1, . . . , n, 3.5

∂1

u

Z1u1

2u, 3.6

∂j

u

Zju

, j 2, . . . , n, 3.7

wheneveru∈ DRⁿandψ∈C^∞Rⁿ in3.4u∈L²Rⁿandψ∈L^∞Rⁿare even allowed.

From formulas 3.6, 3.7 and the L²-boundedness of , it also follows that : H_tan,γ^m Rⁿ → H_γ^mRⁿis a topological isomorphism, for each integerm≥1 and realγ≥1.

Following 9 see also 2, in the next subsection the lastly mentioned property of will be exploited to shift some remarkable properties of the ordinary Sobolev spaces inRⁿto the functional framework of conormal Sobolev spaces over the half-spaceRⁿ.

In the end, we observe that the operatorcontinuously maps the spaceC^∞₀Rⁿinto the Schwartz spaceSRⁿof rapidly decreasing functions inRⁿ note also that the same is no longer true for the image ofC^∞₀Rⁿunder the operator , which is only included into the spaceC^∞_b Rⁿof infinitely smooth functions inRⁿ, with bounded derivatives of all orders.

3.1. Parameter-Depending Norms on Sobolev Spaces

We recall a classical characterization of ordinary Sobolev spaces in Rⁿ, according to H ¨ormander’s 8, based upon the uniform boundedness of a suitable family of parameter- depending norms.

For givens∈R,γ≥1 and for eachδ∈0,1a norm inH^s−1Rⁿis defined by setting

u²_s−1,γ,δ: 2π⁻ⁿ

Rⁿλ^2s,γξλ^−2,γδξ|uξ|²dξ. 3.8

According toSection 2, forγ 1 and any 0 < δ ≤ 1, we set · _s−1,δ : · _s−1,1,δ; the family of δ-weighted norms { · _s−1,δ}_0<δ≤1 was deeply studied in 8; easy arguments relying essentially on a γ-rescaling of functions lead to get the same properties for the norms { · _s−1,γ,δ}_0<δ≤1defined in3.8with an arbitraryγ≥1.

Of course, one has · _s−1,γ,1 · _s−1,γ cf. 2.4, withs−1 instead ofs. It is also clear that, for each fixedδ ∈0,1 , the norm · _s−1,γ,δ is equivalent to · _s−1,γ inH_γ^s−1Rⁿ, uniformly with respect toγ; notice, however, that the constants appearing in the equivalence inequalities will generally depend onδsee3.18.

The next characterization of Sobolev spaces readily follows by taking account of the parameterγinto the arguments used in 8, Theorem 2.4.1.

Proposition 3.1. For everys∈Randγ≥1,u∈H_γ^sRⁿif and only ifu∈H_γ^s−1Rⁿ, and the set {u_s−1,γ,δ}_0<δ≤1is bounded. In this case, one has

u_s−1,γ,δ ↑ us,γ, asδ↓0. 3.9

In order to show the regularity result stated inTheorem 1.1, it is useful to provide the conormal Sobolev spaceH_tan,γ^m−1Rⁿ,m ∈ N,γ ≥ 1, with a family of parameter-depending norms satisfying analogous properties to those of norms defined in 3.8. Nishitani and Takayama 9 introduced such norms in the “unweighted” case γ 1, just applying the ordinary Sobolev norms · _m−1,δin3.8to the pull-back of functions onRⁿ, by theoperator;

then these norms were used in 2to characterize the conormal regularity of functions.

Following 9, forγ ≥1,δ∈0,1, and allu∈H_tan^m−1Rⁿ, we set

u²_Rⁿ, m−1,tan,γ,δ:u²

m−1,γ,δ 2π⁻ⁿ

Rⁿλ^2m,γξλ^−2,γδξuξ²dξ. 3.10 Because is an isomorphism of H_tan,γ^m−1Rⁿ onto H_γ^m−1Rⁿ, the family of norms { · _Rⁿ,m−1,tan,γ,δ}_0<δ≤1keeps all the properties enjoyed by the family of norms defined in3.8.

In particular, the same characterization of ordinary Sobolev spaces onRⁿ, given by Proposition 3.1, applies also to conormal Sobolev spaces inRⁿcf. 2,9.

Proposition 3.2. For every positive integer m and γ ≥ 1, u ∈ H_tan,γ^m Rⁿ if and only if u ∈ H_tan,γ^m−1Rⁿ, and the set{u_Rⁿ,m−1,tan,γ,δ}_0<δ≤1is bounded. In this case, one has

u_Rⁿ,m−1,tan,γ,δ ↑ u_Rⁿ,m,tan,γ, asδ↓0. 3.11

As regards to the mixed space H^m,r_tanRⁿ, it is worthwhile noticing that it can be endowed with theγ-weighted norm defined, by the Fourier transformation, as

u²_Rⁿ_,m,r,tan,γ : 2π⁻ⁿ

Rⁿλ^2r,γξλ

ξ_2m−r,γuξ²dξ; 3.12

here and belowξ : ξ2, . . . , ξndenotes the Fourier dual variables of the tangential space variablesx x2, . . . , xn, and, with a slight abuse of notation, we writeλ^r,γξto mean in factλ^r,γ0, ξ.

Of course, the norm in3.12is equivalent, uniformly with respect toγ, to the norm 2.8.

3.2. A Class of Conormal Operators

Theoperator, defined at the beginning ofSection 3, can be used to allow pseudodifferential operators inRⁿ acting conormally on functions only defined over the positive half-spaceRⁿ. Then the standard machinery of pseudodifferential calculus in the parameter depending

version introduced in 10,11can be rearranged into a functional calculus properly behaved on conormal Sobolev spaces described inSection 2. InSection 4, this calculus will be usefully applied to study the conormal regularity of the stationary BVP1.2-1.3.

Let us introduce the pseudodifferential symbols, with a parameter, to be used later;

here we closely follow the terminology and notations of 12.

Definition 3.3. A parameter-depending pseudodifferential symbol of orderm∈Ris a real-or complex-valued measurable functionax, ξ, γonRⁿ×Rⁿ× 1,∞ , such thataisC^∞with respect to xand ξ, and for all multi-indices α, β ∈ Nⁿ there exists a positive constantCα,β

satisfying

∂^α_ξ∂^βxa

x, ξ, γ≤Cα,βλ^m−|α|,γξ, 3.13

for allx, ξ∈Rⁿandγ≥1.

The same definition as above extends to functionsax, ξ, γtaking values in the space R^N×N resp.,C^N×NofN×N realresp., complexvalued matrices, for all integersN >1 where the module| · |is replaced in3.13by any equivalent norm inR^N×Nresp.,C^N×N. We denote byΓ^mthe set ofγ-depending symbols of orderm ∈ Rthe same notation being used for both scalar-or matrix-valued symbols.Γ^mis equipped with the obvious norms

|a|_m,k: max

|α||^β|^≤k sup

x,ξ∈Rⁿ×Rⁿ, γ≥1λ^−m|α|,γξ∂^α_ξ∂^βxa

x, ξ, γ, ∀k∈N, 3.14

which turn it into a Fr´echet space. For allm, m∈R, withm≤m, the continuous imbedding Γ^m⊂Γ^mcan be easily proven.

For allm∈R, the functionλ^m,γ is of course ascalar-valuedsymbol inΓ^m.

To perform the analysis ofSection 4, it is important to consider the behavior of the weight functionλ^m,γ·λ^−1,γδ·, involved in the definition of the parameter-depending norms in3.8,3.10, as aγ-depending symbol according toDefinition 3.3.

In order to simplify the forthcoming statements, henceforth the following short notations will be used:

λ^m−1,γ_δ ξ:λ^m,γξλ^−1,γδξ, λ^−m1,γ_δ ξ:

λ^m−1,γ_δ ξ₋₁

λ^−m,γξλ^1,γδξ

, 3.15

for all real numbersm∈ R,γ ≥ 1, andδ ∈0,1. One has the obvious identitiesλ^m−1,γ₁ ξ ≡ λ^m−1,γξ,λ^−m1,γ₁ ξ≡ λ^−m1,γ₁ ξ ≡ λ^−m1,γξ. However, to avoid confusion in the following, it is worthwhile to remark that functionsλ^−m1,γ_δ ξandλ^−m1,γ_δ ξare no longer the same as soon asδbecomes strictly smaller than 1; indeed3.15givesλ^−m1,γ_δ ξ λ^−m2,γξλ^−1,γδξ.

A straightforward application of Leibniz’s rule leads to the following result.

Lemma 3.4. For everym∈Rand allα∈Nⁿ, there exists a positive constantCm,αsuch that

∂^α_ξλ^m−1,γ_δ ξ≤Cm,αλ^{m−1−|α|,γ}_δ ξ, ∀ξ∈Rⁿ, ∀γ≥1, ∀δ∈ 0,1. 3.16 Because of estimates3.16,λ^m−1,γ_δ ξcan be regarded as aγ-depending symbol, in two different ways. On one hand, combining estimates3.16with the trivial inequality

λ^−1,γδξ≤1 3.17

immediately gives that{λ^m−1,γ_δ }_0<δ≤1is a bounded subset ofΓ^m. On the other hand, the left inequality in

δλ^1,γξ≤λ^1,γδξ≤λ^1,γξ, ∀ξ∈Rⁿ, ∀δ∈ 0,1, 3.18 together with3.16, also gives

∂^α_ξλ^m−1,γ_δ ξ≤Cm,αδ⁻¹λ^{m−1−|α|,γ}ξ, ∀ξ∈Rⁿ, ∀γ≥1. 3.19

According toDefinition 3.3,3.19means thatλ^m−1,γ_δ actually belongs, for each fixedδ, toΓ^m−1; nevertheless, the family{λ^m−1,γ_δ }_0<δ≤1is unbounded as a subset ofΓ^m−1.

For later use, we also need to study the behavior of functionsλ^−m1,γ_δ asγ-depending symbols.

Analogously toLemma 3.4, one can prove the following result.

Lemma 3.5. For allm∈Randα∈Nⁿ, there existsCm,α>0 such that

∂^α_ξλ^−m1,γ_δ ξ≤Cm,αλ^{−m1−|α|,γ}_δ ξ, ∀ξ∈Rⁿ, ∀γ≥1, ∀δ∈0,1. 3.20

In particular,Lemma 3.5implies that the family{λ^−m1,γ_δ }_0<δ≤1 is a bounded subset of Γ^−m1it suffices to combine3.20with the right inequality in3.18.

Any symbolaax, ξ, γ∈Γ^mdefines a pseudodifferential operator Op^γa ax, D, γ on the Schwartz spaceSRⁿ, by the standard formula

∀u∈ SRⁿ, ∀x∈Rⁿ, Op^γaux a x, D, γ

ux :2π⁻ⁿ

Rⁿe^ix·ξa x, ξ, γ

uξdξ,

3.21

where, of course, we denotex·ξ : _n

j1xjξj.ais called the symbol of the operator3.21, andmis its order. It comes from the classical theory that Op^γadefines a linear-bounded operator

Op^γa:SRⁿ−→ SRⁿ; 3.22

moreover, the latter extends to a linear-bounded operator on the spaceSRⁿof tempered distributions inRⁿ.

An exhaustive account of the symbolic calculus for pseudodifferential operators with symbols inΓ^mcan be found in 11 see also 12. Here, we just recall the following result, concerning the product and the commutator of two pseudodifferential operators.

Proposition 3.6. Leta∈ Γ^mandb∈Γ^l, forl, m∈R. Then Op^γaOp^γbis a pseudodifferential operator with symbol inΓ^ml; moreover, if one letsa#bdenote the symbol of the product, one has for every integerN≥1

a#b−

|α|<N

−i^|α|

α! ∂^α_ξa∂^α_xb∈Γ^ml−N. 3.23

Under the same assumptions, the commutator Op^γa,Op^γb:Op^γaOp^γb−Op^γbOp^γa is again a pseudodifferential operator with symbolc∈Γ^ml. If one further assumes that one of the two symbolsaorbis scalar-valued (so thataandbcommute in the pointwise product), then the symbolc of Op^γa,Op^γbhas orderml−1.

We point out that when the symbolb∈Γ^lof the preceding statement does not depend on thexvariablesi.e.,bbξ, γ, then the symbola#bof the product Op^γaOp^γbreduces to the pointwise product of symbolsa andb; in this case, the asymptotic formula3.23is replaced by the exact formula

a#b x, ξ, γ

a x, ξ, γ

b ξ, γ

. 3.24

According to3.15,3.21, we write

λ^m−1,γ_δ D:Op^γ λ^m−1,γ_δ

, λ^−m1,γ_δ D:Op^γ

λ^−m1,γ_δ

. 3.25

In view of3.15and3.24, the operatorλ^m−1,γ_δ Dis invertible, and its two-sided inverse is given byλ^−m1,γ_δ D.

Starting from the symbolic classesΓ^m,m∈R, we introduce now the class of conormal operators inRⁿ, to be used in the sequel.

Letax, ξ, γbe aγ-depending symbol inΓ^m,m∈R. The conormal operator with symbol a, denoted by Op^γa or equivalentlyax, Z, γ, is defined by setting

∀u∈C^∞₀Rⁿ,

Op^γau

Op^γa u

. 3.26

In other words, the operator Op^γais the composition of mappings

Op^γa ⁻¹◦Op^γa◦. 3.27

As we already noted,u ∈ SRⁿwheneveru∈C^∞₀Rⁿ; hence formula3.26makes sense and gives that Op^γauis aC^∞-function inRⁿ. Also Op^γa:C^∞₀Rⁿ → C^∞Rⁿis a linear- bounded operator that extends to a linear-bounded operator from the space of distributions u ∈ DRⁿsatisfyingu ∈ SRⁿintoDRⁿitself.In principle, Op^γacould be defined by3.26over all functionsu∈C^∞Rⁿ, such thatu ∈ SRⁿ. Then Op^γadefines a linear- bounded operator on the latter function space, provided that it is equipped with the topology induced, via, from the Fr´echet topology ofSRⁿ.Throughout the paper, we continue to denote this extension by Op^γa orax, Z, γequivalently.

As an immediate consequence of3.27, we have that for all symbolsa∈ Γ^m,b∈ Γ^l, withm, l∈R, there holds

∀u∈C^∞₀Rⁿ, Op^γaOp^γbu

Op^γaOp^γb u⁻¹

. 3.28

Then, it is clear that a functional calculus of conormal operators can be straightforwardly borrowed from the corresponding pseudodifferential calculus inRⁿ; in particular we find that products and commutators of conormal operators are still operators of the same type, and their symbols are computed according to the rules collected inProposition 3.6.

Below, let us consider the main examples of conormal operators that will be met in Section 4.

As a first example, we quote the multiplication by a matrix-valued function B ∈ C^∞₀Rⁿ. It is clear that this makes an operator of order zero according to3.26; indeed3.4 gives for any vector-valuedu∈C^∞₀Rⁿ

Bux Bxux, 3.29

andB is aC^∞-function inRⁿ, with bounded derivatives of any order, hence a symbol inΓ⁰. We remark that, when computed forB, the norm of orderk ∈N, defined on symbols by3.14, just reduces to

B

0,kmax

|α|≤k

∂^αB

L^∞Rⁿmax

|α|≤kZ^αB_L∞Rⁿ, 3.30 where the second identity above exploits formulas3.5and that maps isometricallyL^∞Rⁿ ontoL^∞Rⁿ.

Now, letL : γIN_n

j1AjxZj be a first-order linear partial differential operator, with matrix-valued coefficientsAj∈C^∞₀Rⁿforj1, . . . , nandγ≥1. Since the leading part ofLonly involves conormal derivatives, applying3.4,3.6, and3.7then gives

⎛

⎝γuⁿ

j1

AjZju

⎞

⎠

γI−1

2A₁

uⁿ

j1

A_j∂juOp^γau, 3.31

wherea ax, ξ, γ : γIN −1/2A₁x i_n

j1A_jxξj is a symbol in Γ¹. ThenLis a conormal operator of order 1, according to3.26.

InSection 4, we will be mainly interested in the family of conormal operators

λ^m−1,γ_δ Z:Op^γ λ^m−1,γ_δ

, λ^−m1,γ_δ Z:Op^γ

λ^−m1,γ_δ

. 3.32

The operatorsλ^m−1,γ_δ Zare involved in the characterization of conormal regularity provided byProposition 3.2remember that, afterLemma 3.4,λ^m−1,γ_δ ∈Γ^m−1. Indeed, from Plancherel’s formula and the fact that the operatorpreserves theL²-norm, the following identities

u_Rⁿ,m−1,tan,γ,δ≡λ^m−1,γ_δ Zu

L²Rⁿ 3.33

can be straightforwardly established; hence,Proposition 3.2can be restated in terms of the boundedness, with respect toδ, of theL²-norms of functionsλ^m−1,γ_δ Zu. This observation is the key point that leads to the analysis performed inSection 4.

Another main feature of the conormal operators 3.32 is that λ^−m1,γ_δ Z provides a two-sided inverse of λ^m−1,γ_δ Z; this comes at once from the analogous property of the operators in3.25and formulas3.26,3.28.

3.3. Sobolev Continuity of Conormal Operators

Proposition 3.7. Ifs, m∈Rthen for alla∈Γ^mthe pseudodifferential operator Op^γaextends as a linear-bounded operator fromH_γ^smRⁿintoH_γ^sRⁿ, and the operator norm of such an extension is uniformly bounded with respect toγ.

We refer the reader to 11for a detailed proof ofProposition 3.7. A thorough analysis shows that the norm of Op^γa, as a linear-bounded operator from H_γ^smRⁿ to H_γ^sRⁿ, actually depends only on a norm of type3.14of the symbola, besides the Sobolev order s and the symbolic order m cf. 11 for detailed calculations. This observation entails, in particular, that the operator norm is uniformly bounded with respect to γ and other additional parameters from which the symbol of the operator might possibly depend, as a bounded mapping.

From the Sobolev continuity of pseudodifferential operators quoted above, and using that the operator maps isomorphically conormal Sobolev spaces in Rⁿ onto ordinary Sobolev spaces inRⁿ, we easily derive the following result.

Proposition 3.8. If m ∈ Zand a ∈ Γ^m, then the conormal operator Op^γaextends to a linear- bounded operator from H_tan,γ^smRⁿ to H_tan,γ^s Rⁿ, for every integers ≥ 0, such that sm ≥ 0;

moreover the operator norm of such an extension is uniformly bounded with respect toγ.

Remark 3.9. We point out that the statement above only deals with integer orders of symbols and conormal Sobolev spaces. The reason is that, inSection 2, conormal Sobolev spaces were only defined for positive integer orders. In principle, this lack could be removed by extending the definition of conormal spacesH_tan^s Rⁿto any real order s: this could be trivially done, just definingH_tan^s Rⁿto be the pull-back, by the operator, of functions inH^sRⁿ. However, this extension to fractional exponents seems to be useless for the subsequent developments.

As regards to the action of conormal operators on the mixed spacesH^s,r_tan,γRⁿ, similar arguments to those used in the proof ofProposition 3.8lead to the following.

Proposition 3.10. Leta ax, ξ, γbe a symbol inΓ^m, form ∈Z. Then for all integersr, s ∈N, such thats≥r,s >0, andrm≥0, Op^γa ax, Z, γextends by continuity to a linear-bounded operator

Op^γa:H^sm,rm_tan,γ Rⁿ−→ H^s,r_tan,γRⁿ. 3.34

Moreover, the operator norm of such an extension is uniformly bounded with respect toγ.

4. Proof of Theorem 1.1

This section is entirely devoted to the proof ofTheorem 1.1.

4.1. The Strategy of the Proof:

Comparison with the Strongly Well-Posed Case

As it was already pointed out in the Introduction, in order to solve the BVP1.2-1.3 in L²,Theorem 1.1requires an additional tangential/conormal regularity of the corresponding data. The precise increase of regularity needed for the data is prescribed by the energy inequality1.7: to estimate theL²-norm of the solution, in the interior and on the boundary of the domain, we loserconormal derivatives ands−rtangential derivatives with respect to the interior source termF, andstangentialderivatives with respect to the boundary datum G.

In 2, the conormal regularity of weak solutions to the BVP1.2-1.3was studied, under the assumption that no loss of derivatives occurs in the estimate of the solution by the data. To prove the result of 2, Theorem 15, the solutionuto1.2-1.3was regularized by a family of tangential mollifiersJε, 0 < ε < 1, defined by Nishitani and Takayama in 9as a suitable combination of the operatorand the standard Friedrichs’mollifiers. The essential point of the analysis performed in 2was to notice that the mollified solutionJεusolves the same problem1.2-1.3, as the original solutionu. The data of the problem forJεucome from the regularization, byJε, of the data involved in the original problem foru; furthermore, an additional term Jε, Lu, where Jε, Lis the commutator between the differential operatorL and the tangential mollifierJε, appears into the equation satisfied byJεu. Because the energy estimate associated to a strongL²-well-posed problem does not lose derivatives, actually this term can be incorporated into the source term of the equation satisfied byJεu.

In the case ofTheorem 1.1, where the a priori estimate1.7exhibits a finite loss of regularity with respect to the data, this technique seems to be unsuccessful, since Jε, Lu cannot be absorbed into the right-hand side without losing derivatives on the solutionu; on the other hand, it seems that the same term cannot be merely reduced to a lower-order term involving the smoothed solutionJεu, as wellthis should require a sharp control of the error termu−Jεu.

These observations lead to develop another technique, where the tangential mollifier Jεis replaced by the family of operators3.32, involved in the characterization of regularity given by Proposition 3.2. Instead of studying the problem satisfied by the smoothed

solutionJεu, here we consider the problem satisfied byλ^m−1,γ_δ Zu. As before, a new term λ^m−1,γ_δ Z, Lu appears which takes account of the commutator between the differential operatorLand the conormal operatorλ^m−1,γ_δ Z. Since we assume the weak well-posedness of the BVP 1.2-1.3 to be preserved under lower order terms, the approach consists of treating the commutator λ^m−1,γ_δ Z, Luas a lower-order term within the interior equation forλ^m−1,γ_δ Zusee4.10 differently from the stronglyL²-well-posed case studied in 2, the stability of problem 1.2-1.3 under lower-order perturbations is no longer a trivial consequence of the well-posedness itself. In Theorem 1.1, this stability is required as an additional hypothesis about the BVP; this is made possible by taking advantage from the invertibility of the operatorλ^m−1,γ_δ Z.

We argue by induction on the integer order m ≥ 1. Let us take arbitrary data F ∈ H^sm,rm_tan,γ Rⁿ,G∈H_γ^smRⁿ⁻¹, and fix an arbitrary matrix-valued functionB∈C^∞₀Rⁿ as the lower order term in the interior equation1.2.

Because of the inductive hypothesis, we already know that the uniqueL²-solutionuto 1.2-1.3actually belongs toH_tan,γ^m−1Rⁿandu^I_|x10belongs toH_γ^m−1Rⁿ⁻¹, provided thatγis taken to be larger than a certain constantγ_m−1≥1; moreover the solutionuobeys the estimate 1.8of orderm−1

γu²_Hm−1

tan,γRⁿu^I_|x10²

Hγ^m−1Rn−1

≤C_m−1 1

γ^2s1F²_Hsm−1,rm−1

tan,γ Rⁿ 1

γ^2sG²_Hsm−1

γ ^Rn−1

,

4.1

where the positive constantCm−1, as well asγm−1, only depends on the smoothness orderm and theL^∞-norm of a finite numberdepending onmitselfof conormal derivatives ofBcf.

3.30, besides the coefficientsAj 1≤j≤nofLand the integer numbersrands.

In order to increase the conormal regularity of the solutionuby order one, we are going to act onuby the conormal operatorλ^m−1,γ_δ Z; then we consider the analogue of the original problem1.2-1.3satisfied byλ^m−1,γ_δ Zu.

4.2. A Modified Version of the Conormal Operatorλ^m−1,γ_δ Z

Due to some technical reasons that will be clarified inSection 4.3, we need to slightly modify the conormal operatorλ^m−1,γ_δ Zto be applied to the solutionuof the original BVP1.2-1.3, as was described in the preceding section.

The first step is to decompose the weight function λ^m−1,γ_δ as the sum of two contributions. To do so, we proceed as follows. Firstly, we take an arbitrary positive, even functionχ∈C^∞Rⁿwith the following properties:

0≤χx≤1, ∀x∈Rⁿ, χx≡1, for |x| ≤ 1

2, χx≡0, for|x|>1. 4.2