New global logarithmic stability of the Cauchy problem for elliptic equations
aa r X i v : . [ m a t h . A P ] M a r NEW GLOBAL LOGARITHMIC STABILITY OF THE CAUCHYPROBLEM FOR ELLIPTIC EQUATIONS
MOURAD CHOULLI
Abstract.
In this short paper we prove a global logarithmic stability of theCauchy problem for H -solutions of an anisotropic elliptic equation in a Lip-schitz domain. The result we obtained is based on tools borrowed from theexisting technics to establish stability estimate for the Cauchy problem [4] (seealso [1]) combined with tools we already used in [6] to study an inverse mediumproblem. Throughout this text, Ω is a Lipschitz bounded domain of R n , n ě
2, and Γ is anonempty open subset of B Ω. Consider then the divergence form elliptic operator L that acts as follows Lu p x q “ div p A p x q ∇ u p x qq , where A “ p a ij q is a symmetric matrix, with coefficients in W , p Ω q , so that thereexist κ ą λ ě λ ´ | ξ | ď A p x q ξ ¨ ξ ď λ | ξ | , x P Ω , ξ P R n , and(2) n ÿ k “ ˇˇˇˇˇ n ÿ i,j “ B k a ij p x q ξ i ξ j ˇˇˇˇˇ ď κ | ξ | , x P Ω , ξ P R n . The Cauchy problem we consider can be stated as follows: given p F, f, g q P L p Ω q ˆ L p Γ q ˆ L p Γ q n , find u P H p Ω q obeying to the boundary value problem(3) $&% Lu p x q “ F p x q a.e. in Ω .u p x q “ f a.e. on Γ , ∇ u p x q “ g a.e. on Γ . It is well known that this problem may not have a solution and, according tothe classical uniqueness of continuation from Cauchy data, the boundary valueproblem (3) has at most one solution. Moreover, even if the solution of (3) exists,the continuous dependence of the solution on the data p F, f, g q is not in generalLipschitz. In other words, the Cauchy problem is ill-posed in Hadamard’s sense. Asit is shown by Hadamard [8], the modulus of continuity of the mapping p F, f, g q ÞÑ u can be of logarithmic type. Therefore, for the general Cauchy problem, thelogarithmic type stability estimate is the best possible one that we can expect.We aim here to prove the following result. Theorem 1.
Let ă s ă . Then there exist two constants c ą and C ą ,only depending on s , Ω , Γ , λ and κ , and δ only depending on Ω , so that, for any The author is supported by the grant ANR-17-CE40-0029 of the French National ResearchAgency ANR (project MultiOnde). u P H p Ω q , ă δ ă δ and j “ , , we have C } u } H j p Ω q ď δ sj ` } u } H j ` p Ω q (4) ` e e c { δ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . As usual, the interpolation inequality (4) yields a double logarithmic stabilityestimate. Precisely, we have the following corollary in whichΨ cs,j p ρ q “ " p ln ln ρ q ´ sj ` if ρ ą c,ρ if 0 ă ρ ă c, j “ , , extended by continuity at ρ “ cs,j p q “
0, where c ą e . Corollary 1.
Let ă s ă . Then there exist two constants c ą e and C ą ,only depending on s , Ω , Γ , λ and κ so that, for any u P H p Ω q , u ‰ and j “ , ,we have C } u } H j p Ω q ď } u } H j ` p Ω q Ψ cs,j ˆ } u } H j ` p Ω q } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˙ . As we observed above, according to the classical uniqueness of continuation fromCauchy data for elliptic equations, if u P H p Ω q satisfies Lu “ u “ ∇ u “ u “ C ,α -solutionsand (ii) C , domain and H -solutions. This optimal stability estimate is of singlelogarithmic type. For the case (i), we refer to [4] under an additional geometriccondition on the domain. This condition was removed in [1] (see also [5]). A similarresult was obtained in [3] for the Laplace operator. The case (ii) was establishedin [2] for the Laplace operator. However the results in [2] can be extended to ananisotropic elliptic operator in divergence form. In the present paper we deal withthe case of Lipschitz domain and H -solutions. For this case we are only able toget a stability estimate of double logarithmic type (Corollary 1). We do not knowwhether this result can be improved to a single logarithmic type.Let us explain briefly the main steps to obtain the global stability estimate forthe Cauchy problem. The first step consists in continuing a well chosen interiordata to the boundary. In the second step we continue the data from an interiorsubdomain to another subdomain. The continuation of the Cauchy data to someinterior subdomain constitute the third step. For the last two steps it is sufficient toassume that the domain is Lipschitz and the solutions have H -regularity. Whilein the first step, it is necessary to assume that either the domain is C , or thesolutions have C ,α -regularity. Apart from these two cases we do not know how toprove the continuation result in the first step. It is worth mentioning that the lasttwo steps give rise to a stability estimate of Hölder type and for the first step thestability estimate we obtain is of logarithmic type.Since we can not use this classical scheme to prove Theorem 1, we modify itslightly to avoid the use of the first step. The main idea consists in refining thesecond step. Precisely, we show that we can continue the data, away from theboundary, from a ball with arbitrary small radius to another ball with the sameradius, with an exact dependence of the constants on the radius. This new stepyields a stability estimate of double logarithmic type. It turn out that this result isoptimal if one uses to prove it three-ball inequalities. For this reason we think that LLIPTIC CAUCHY PROBLEM 3 the actual technics, based on three-ball inequalities, can not be used to improveTheorem 1.As we already mentioned, the proof of Theorem 1 consists in an adaptation ofexisting results. The proposition hereafter is proved in [4] under an additionalgeometric condition and for a Lipschitz domain in [1] (see also [5]).Henceforward, C is a generic constant only depending on Ω, λ and κ , while C is a generic constant only depending Ω, Γ, λ and κ . Proposition 1.
There exist a constant γ ą and a ball B in R n satisfying B X Ω ‰H , B X p R n z Ω q ‰ H and B X B Ω Ť Γ , only depending on Ω , Γ , λ and κ , so that,for any u P H p Ω q and ǫ ą , we have (5) C } u } H p B X Ω q ď ǫ γ } u } H p Ω q ` ǫ ´ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . Proof of Theorem 1.
Let B as in the preceding proposition. Pick then ˜ x P B X B
Ω.As B X Ω is Lipschitz, it contains a cone with vertex at ˜ x . That is we can find R ą θ Ps , π { r and ξ P S n ´ so that C p ˜ x q “ t x P R n ; 0 ă | x ´ ˜ x | ă R, p x ´ ˜ x q ¨ ξ ą | x ´ ˜ x | cos θ u Ă B X Ω . Let x δ “ ˜ x ` δ θ ξ , with δ ă R sin θ . Then dist p x δ , Bp B X Ω qq ą δ .Define, for δ ą
0, Ω δ “ t x P Ω; dist p x, B Ω q ą δ u , Ω δ “ t x P Ω; dist p x, B Ω q ă δ u and set δ ˚ “ sup t δ ą
0; Ω δ ‰ Hu . Let 0 ă δ ď δ ˚ {
3. Then a slight modification of the proof of [6, Theorem 2.1,step 1] yields, for any u P H p Ω q , y, y P Ω δ and ǫ ą C } u } L p B p y,δ qq ď ǫ ´ ψ p δ q } u } L p Ω q ` ǫ ´ ψ p δ q ` } Lu } L p Ω q ` } u } L p B p y ,δ qq ˘ . Here ψ is of the form ψ p δ q “ se ´ C { δ , with 0 ă s ă λ and κ .Putting together (5) and (6) with y “ x δ , we find, for any u P H p Ω q , y P Ω δ ,0 ă δ ă δ : “ min ´ δ ˚ , R sin θ ¯ , ǫ ą η ą C } u } L p B p y,δ qq ď ǫ ´ ψ p δ q } u } L p Ω q (7) ` ǫ ´ ψ p δ q “ } Lu } L p Ω q ` η γ } u } H p Ω q ` η ´ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘‰ . In (7), take η “ ǫ γψ p δ qp ´ ψ p δ qq in order to obtain C } u } L p B p y,δ qq ď φ p ǫ, δ q} u } H p Ω q (8) ` φ p ǫ, δ q ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ , where φ p ǫ, δ q “ ǫ ´ ψ p δ q ,φ p ǫ, δ q “ ǫ ´ ψ p δ q max ´ , ǫ ´ γψ p δ qp ´ ψ p δ qq ¯ . MOURAD CHOULLI
On the other hand, it is straightforward to check that Ω δ can be recovered byat most k n balls with center in Ω δ and radius δ , where k “ r c { δ s , the constant c only depends on n and the diameter of Ω. Whence we obtain from (8) C } u } L p Ω δ q ď δ ´ n φ p ǫ, δ q} u } H p Ω q (9) ` δ ´ n φ p ǫ, δ q ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . Now, according to Hardy’s inequality (see for instance [7, Theorem 1.4.4.4, page29]), for 0 ă s ă , there exists κ , only depending on Ω and s so that } u } L p Ω δ q ď p δ q s ›››› u dist p x, B Ω q s ›››› L p Ω q ď κ δ s } u } H s p Ω q . As H p Ω q is continuously embedded in H s p Ω q , changing if necessary κ , we have(10) } u } L p Ω δ q ď κ δ s } u } H p Ω q . Henceforward, 0 ă s ă is fixed and C is a generic constant that only dependson Ω, Γ, λ , κ and s .Putting together (9) and (10), we get C } u } L p Ω q ď ` δ ´ n φ p ǫ, δ q ` δ s ˘ } u } H p Ω q (11) ` δ ´ n φ p ǫ, δ q ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . We take in (11) ǫ so that δ ´ n φ p ǫ, δ q “ δ s or equivalently ǫ “ δ p n ` s q ψ p δ q . Thenelementary computations yield C } u } L p Ω q ď δ s } u } H p Ω q (12) ` e e c { δ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . This inequality corresponds to (4) when j “ H p Ω q can be seen as an interpolated space between L p Ω q and H p Ω q , we have pour ǫ ą C Ω } u } H p Ω q ď ǫ } u } H p Ω q ` ǫ ´ } u } L p Ω q , the constant C Ω only depends on Ω.This last inequality with ǫ “ δ s { and (12) entail C } u } H p Ω q ď δ s { } u } H p Ω q ` e e c { δ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . That is we proved (4) in the case j “ (cid:3) Remark 1. (i) It is worth mentioning that Theorem 1 still holds if L is substitutedby L ` L , where L is a first order partial differential operator with boundedcoefficients. In that case the constants c and C in the statement of Theorem 1 mayalso depend on bounds on the coefficients of L .(ii) For 0 ă t ă
2, we have the following interpolation inequality, with u P H p Ω q and ǫ ą C Ω } u } H t p Ω q ď ǫ t ´ t } u } H p Ω q ` ǫ ´ } u } L p Ω q . LLIPTIC CAUCHY PROBLEM 5
We can then proceed as in the preceding proof in order to get, for u P H p Ω q and0 ă δ ă δ , C } u } H t p Ω q ď δ st } u } H p Ω q ` e e c { δ ` } u } L p Γ q ` } ∇ u } L p Γ q ` } Lu } L p Ω q ˘ . Here the constants c and C only depend on s , t , Ω, Γ, λ and κ , and δ only dependson Ω. References [1] M. Bellassoued and M. Choulli, Global logarithmic stability of the Cauchy problem foranisotropic wave equations, arXiv:1902.05878.[2] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: thecase of C , domains, M2AN Math. Model. Numer. Anal. 44 (2010), no. 4, 715-735.[3] L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchyproblems: the case of Lipschitz domains, Appl. Anal. 89 (2010), no. 11, 1745-1768.[4] M. Choulli, Applications of elliptic Carleman inequalities to Cauchy and inverse problems,SpringerBriefs in Mathematics. BCAM SpringerBriefs. Springer, [Cham]; BCAM Basque Cen-ter for Applied Mathematics, Bilbao, 2016. ix+81 pp.[5] M. Choulli, An introduction to the analysis of elliptic partial differential equations, bookunder review.[6] M. Choulli and F. Triki, Hölder stability for an inverse medium problem with internal data,Res. Math. Sci. 6 (2019), no. 1, Paper No. 9, 15 pp.[7] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathe-matics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp.[8] J. Hadamard, Lectures in Cauchy’s problem in linear partial differential equations, YaleUniversity Press, New Haven, 1923. Université de Lorraine, 34 cours Léopold, 54052 Nancy cedex, France
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