On the Inverse Source Identification Problem in L ∞ $L^{\infty }$ for Fully Nonlinear Elliptic PDE

Vietnam Journal of Mathematics (2021) 49:815–829 https://doi.org/10.1007/s10013-021-00515-6 ORIGINAL ARTICLE On the Inverse Source Identification Problem in L ∞ for Fully Nonlinear Elliptic PDE Birzhan Ayanbayev1 · Nikos Katzourakis2 Received: 20 May 2020 / Accepted: 13 April 2021 / Published online: 22 July 2021 © The Author(s) 2021 Abstract In this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the L∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals. Keywords Regularisation strategy · Tykhonov regularisation · Inverse source identification problem · Fully nonlinear elliptic equations · Calculus of Variations in L∞ Mathematics Subject Classification (2010) 35R25 · 35R30 · 35J60 · 35J70 1 Introduction Let n, k ∈ N with k, n ≥ 2 and let Ω ⊆ Rn be a bounded connected domain with C 1,1 regular boundary ∂Ω. Let also ⊗2 F : Ω × R × Rn × Rns −→ R be a Carathéodory function, namely x  → F (x, r, p, X) is Lebesgue measurable for all ⊗2 (r, p, X) ∈ R × Rn × Rns and (r, p, X)  → F (x, r, p, X) is continuous for a.e. x ∈ Ω. Dedicated to Enrique Zuazua on the occasion of his 60th birthday.  Nikos Katzourakis Birzhan Ayanbayev 1 Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany 2 Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK 816 B. Ayanbayev, N. Katzourakis ⊗k In this paper, the notation Rns stands for the vector space of fully symmetric k-th order tensors in Rn ⊗ · · · ⊗ Rn (k-times). Given g ∈ W 2,∞ (Ω), consider the Dirichlet problem  F[u] = f in Ω, (1.1) u=g on ∂Ω, for some appropriate source f : Ω −→ R. Here F[u] denotes the induced fully nonlinear 2nd order differential operator, defined on smooth functions u as F[u] := F (·, u, Du, D2 u). (1.2) Evidently, we are employing the standard symbolisations Du = (Di u)i=1...n , D2 u = (D2ij u)i,j =1...n and Di ≡ ∂/∂xi . The above direct Dirichlet problem for F asks to determine u, given a source f and boundary data g. (In fact the source f is obsolete and can be absorbed into F , but for the problem we are interested in this paper it is more convenient to write it in this separated form). This is a semi-classical problem which is essentially standard material, see e.g. [22]. In particular, it is known that under various sets of assumptions on F that (1.1) is well-posed and, given f ∈ L∞ (Ω) and g ∈ W 2,∞ (Ω), for any p > n there exists a unique solution u in the locally convex (Fréchet) space    1,p Wg2,∞ (Ω) := W 2,p ∩ Wg (Ω). 1<p<∞ In general, the solution u is not in the smaller space (W 2,∞ ∩ Wg1,∞ )(Ω) (not even locally), due to the failure of the W 2,p estimates for p = ∞, which happens even in the linear case (see e.g. [21]). Additionally, (1.1) satisfies for any p > n the fully nonlinear Lp global estimate F (·, v, Dv, D2 v) Lp (Ω) ≥ C1 v W 2,p (Ω) − C1 g W 2,p (Ω) − C2 (1.3) for some constants C1 , C2 > 0 depending only on the parameters and any v ∈ (W 2,p ∩ 1,p Wg )(Ω). For sufficient conditions on F which guarantee the satisfaction of solvability of (1.1) in the strong sense and of the uniform estimate (1.3) we refer to [12, 13, 16, 28, 30]. Note that the above problem contains as a special case the archetypal instance of divergence operators with C 1 matrix coefficient A, as well as the non-divergence linear case with continuous coefficient:  L1 [u] = div(ADu) + b · Du + cu, (1.4) L2 [u] = A : D2 u + b · Du + cu. In the above, the notations “:” and “·” symbolise the Euclidean inner products in the space ⊗2 of symmetric matrices Rns and in Rn respectively. More generally, the inner product of ⊗k two tensors T , S ∈ Rns will also be denoted by “:”, that is  Ta1 ···ak Sa1 ···ak . T : S := 1≤a1 ,...,ak ≤n The inverse problem relating to (1.1) asks the question of perhaps determining f , given the boundary data g and some other partial information on the solution u, typically some approximate experimental measurements of some function of it known only up to some error. The inverse problem is severely ill-posed even in the linear case of the Laplacian operator F = Δ, as the noisy data measured on a subset of Ω might either not be compatible with any exact solution, or they may not suffice to determine a unique f even if compatibility holds true. On the Inverse Source Identification Problem in L∞... 817 The above type of inverse problems are especially crucial for various applications, even in the model case of the Poisson equation, see e.g. [1, 8, 17, 23, 31, 32, 34, 36–40]. In this paper we will assume that the approximate information on u takes the form on K, K[u] = k γ (1.5) where K is an observation operator, taken to be a first order fully nonlinear differential operator of the form K[u] := K(·, u, Du), (1.6) where K and its partial derivates Kr , Kp satisfy K, Kr ∈ C(K × R × Rn ), Kp ∈ C(K × R × Rn ; Rn ). (1.7) In (1.5) and (1.7), K symbolises the set on which we take measurements, which will be assumed to satisfy K ⊆ Ω is compact and exists κ ∈ [0, n] : Hκ (K) < ∞. (1.8) In the above, Hκ denotes the Hausdorff measure of dimension κ. Our general measure and functional notation will be either standard or self-explanatory, e.g. as in [15, 18, 27]. Finally, k γ ∈ L∞ (K, Hκ ) is the function of approximate (deterministic) measurements taken on K, at noise level at most γ > 0: kγ − k0 L∞ (K,Hκ ) ≤ γ, (1.9) where k 0 = K[u0 ] corresponds to ideal error-free measurements of an exact solution to (1.1) with source f = F[u0 ]. To recapitulate, in this paper we study the following ill-posed inverse source identification problem for fully nonlinear elliptic PDEs: ⎧ in Ω, ⎨ F[u] = f u=g on ∂Ω, (1.10) ⎩ on K. K[u] = k γ This means that we are searching for a selection process of a suitable approximation for f from the data k γ on K through the observation K[u] of the solution u. To the best of our knowledge, (1.10) has not been studied before, at least in this generality. Our approach does not exclude the extreme cases of K = Ω (full information) or of K = ∅ (no information), although trivial changes are required in the proofs. Sadly, an exact solution may not exist as the constraint may be incompatible with the solution of (1.1), owing to the errors in measurements. On the other hand, it is not possible to have a uniquely determined source on the constraint-free region Ω \ K. Instead, our goal is a strategy to determine an optimally fitting uγ (and respective source f γ : (...truncated)