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

In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 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 $L^2$"viscosity term"for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.


Introduction
Let n, k ∈ N with k, n ≥ 2 and let Ω ⊆ R n be a bounded connected domain with C 1,1 regular boundary ∂Ω. Let also F : Ω × R × R n × R n ⊗2 s −→ R be a Carathéodory function, namely x → F (x, r, p, X) is Lebesgue measurable for all (r, p, X) ∈ R × R n × R n ⊗2 s and (r, p, X) → F (x, r, p, X) is continuous for a.e. x ∈ Ω. In this paper, the notation R n ⊗k s stands for the vector space of fully symmetric k-th order tensors in R n ⊗ · · · ⊗ R n (k-times). Given g ∈ W 2,∞ (Ω), consider the Dirichlet problem Evidently, we are employing the standard symbolisations Du = (D i u) i=1...n , D 2 u = (D 2 ij u) i,j=1...n and D i ≡ ∂/∂x i . 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above direct Dirichlet problem for F asks to determine u, given a source f and boundary data g. (In fact the source f is obsolete and can be absorbed into F , but for the problem we are interested in this paper it is more convenient to write it in this separated form).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 In general, the solution u is not in the smaller space (W 2,∞ ∩ W 1,∞ g )(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 L p global estimate for some constants C 1 , C 2 > 0 depending only on the parameters and any v ∈ W 2,p ∩ W 1,p g (Ω). For sufficient conditions on F which guarantee the satisfaction of solvability of (1.14) 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: In the above, the notations ":" and "·" symbolise the Euclidean inner products in the space of symmetric matrices R n ⊗2 s and in R n respectively. More generally, the inner product of two tensors T, S ∈ R n ⊗k s will also be denoted by ":", that is 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 = ∆, 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.
The above type of inverse problems are especially crucial for various applications, even in the model case of the Poisson equation, see e.g. [1,8,17,23,31,32,34,36,37,38,39,40]. In this paper we will assume that the approximate information on u takes the form where K is an observation operator, taken to be a first order fully nonlinear differential operator of the form where K and its partial derivates K r , K p satisfy In (1.5) and (1.7), K symbolises the set on which we take measurements, which will be assumed to satisfy In the above, H κ 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, H κ ) is the function of approximate (deterministic) measurements taken on K, at noise level at most γ > 0: To recapitulate, in this paper we study the following ill-posed inverse source identification problem for fully nonlinear elliptic PDEs: 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 u γ (and respective source f γ := F[u γ ]) to the ill-posed problem (1.10). A popular choice of operator K in the literature (when L = ∆) consist of some term of the separation of variables formula, as e.g. in [39].
Herein we follow an approach based on recent advances in Calculus of Variations in the space L ∞ (see e.g. [26] and references therein) developed for functionals involving higher order derivatives, which has already been applied to the special case of the inverse source problem for linear PDEs (1.4) in [25]. This relatively new field field was pioneered by Gunnar Aronsson in the 1960s (see e.g. [3,4,5,6,7]) and is still a very active area of research; for a review of the by-now classical theory involving scalar first order functionals we refer to [24].
Following [25], we aim at providing a regularisation strategy inspired by the classical Tykhonov regularisation strategy in L 2 (see e.g. [29,33]). As a first possible step, consider the next putative L ∞ "error" functional: for some fixed parameter α > 0. The advantage of searching for a best fitting solution in L ∞ is evident: we can keep the error term |K[u] − k γ | uniformly small and not merely small on average, as would happen if one chose to minimise some integral of a power of the error instead. As in [25], the goal would be to minimise E ∞,α over W 2,∞ g (Ω), and then any minimiser of (1.11) would provide a candidate solution for our problem. Then, for any fixed α, this would be the best fitting solution with the least possible uniform error, namely F[u] ∼ = f uniformly on Ω and K[u] ∼ = k γ uniformly on K. Unfortunately, even if one momentarily ignores the problem of lack of regularity for (1.11) and the fact that W 2,∞ g (Ω) is not a Banach space, the main problem is that in general minimisers do not exist in the genuine fully nonlinear case of operator F (namely when X → F(x, r, p, X) is nonlinear) as (1.11) is not weakly lower semicontinuous in the Fréchet space W 2,∞ g (Ω), as the highest order term may be nonconvex/non-quasiconvex. In the special linear case of [25], this problem was not present as the linearity of the differential operator was implying the desired weak lower semi-continuity.
In this work we resolve the problem explained above by proposing a double approximation method (or rather triple, as we will see shortly) which involves an additional Tykhonov or "viscosity" term which effectively is a weakly lowersemicontinuous approximation of (1.11). Hence, we will consider instead .
It is well known in the Calculus of Variations in L ∞ that (global) minimisers of supremal functionals, although usually simple to obtain with a standard direct minimisation ( [15,19]), they are not genuinely minimal as they do not share the nice "local" optimisation properties of minimisers of their integral counterparts (see e.g. [10,35]). The case of (1.12) studied herein is no exception to this rule. A relatively standard method is bypass these obstructions is to employ minimisers of L p approximating functionals as p → ∞, establishing appropriate convergence of L p minimisers to a limit L ∞ minimiser. The idea underlying this approximation technique is based on the simple measure theory fact that the L p norm (of a function in L 1 ∩ L ∞ ) converges to the L ∞ norm as p → ∞. This method is quite standard in the field and furnishes a selection principle of L ∞ minimisers with additional desirable properties (see e.g. [9,11,14,20,26]). In this fashion one is also able to bypass the lack of differentiability of supremal functionals and derive necessary PDE conditions satisfied by L ∞ extrema. This is indeed the method that is employed in this work as well, along the lines of [25].
We now present the main results to be established in this paper. As already explained, we will obtain special minimisers of (1.12) as limits of minimisers of (1.13) In (1.13) we have used the normalised L p norms and the integral signs with slashes symbolises the average with respect to the Hausdorff measure H κ and the Lebesgue measure L n respectively. Further, | · | (p) symbolises the next p-regularisation of the absolute value away from zero: Let us also note that, due to our L p -approximation method, as an auxiliary result we also provide an L p regularisation strategy for finite p as well, which is of independent interest. For the proof we will need to assume that the Dirichlet problem (1.1) for the fully nonlinear operator F satisfies the W 2,p elliptic estimates (1.3) for all large enough (finite) p > n, as well the following: F and its partial derivates F r , F p , F X satisfy (1.14) We note that sufficient general conditions of when such operators satisfy W 2,p elliptic estimates can be found for instance in the papers [12,13,16,28,30].
Let Ω ⊆ R n be a bounded C 1,1 domain and g ∈ Wn ,2 (Ω).  .8) and (1.14). Suppose further a function k γ ∈ L ∞ (K, H κ ) is given which satisfies (1.9) for γ > 0. Let finally α, β > 0 be fixed. Then, we have the following results for the inverse problem associated to (1.10): and Further, there exist signed Radon measures is satisfied by the triplet (u ∞ , µ ∞ , ν ∞ ) in the sense of distributions (see (2.4)). In (1.15), the operator ((dF) u∞ ) * is the formal adjoint of the linearisation of F at u ∞ , defined via duality as F r , K r , F p , K p , F X denote the partial derivatives of F, K with respect to the respective variables and , Dv), ... etc. Additionally, the error measure ν ∞ is supported in the closure of the subset of K of maximum error, namely , where " ( · ) " denotes the "essential limsup" with respect to H κ K on K (see Definition 3 that follows). If additionally the data function k γ is continuous on K, (1.16) can be improved to (ii) For any α, β, γ > 0, the minimiser u ∞ can be approximated by a family of minimisers (u p ) p>n ≡ (u α,β,γ p ) p>n of the respective L p functionals (1.13) and the pair of measures (µ ∞ , ν ∞ ) ∈ M(Ω) × M(K) can be approximated by respective signed measures (µ p , ν p ) p>n ≡ (µ α,β,γ p , ν α,β,γ p ) p>n , as follows: For any p > n, (1.13) has a global minimiser u p ≡ u α,β,γ p in Wn ,2 ∩ W 1,2 g (Ω) and there exists a sequence p j −→ ∞ as j → ∞, such that as p j → ∞. Further, for each p > n, the triplet (u p , µ p , ν p ) solves the PDE in the sense of distributions (see (2.3)).
We note that the estimate in part (iv) is useful if we have merely that F[u 0 ] ∈ L p (Ω) for p < ∞ (namely when F[u 0 ] ∈ L ∞ (Ω)). We close this section by noting that the reader may find in [25] various comments and counter-examples regarding the optimality of Theorem 1 (therein stated for the case of a linear differential operator F).

Discussion, auxiliary results and proofs
We begin with some clarifications on Theorem 1.

Remark 2.
In index form, the definition of the formal adjoint ((dF) u∞ ) * of the linearisation of F at u ∞ can be written as and its distributional interpretation via duality reads (2.2) for all φ ∈ Cn c (Ω). Hence, by taking into account the definitions of the measures µ p , ν p in (1.19), the distributional interpretation of (1.20) is for all φ ∈ Cn c (Ω). Similarly, the distributional interpretation of (1.15) is for all φ ∈ Cn c (Ω).
We now state a definition and a result taken from [25] which are required for our proofs.
Definition 3 (The essential limsup, [25]). Let X ⊆ R n be a Borel set and let ν ∈ M(X) be a finite positive Radon measure on X. Given f ∈ L ∞ (X, ν), we define f ∈ L ∞ (X, ν) by The following result studies what we call "concentration measures" of the approximate L p minimisation problems as p → ∞.
Proposition 4 (L p concentration measures as p → ∞). Let X be a compact metric space, endowed with a positive finite Borel measure ν which gives positive values to any open subset of X except ∅. Consider (f p ) ∞ 1 ⊆ L ∞ (X, ν) and the sequence of signed Radon measures (ν p ) ∞ 1 ⊆ M(X), given by: where | · | (p) = (| · | 2 + p −2 ) 1/2 . Then: (i) There exists a subsequence (p i ) ∞ 1 and a limit measure ν ∞ ∈ M(X) such that then ν ∞ is supported in the set where |f ∞ | is maximised: (iii) If additionally to (ii) the modulus |f ∞ | of f ∞ is continuous on X, then the following stronger assertion holds: Now we establish Theorem 1. The proof consists of several lemmas. We note that some details might be quite well known to the experts of Calculus of Variations, but we chose to give most of the niceties for the convenience of the readers and for the sake of completeness of the exposition.
Note that the proof above reveals the fact that E p,α,β is weakly lower semicontinuous on the space (Wn ,2 ∩ W 1,2 g )(Ω), even though it is not explicitly stated.
Proof. For each p > n, let u p ∈ (Wn ,2 ∩ W 1,2 g )(Ω) be the minimiser of E p,α,β given by Lemma 5. (We will follow a similar method and utilise the estimates appearing therein.) For any fixed q ∈ (n, ∞) and p ≥ q, Hölder's inequality and minimality yield By employing again the coercivity of E q,α,β , we have the estimate which by Poincaré's inequality and the interpolation inequalities in Wn ,2 (Ω), yield sup p≥q u p Wn ,2 (Ω) ≤ C for some C > 0 depending on q and all the parameters, but independent of p. By a standard diagonal argument, for any sequence (p i ) ∞ 1 with p i −→ ∞ as i → ∞, there exists a function u ∞ ∈ (Wn ,2 ∩ W 1,2 g )(Ω) and a subsequence (denoted again by (p i ) ∞ 1 ) along which (1.18) holds true. It remains to show that u ∞ is in fact a minimiser of E ∞ over the same space. To this end, note that for any fixed q ∈ (n, ∞) and p ≥ q, we have for any v ∈ (Wn ,2 ∩ W 1,2 g )(Ω). By the weak lower semi-continuity of E q,α,β in the space (Wn ,2 ∩ W 1,2 g )(Ω) demonstrated in Lemma 5, we have for any v ∈ (Wn ,2 ∩ W 1,2 g )(Ω). By letting q → ∞ in the estimate above, we deduce that as desired.
Proof. We involve a standard Gateaux differentiability argument. Let us begin by checking that µ p , ν p as defined in (1.19) are uniformly bounded Radon measures when u p ∈ Wn ,2 (Ω). Since by the regularity of F, K, u p they obviously define absolutely continuous measures, it suffices to check that by Hölder inequality's, we have the total variation estimates and similarly Next, fix φ ∈ Cn c (Ω). By using the regularity assumptions on F, K, we formally compute, recalling the abbreviations F By invoking the definitions of µ p , ν p , the above yields that Since u p is the minimiser of E p,α,β , we have that E p,α,β (u p ) ≤ E p,α,β (u p +εφ) for all ε ∈ R and all φ ∈ Cn c (Ω). Hence, our computation implies that the PDE (1.20) is indeed satisfied as claimed, once we confirm that the formal computation is rigorous, and that therefore E p is Gateaux differentiable at the minimiser u p for any direction φ ∈ Cn c (Ω) because by the continuity of F, K and the fact that u p ∈ (C 2 ∩Wn ,2 )(Ω), F[u p ] ∈ C(Ω) and K[u p ] − k γ ∈ L ∞ (K, H κ ), Hölder's inequality implies that and K r (·, u p , Du p ) φ + K p (·, u p , Du p ) · Dφ ∈ C(K), for any φ ∈ Cn c (Ω). Finally, Dnu p : Dnφ ∈ L 1 (Ω) since Dnu p ∈ L 2 Ω, R n ⊗n s . Lemma 8. For any α, β, γ > 0, consider the minimiser u ∞ of E ∞,α,β constructed in Lemma 6 and the minimisers (u p ) p>n of the functionals (E p,α,β ) p>n . Then, there exist signed Radon measures µ ∞ ∈ M(Ω) and ν ∞ ∈ M(K) such that the triplet (u ∞ , µ ∞ , ν ∞ ) satisfies the PDE (1.15) in the distributional sense, that is (2.4) holds true. Additionally, there exists a further subsequence along which the weak* modes of convergence of (1.19) hold true as p → ∞.
Proof. By the proof of Lemma 7, we have the uniform in p total variation bounds µ p (Ω) ≤ 1 and ν p (K) ≤ 1. Hence, by the sequential weak* compactness of the spaces of Radon measures there exists a further subsequence denoted again by (p i ) ∞ 1 such that µ p * − − µ ∞ in M(Ω) and ν p * − − ν ∞ in M(K), as p i → ∞. Fix now φ ∈ Cn c (Ω). By Lemma 7, we have that the triplet (u p , µ p , ν p ) satisfies (1.20), that is (2.3) holds true for any fixed test function. Since u p −→ u ∞ in C 2 (Ω) as p i → ∞, F, F r , F p , F X , K r , K p are all continuous up to the boundary, we have as p i → ∞ that uniformly on K and on Ω respectively, for any fixed φ ∈ Cn c (Ω). Further, we have µ p * − − µ ∞ and ν p * − − ν ∞ in M(Ω) and in M(K) respectively as p j → ∞. By standard properties of the weak*-strong continuity of the duality pairings (2.5), we have that the first two terms of (2.4) converge to the respective first two terms of (2.3), as p i → ∞. Finally, since Dnu p − − Dnu p in L 2 Ω, R n ⊗n s and Dnφ ∈ L 2 Ω, R n ⊗n s , we also have that the last term of (2.4) converges to the respective last term of (2.3), as p i → ∞. Hence, we have indeed obtain (1.15) by passing to the limit as p i → ∞ in (1.20).
Proof. Let us denote k 0 := K[u 0 ], noting that k 0 ∈ C(K) and that k γ − k 0 L ∞ (K,H κ ) ≤ γ. Recall that for any p > n, u p is a global minimiser of E p,α,β in (Wn ,2 ∩ W 1,2 g )(Ω). Therefore, E p,α,β (u p ) ≤ E p,α,β (u 0 ). The above together with Minkowski's and Hölder's inequalities yields as claimed. To obtain the corresponding estimate for u ∞ in the case that additionally u 0 ∈ (Wn ,2 ∩ W 1,2 g )(Ω), we pass to the limit as p i → ∞ in the estimate above and the conclusion follows by letting p i → ∞ due to the strong convergence u p −→ u ∞ in C 2 (Ω) as p i → ∞. The lemma ensues.