for Fully Nonlinear Elliptic PDE

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 /

©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 orderL²“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 inL^∞

Mathematics Subject Classification (2010) 35R25·35R30·35J60·35J70

1 Introduction

Letn, k ∈ Nwithk, n ≥ 2 and letΩ ⊆ Rⁿ be a bounded connected domain withC^1,1 regular boundary∂Ω. Let also

F : Ω×R×Rⁿ×Rⁿs^⊗2 −→R

be a Carath´eodory function, namely x → F (x,r,p,X) is Lebesgue measurable for all (r,p,X) ∈ R×Rⁿ×Rⁿs^⊗2 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 n.katzourakis@reading.ac.uk Birzhan Ayanbayev

bayanbayev@zedat.fu-berlin.de

1 Freie Universit¨at Berlin, Arnimallee 6, 14195 Berlin, Germany

2 Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK

Published online: 22 July 2021

In this paper, the notationRⁿs^⊗k stands for the vector space of fully symmetrick-th order tensors inRⁿ⊗ · · · ⊗Rⁿ(k-times). Giveng∈W^2,∞(Ω), consider the Dirichlet problem

F[u] =f inΩ,

u=g on∂Ω, (1.1)

for some appropriate sourcef :Ω −→R. Here F[u]denotes the induced fully nonlinear 2nd order differential operator, defined on smooth functionsuas

F[u] :=F (·, u,Du,D²u). (1.2) Evidently, we are employing the standard symbolisations Du = (Diu)i=1...n, D²u = (D²_iju)_i,j=1...n and D_i ≡ ∂/∂x_i. The above direct Dirichlet problem for F asks to deter- mineu, given a sourcef and boundary datag. (In fact the sourcef is obsolete and can be absorbed intoF, but for the problem we are interested in this paper it is more convenient to write it in this separated form).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 stan- dard material, see e.g. [22]. In particular, it is known that under various sets of assumptions onF that (1.1) is well-posed and, givenf ∈ L^∞(Ω)andg ∈ W^2,∞(Ω), for anyp > n there exists a unique solutionuin the locally convex (Fr´echet) space

Wg^2,∞(Ω):=

1<p<∞

W^2,p∩W_g^1,p (Ω).

In general, the solutionuis not in the smaller space(W^2,∞∩W_g^1,∞)(Ω)(not even locally), due to the failure of theW^2,pestimates forp= ∞, which happens even in the linear case (see e.g. [21]). Additionally, (1.1) satisfies for anyp > nthe fully nonlinearL^p global estimate

F (·, v,Dv,D²v)L^p(Ω)≥C1vW^2,p(Ω)−C1gW^2,p(Ω)−C2 (1.3) for some constantsC₁, C₂ > 0 depending only on the parameters and anyv ∈ (W^2,p∩ W_g^1,p)(Ω). For sufficient conditions onF 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 diver- gence operators withC¹matrix coefficientA, as well as the non-divergence linear case with continuous coefficient:

L1[u] =div(ADu)+b·Du+cu,

L₂[u] =A:D²u+b·Du+cu. (1.4) In the above, the notations “:” and “·” symbolise the Euclidean inner products in the space of symmetric matricesRⁿs^⊗2 and inRⁿ respectively. More generally, the inner product of two tensorsT , S∈Rⁿs^⊗kwill also be denoted by “:”, that is

T :S:=

1≤a1,...,ak≤n

Ta₁···a_kSa₁···a_k.

Theinverse problemrelating to (1.1) asks the question of perhaps determiningf, given the boundary datag and some otherpartial information on the solutionu, typically some approximate experimental measurements of some function of it known only up to some error.

The inverse problem isseverely ill-posedeven in the linear case of the Laplacian operator F= Δ, as the noisy data measured on a subset ofΩmight either not be compatible with anyexact solution, or they may not suffice to determine a uniquef 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. [1,8,17,23,31,32,34,36–40]. In this paper we will assume that the approximate information onutakes the form

K[u] =k^γ onK, (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)

whereKand its partial derivatesK_r,K_psatisfy

K, K_r∈C(K×R×Rⁿ), K_p ∈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⊆Ωis compact and existsκ∈ [0, n] : H^κ(K) <∞. (1.8) In the above,H^κdenotes the Hausdorff measure of dimensionκ. 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 onK, at noise level at mostγ >0:

k^γ −k⁰L^∞(K,H^κ)≤γ , (1.9)

wherek⁰ = K[u⁰]corresponds to ideal error-free measurements of an exact solution to (1.1) with sourcef =F[u⁰].

To recapitulate, in this paper we study the following ill-posed inverse source identifica- tion problem for fully nonlinear elliptic PDEs:

⎧⎨

⎩

F[u] =f inΩ, u=g on∂Ω, K[u] =k^γ onK.

(1.10) This means that we are searching for a selection process of a suitable approximation forf from the datak^γ onKthrough the observation K[u]of the solutionu. 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 ofK=Ω(full information) or ofK= ∅(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 regionΩ\K. Instead, our goal is a strategy to determine an optimally fittingu^γ (and respective sourcef^γ := F[u^γ]) to the ill-posed problem (1.10). A popular choice of operator K in the literature (when L=Δ) 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 spaceL^∞ (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 was pioneered by

Gunnar Aronsson in the 1960s (see e.g. [3–7]) 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 aregularisation strategyinspired by the classical Tykhonov regularisation strategy inL²(see e.g. [29,33]). As a first possible step, consider the next putativeL^∞“error” functional:

E_∞,α(u):= K[u] −k^γL^∞(K,H^κ)+αF[u]L^∞(Ω), u∈W_g^2,∞(Ω), (1.11) for some fixed parameterα > 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 overWg^2,∞(Ω), and then any minimiser of (1.11) would provide a candidate solution for our problem. Then, for any fixedα, this would be the best fitting solution with the least possible uniform error, namely F[u] ∼= f uniformly onΩand K[u] ∼= k^γ uniformly onK. Unfortunately, even if one momentarily ignores the problem of lack of regularity for (1.11) and the fact that Wg^2,∞(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 Fr´echet spaceWg^2,∞(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 approxima- tion method (or rather triple, as we will see shortly) which involves an additional Tykhonov or “viscosity” term which effectively is a weakly lower-semicontinuous approximation of (1.11). Hence, we will consider instead

⎧⎨

⎩

E_∞,α,β(u):= K[u] −k^γL^∞(K,H^κ)+αF[u]L^∞(Ω)+^β₂D^n¯u²_L2(Ω), u∈

W^n,2¯ ∩W_g^1,2

(Ω), (1.12)

wheren¯:= [n/2] +3. In the aboveβ >0 is a fixed parameter,[·]symbolises the integer part and D^n¯uis then-th order weak derivative of¯ u.

It is well known in the Calculus of Variations inL^∞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 ofL^papproximating functionals asp→ ∞, establishing appro- priate convergence ofL^p minimisers to a limitL^∞ minimiser. The idea underlying this approximation technique is based on the simple measure theory fact that theL^pnorm (of a function inL¹∩L^∞) converges to theL^∞norm asp→ ∞. This method is quite standard in the field andfurnishes a selection principle ofL^∞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 byL^∞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 obtainspecialminimisers of (1.12) as limits of minimisers of

⎧⎨

⎩

E_p,α,β(u):= |K[u] −k^γ|(p)

L^p(K,H^κ)+α|F[u]|(p)

L^p(Ω)+^β₂D^n¯u²_L2(Ω), u∈

W^n,2¯ ∩W_g^1,2 (Ω).

(1.13) In (1.13) we have used the normalisedL^pnorms

fL^p(K,H^κ):=

−

K|f|^pdH^κ1/p

, fL^p(Ω):=

−

Ω|f|^pdLⁿ1/p

and the integral signs with slashes symbolise the average with respect to the Hausdorff measureH^κand the Lebesgue measureLⁿ, respectively. Further,|·|(p)symbolises the next p-regularisation of the absolute value away from zero:

|a|(p):=

|a|²+p⁻².

Let us also note that, due to ourL^p-approximation method, as an auxiliary result we also provide anL^pregularisation strategy for finitepas 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 theW^2,pelliptic estimates (1.3) for all large enough (finite)p > n, as well the following:F and its partial derivatesF_r, F_p, F_Xsatisfy

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

F, F_r ∈C

Ω×R×Rⁿ×Rⁿs^⊗2

, Fp ∈C

Ω×R×Rⁿ×Rⁿs^⊗2,Rⁿ , FX ∈C

Ω×R×Rⁿ×Rⁿs^⊗2;Rⁿs^⊗2

.

(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].

Theorem 1 LetΩ ⊆Rⁿbe a boundedC^1,1 domain andg ∈W^n,2¯ (Ω). Suppose also the operators(1.2)and(1.6)are given, satisfying the assumptions(1.3),(1.7),(1.8)and(1.14).

Suppose further a function k^γ ∈ L^∞(K,H^κ)is given which satisfies (1.9)forγ > 0.

Let finallyα, β >0be fixed. Then, we have the following results for the inverse problem associated to(1.10):

(i) There exists a global minimiseru_∞ ≡u^α,β,γ_∞ ∈(W^n,2¯ ∩Wg^1,2)(Ω)of the functional E_∞,α,β defined in (1.12). In particular, we haveE_∞(u_∞) ≤ E_∞(v) for all v ∈ (W^n,2¯ ∩W_g^1,2)(Ω)and

f_∞≡f_∞^α,β,γ :=F[u^α,β,γ_∞ ] ∈L^∞(Ω).

Further, there exist signed Radon measures

μ_∞≡μ^α,β,γ_∞ ∈M(Ω), ν_∞≡ν_∞^α,β,γ ∈M(K) such that the nonlinear divergence PDE

Kr[u_∞]ν_∞−div

Kp[u_∞]ν_∞

+α((dF)_u∞)^∗[μ_∞] +β(−1)^n¯(D^n¯ :D^n¯u_∞)=0, (1.15)

is satisfied by the triplet(u_∞, μ_∞, ν_∞)in the sense of distributions (see(2.2)). In (1.15), the operator((dF)u_∞)^∗is the formal adjoint of the linearisation ofFatu_∞, defined via duality as

((dF)_u∞)^∗[v] :=div(div(F_X[u_∞]v))−div(F_p[u_∞]v)+F_r[u_∞]v,

F_r,K_r,F_p,K_p,F_Xdenote the partial derivatives ofF, Kwith respect to the respec- tive variables andF_r[v],K_r[v],F_p[v],K_p[v],F_X[v]denote the respective differential operatorsFr(·, v,Dv,D²v), Kr(·, v,Dv), . . . etc. Additionally, the error measureν_∞ is supported in the closure of the subset ofKof maximum error, namely

supp(ν_∞)⊆

|K[u_∞] −k^γ|= K[u_∞] −k^γL^∞(K,H^κ)

, (1.16)

where“(·)”denotes the “essential limsup” with respect toH^κKonK(see Defi- nition 3 that follows). If additionally the data functionk^γ is continuous onK,(1.16) can be improved to

supp(ν_∞)⊆

|K[u_∞] −k^γ| = K[u_∞] −k^γL^∞(K,H^κ)

. (1.17)

(ii) For anyα, β, γ >0, the minimiseru_∞can be approximated by a family of minimis- ers(up)p>n ≡ (u^α,β,γ_p )p>n of the respectiveL^pfunctionals(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_p^α,β,γ)_p>n, as follows:

For anyp > n,(1.13)has a global minimiserup≡u^α,β,γp in(W^n,2¯ ∩Wg^1,2)(Ω) and there exists a sequence(pj)^∞₁ , such that

⎧⎪

⎨

⎪⎩

up−→u_∞ inC²(Ω), D^kup−→D^ku_∞ inL²

Ω,Rⁿs^⊗k

, for allk∈ {3, . . . ,n¯−1}, D^n¯upD^n¯u_∞ inL²

Ω,Rⁿs^⊗¯n

,

(1.18) asp_j→ ∞. Moreover, we have

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

ν_p:= K[u_p] −k^γp−2

(p)

K[u_p] −k^γ H^κ(K)|K[up] −k^γ|(p)^p−1

L^p(K,H^κ)

H^κK ∗

ν_∞ inM(K), μ_p:= |F[u_p]|^p−2(p) F[u_p]

Lⁿ(Ω)|F[u_p]|(p)^p−1

L^p(Ω)

LⁿΩ

μ∗ _∞ inM(Ω),

(1.19)

aspj→ ∞. Further, for eachp > n, the triplet(up, μp, νp)solves the PDE Kr[up]νp−div(Kp[up]νp)+α((dF)up)^∗[μp] +β(−1)^n¯(D^n¯ :D^n¯up)=0, (1.20) in the sense of distributions (see(2.1)).

(iii) For any exact solutionu⁰ ∈ (W^n,2¯ ∩W_g^1,2)(Ω)of (1.10) (with f = F[u⁰]and K[u⁰] =k⁰) corresponding to measurements with zero error, we have the estimate:

K[u^α,β,γ_∞ ] −K[u⁰]

L^∞(K,H^κ)≤2γ +αF[u⁰]L^∞(Ω)+β

2D^n¯u⁰²_L2(Ω), (1.21) for anyα, β, γ >0.

(iv) For any exact solutionu⁰ ∈ (W^n,2¯ ∩W_g^1,2)(Ω)of (1.10) (with f = F[u⁰]and F[u⁰] =k⁰) corresponding to measurements with zero error and forp > n, we have the estimate:

K[u^α,β,γp ] −K[u⁰]

L^p(K,H^κ)≤2γ +αF[u⁰]L^p(Ω)+β

2D^n¯u⁰²_L2(Ω), (1.22)

for anyα, β, γ >0.

We note that the estimate in part (iv) is useful if we have merely that F[u⁰] ∈L^p(Ω) forp < ∞(namely, when F[u⁰] ∈ 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).

2 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 atu_∞can be written as

((dF)u_∞)^∗[v] = n i,j=1

D²_ij

FX_ij[u_∞]v

− n k=1

Dk

Fp_k[u_∞]v

+Fr[u_∞]v

and its distributional interpretation via duality reads ((dF)u_∞)^∗[v], φ =

Ω

⎧⎨

⎩ n i,j=1

(D²_ijφ)FX_ij[u_∞] + n k=1

(Dkφ)Fp_k[u_∞] +φFr[u_∞]

⎫⎬

⎭vdLⁿ, for allφ∈C_c^n¯(Ω). Hence, by taking into account the definitions of the measuresμp,νpin (1.19), the distributional interpretation of (1.20) is

−

K

K_r[u_p]φ+K_p[u_p] ·Dφ|K[u_p] −k^γ|^p_(p)⁻²(K[u_p] −k^γ) |K[u_p] −k^γ|(p)^p−1

L^p(K,H^κ)

dH^κ

+α−

Ω

F_r[u_p]φ+F_p[u_p] ·Dφ+F_X[u_p] :D²φ |F[u_p]|^p_(p)⁻²F[up] |F[u_p]|(p)^p−1

L^p(Ω)

dLⁿ +β−

Ω

D^n¯u_p:D^n¯φdLⁿ=0, (2.1)

for allφ∈C_c^n¯(Ω). Similarly, the distributional interpretation of (1.15) is

−

K

Kr[u_∞]φ+Kp[u_∞] ·Dφ dν_∞ +α−

Ω

F_r[u_∞]φ+F_p[u_∞] ·Dφ+F_X[u_∞] :D²φ dμ_∞ +β−

Ω

D^n¯u_∞:D^n¯φdLⁿ=0, (2.2)

for allφ∈C_c^n¯(Ω).

Further, we note that the physical interpretation of the measuresμ_∞andν_∞arising in the PDE is that they essentially “charge” the sets whereon|F[u_∞]|and|K[u_∞] −k^γ|are maximised overΩandK, respectively.

We now state a definition and a result taken from [25] which are required for our proofs.

Definition 3 (The essential limsup, [25]) LetX⊆Rⁿbe a Borel set and letν∈M(X)be a finite positive Radon measure onX. Givenf ∈L^∞(X, ν), we definef∈L^∞(X, ν)by

f(x):= lim

ε→0

ν−ess sup

y∈Bε(x)

f (y)

, ∀x∈X,

and callf theν-essential limsup off. Here, Bε(x)denotes the open ball of radiusε centred atx∈X.

The following result studies what we call “concentration measures” of the approximate L^pminimisation problems asp→ ∞.

Proposition 4 (L^pconcentration measures asp→ ∞)LetXbe a compact metric space, endowed with a positive finite Borel measureν which gives positive values to any open subset ofX except∅. Consider(fp)^∞₁ ⊆ L^∞(X, ν) and the sequence of signed Radon measures(ν_p)^∞₁ ⊆M(X), given by:

ν_p:= 1 ν(X)

|fp|(p)

_p−2 fp

|f_p|(p)^p−1

L^p(X,ν)

ν, p∈N,

where| · |(p)=(| · |²+p⁻²)^1/2. Then:

(i) There exists a subsequence(pi)^∞₁ and a limit measureν_∞∈M(X)such that ν_p ν^∗ _∞ in M(X),

aspi→ ∞.

(ii) If there existsf_∞∈L^∞(X, ν)\ {0}such that sup

X |fp−f_∞| −→0 asp→ ∞, thenν_∞is supported in the set where|f_∞|is maximised:

supp(ν_∞)⊆

|f_∞|= f_∞L^∞(X,ν)

.

(iii) If additionally to (ii) the modulus|f_∞|off_∞is continuous onX, then the following stronger assertion holds:

supp(ν_∞)⊆

|f_∞| = f_∞L^∞(X,ν)

.

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.

Lemma 5 For anyp > n andα, β, γ >0, the functional(1.13)has a minimiseru_p ∈ (W^n,2¯ ∩W_g^1,2)(Ω):

Ep,α,β(up)=inf

Ep,α,β(v):v∈(W^n,2¯ ∩W_g^1,2)(Ω)

.

Proof Sinceg∈W^2,∞(Ω)

E_p,α,β(g) ≤ E_∞,α,β(g)

≤ k^γL^∞(K,H^κ)+ K(·, g,Dg)L^∞(K,H^κ)

+αF(·, g,Dg,D²g)L^∞(Ω)+β

2D^n¯g²_L2(Ω)

< ∞. Hence,

0≤inf

E_p,α,β(v):v∈(W^n,2¯ ∩W_g^1,2)(Ω)

≤E_∞,α,β(g) <∞.

Further, Ep,α,βis coercive in the space(W^n,2¯ ∩W_g^1,2)(Ω). Indeed, by our assumption (1.3), H¨older’s inequality and thatp > n≥2, for anyv∈(W^n,2¯ ∩W_g^1,2)(Ω)we have

E_p,α,β(v) ≥ αF[v]L^p(Ω)+ β

2D^n¯v²_L2(Ω)

≥ α

C1vW^2,p(Ω)−C1gW^2,p(Ω)−C2 +β

2D^n¯v²_L2(Ω)

≥ αC₁vW^2,2(Ω)+β

2D^n¯v²_L2(Ω)−αC₁gW^2,p(Ω)−αC₂ which implies

αC1vW^2,2(Ω)+β

2D^n¯v²_L2(Ω)≤E_∞,α,β(g)+αC1gW^2,p(Ω)+αC2, for anyv∈(W^n,2¯ ∩W_g^1,2)(Ω). Now, by Poincar´e inequality inW_g^1,2(Ω)we have

vL²(Ω)≤C

DvL²(Ω)+ gW^1,2(Ω)

for someC > 0, and by the interpolation inequalities in the Sobolev spaceW^n,2¯ (Ω), we have

D^kvL²(Ω)≤C

DvL²(Ω)+ D^n¯vL²(Ω)

,

for someC >0 and anyk ∈ {1, . . . ,n¯}. By putting the last three estimates together, we conclude that

vW^n,2¯ (Ω)≤C,

where the constantC >0 in general depends onpbut is uniform forv∈(W^n,2¯ ∩W_g^1,2)(Ω).

Let now(u^m_p)^∞₁ be a minimising sequence of E_p,α,β: Ep,α,β(u^m_p)−→inf

Ep(v):v∈(W^n,2¯ ∩W_g^1,2)(Ω) ,

asm→ ∞. Then, by the coercivity estimate, we have the uniform bound u^m_pW^n,2¯ (Ω)≤C

for some C > 0 independent of m ∈ N. By standard weak and strong compactness arguments in Sobolev and H¨older spaces, together with the Morrey estimate

v_Ck−[n/2]−1,σ(Ω)≤CvW^k,2(Ω),

applied tok = ¯n, there exists a subsequence(u^m_p^k)^∞₁ andu_p ∈ (W^n,2¯ ∩W_g^1,2)(Ω)such that, along this subsequence we have

⎧⎪

⎨

⎪⎩

u^m_p −→u_p inC²(Ω), D^ku^m_p −→D^ku_p inL²

Ω,Rⁿs^⊗k

, for allk∈ {3, . . . ,n¯−1}, D^n¯u^m_p D^n¯up inL²

Ω,Rⁿs^⊗¯n

,

asmk→ ∞. The above modes of convergence and the continuity of the functionKdefining the operator K imply that K[u^m_p] −→K[up]uniformly onKasmk→ ∞. Therefore,

|K[u^m_p] −k^γ|(p)

L^p(K,H^κ)−→|K[u_p] −k^γ|(p)

L^p(K,H^κ)

asm_k→ ∞. Additionally, by the continuity of the functionF defining the operator F and the uniform convergence of the minimising sequence up to second order derivatives, we have

F[u^m_p] −→F[up] inC(Ω), asm_k → ∞. Finally, by weak lower semi-continuity inL²we have

D^n¯up

L²(Ω)≤lim inf

k→∞

D^n¯u^m_p^k

L²(Ω). By putting all the above together, we infer that

E_p,α,β(u_p)≤lim inf

k→∞ E_p,α,β(u^m_p^k)≤inf

E_p,α,β(v):v∈(W^n,2¯ ∩W_g^1,2)(Ω) ,

which concludes the proof.

Note that the proof above reveals the fact that E_p,α,β is weakly lower semi-continuous on the space(W^n,2¯ ∩W_g^1,2)(Ω), even though it is not explicitly stated.

Lemma 6 For anyα, β, γ > 0, there exists a (global) minimiseru_∞ of E_∞,α,β in the space(W^n,2¯ ∩Wg^1,2)(Ω), as well as a sequence of minimisers(upi)^∞₁ of the respective Ep,α,β-functionals from Lemma 5, such that(1.18)holds true.

Proof For eachp > n, letu_p ∈(W^n,2¯ ∩W_g^1,2)(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 fixedq ∈(n,∞)andp≥q, H¨older’s inequality and minimality yield

E_q,α,β(u_p)≤E_p,α,β(u_p)≤E_p,α,β(g)≤E_∞,α,β(g) <∞. By employing again the coercivity of Eq,α,β, we have the estimate

E_q,α,β(u_p)≥α

C₁u_pW^2,q(Ω)−C₁gW^2,q(Ω)−C₂ +β

2D^n¯u_p²_L2(Ω), which by Poincar´e’s inequality and the interpolation inequalities inW^n,2¯ (Ω), yield

sup

p≥qu_pW^n,2¯ (Ω)≤C

for someC >0 depending onqand all the parameters, but independent ofp. By a standard diagonal argument, for any sequence (pi)^∞₁ with pi −→ ∞asi → ∞, there exists a function

u_∞∈(W^n,2¯ ∩W_g^1,2)(Ω)

and a subsequence (denoted again by(p_i)^∞₁ ) along which (1.18) holds true. It remains to show thatu_∞is in fact a minimiser of E_∞over the same space. To this end, note that for any fixedq∈(n,∞)andp≥q, we have

Eq,α,β(up)≤Ep,α,β(up)≤Ep,α,β(v)≤E_∞,α,β(v)

for anyv∈(W^n,2¯ ∩Wg^1,2)(Ω). By the weak lower semi-continuity of Eq,α,βin the space (W^n,2¯ ∩Wg^1,2)(Ω)demonstrated in Lemma 5, we have

Eq,α,β(u_∞)≤lim inf

i→∞Eq,α,β(up_i)≤E_∞,α,β(v),

for anyv∈(W^n,2¯ ∩W_g^1,2)(Ω). By lettingq→ ∞in the estimate above, we deduce that E_∞,α,β(u_∞)≤inf

E_∞,α,β(v):v∈(W^n,2¯ ∩W_g^1,2)(Ω) ,

as desired.

Lemma 7 For anyα, β, γ >0andp > n, consider the minimiseru_p of the functional Ep,α,β over(W^2,p∩W_g^1,p)(Ω)constructed in Lemma 5. Consider also the signed Radon measuresμ_p∈M(Ω)andν_p ∈M(K), defined in(1.19). Then, the triplet(u_p, μ_p, ν_p) satisfies the PDE (1.20) in the distributional sense, namely(2.1)holds true for all test functionsφ∈C_c^n¯(Ω).

Proof We involve a standard Gateaux differentiability argument. Let us begin by check- ing that μp,νp as defined in (1.19) are uniformly bounded Radon measures whenup ∈ W^n,2¯ (Ω). Since by the regularity ofF, K, up they obviously define absolutely continu- ous measures, it suffices to check that by H¨older inequality’s, we have the total variation estimates

ν_p(K) ≤ |K[u_p] −k^γ|(p)

L^p(K,H^κ)

1−p

−

K

K[u_p] −k^γp−1

(p) dH^κ

≤ |K[up] −k^γ|(p)

L^p(K,H^κ)

_1−p

−

K

K[up] −k^γp

(p)dH^κp−1_p

= 1 and similarly

μp(Ω)≤ |F[up]|(p)

L^p(Ω)

_1−p

−

Ω

F[up]^p−1

(p) dLⁿ

≤ |F[u_p]|(p)

L^p(Ω)

1−p

−

Ω

F[u_p]^p

(p) dL^{n p−1}_p

= 1.

Next, fix φ ∈ C_c^n¯(Ω). By using the regularity assumptions on F, K, we formally compute, recalling the abbreviations F[v] =F (·, v,Dv,D²v)and K[v] =K(·, v,Dv):

d dε

ε=0Ep,α,β(up+εφ)

=

−

K|K[up] −k^γ|^p_(p)dH^κ_p¹₋1

−

K|K[up] −k^γ|^p_(p)⁻²

K[up] −k^γ

K_r[u_p]φ+K_p[u_p] ·Dφ dH^κ +α

−

Ω|F[up]|^p_(p)dLⁿ_p¹₋₁

−

Ω|F[up]|^p−2_(p) F[up]

Fr[up]φ+Fp[up] ·Dφ+FX[up] :D²φ dLⁿ

+β−

Ω

D^n¯u_p:D^n¯φdLⁿ.

By invoking the definitions ofμ_p, ν_p, the above yields that d

dε

ε=0

E_p,α,β(u_p+εφ) =

K

K_r[u_p]φ+K_p[u_p] ·Dφ dν_p +α

Ω

Fr[up]φ+Fp[up] ·Dφ+FX[up] :D²φ dμp

+β−

Ω

D^n¯u_p:D^n¯φdLⁿ.

Sinceu_pis 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 ∈Rand allφ ∈C_c^n¯(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 Epis Gateaux differentiable at the minimiserup for any directionφ ∈ Cⁿ_c^¯(Ω) because by the continuity of F, K and the fact thatup ∈(C²∩W^n,2¯ )(Ω), F[up] ∈C(Ω) and K[up] −k^γ ∈L^∞(K,H^κ), H¨older’s inequality implies that

Fr[up]φ+Fp[up] ·Dφ+FX[up] :D²φ∈C(Ω) and

K_r(·, u_p,Du_p)φ+K_p(·, u_p,Du_p)·Dφ∈C(K) for anyφ∈Cⁿ_c^¯(Ω). Finally, D^n¯up:D^n¯φ∈L¹(Ω)since D^n¯up∈L²

Ω,Rⁿs^⊗¯n

.

Lemma 8 For any α, β, γ > 0, consider the minimiser u_∞ of E_∞,α,β constructed in Lemma 6 and the minimisers (up)p>n of the functionals (Ep,α,β)p>n. Then, there exist signed Radon measuresμ_∞∈M(Ω)andν_∞∈M(K)such that the triplet(u_∞, μ_∞, ν_∞) satisfies the PDE(1.15)in the distributional sense, that is(2.2)holds true. Additionally, there exists a further subsequence along which the weak* modes of convergence of (1.19) hold true asp→ ∞.

Proof By the proof of Lemma 7, we have the uniform in p total variation bounds μ_p(Ω) ≤ 1 andν_p(K) ≤ 1. Hence, by the sequential weak* compactness of the spaces of Radon measures

M(Ω)=

C₀(Ω)_∗

, M(K)= C(K)_∗

, (2.3)