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^{infty } minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.
Highlights
Let n, k ∈ N with k, n ≥ 2 and let Ω ⊆ Rn be a bounded connected domain with C1,1 regular boundary ∂Ω
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
In this paper we study the following ill-posed inverse source identification problem for fully nonlinear elliptic PDEs:
Summary
Let n, k ∈ N with k, n ≥ 2 and let Ω ⊆ Rn be a bounded connected domain with C1,1 regular boundary ∂Ω. [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. In the above β > 0 is a fixed parameter, [·] symbolises the integer part and Dnu is the n-th order weak derivative of u It is well known in the Calculus of Variations in L∞ that (global) minimisers of supremal functionals, 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 (ii) For any α, β, γ > 0, the minimiser u∞ can be approximated by a family of minimisers (up)p>n ≡ (uαp,β,γ )p>n of the respective Lp functionals (1.13) and the pair of measures (μ∞, ν∞) ∈ M(Ω) × M(K) can be approximated by respective signed measures (μp, νp)p>n ≡ (μαp,β,γ , νpα,β,γ )p>n, as follows: For any p > n, (1.13) has a global minimiser up ≡ uαp,β,γ in (W n,2 ∩ Wg1,2)(Ω)
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