Let n, N ∈ N with Ω ⊆ R n open. Given H ∈ C 2 (Ω × R N × R N n ), we consider the functional
The associated PDE system which plays the role of EulerHerein we establish that generalised solutions to (2) can be characterised as local minimisers of (1) for appropriate classes of affine variations of the energy. Generalised solutions to (2) are understood as D-solutions, a general framework recently introduced by one of the authors.

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ
p
subspaces of the sequence space
R
N
. Each measure in the product is a shifted and scaled copy of a reference probability measure on
R
that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C 2 maps u :This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the p-Laplace system for all p ∈ [2, ∞]. For p = ∞, the ∞-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in L ∞ . Herein we show that the image of a solution u is piecewise affine if either the rank of Du is equal to one or n = 2 and u has additively separated form. As a consequence we obtain corresponding flatness results for p-Harmonic maps for p ∈ [2, ∞].

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of ∞-minimal maps introduced more recently by the second author. We prove that C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.

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 $L^{\infty }$
L
∞
minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of ∞-minimal maps introduced more recently by the second author. We prove that C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.

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