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We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the formwith non autonomous integrand F(x, ξ) which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial mapx → D ξ F(x, ξ) has weak derivative in the almost critical Zygmund class L n log α L and the datum f is assumed to belong to the same Zygmund class.

This paper deals with Bayesian estimations of scale parameter of the exponential distribution based on upper record range (R n ). This has been done in two steps; point and interval. In the first step the quadratic, squared error and absolute error, loss functions have been considered to obtain Bayesianpoint estimations. Also in the next step the shortest Bayes interval (Hight Posterior Density interval) and Bayes interval with equal tails based on upper record range have been found. Therefore, the Homotopy Perturbation Method (HPM) has been applied to obtain the limits of Hight Posterior Density intervals. Moreover, efforts have been made to meet the admissibility conditions for linear estimators based on upper record range of the form mR n +d by obtained Bayesian point estimations. So regarding the consideration of loss functions, the prior distribution between the conjunction family has been chosen to be able to produce the linear estimations from upper record range statistics. Finally, some numerical examples and simulations have been presented.

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