If we assume the matrix A satisfies the following condition begin{aligned} D_z A x,z,p) ge c |p|^2I, end{aligned} (3.70)as (|p|rightarrow infty ), (xin Omega ), (|z|le M), (pin {mathbb {R}}^n) for any (M>0) and for some (c>0), we can dispense with the uniformly regular assumption on A in Theorem 1.3 and Theorem 3.1 and the proof is much simpler.
One is our minimal regular assumptions which are concerned with elliptic obstacle problems (1.1) with partially BMO coefficients over the bounded nonsmooth domain.
Indeed, the regular assumptions made in Caristi's theorem imply that any chain (for ≺ α ) has a lower bound, which is stronger than having a minimal element.
It is noted that in this paper we only give the homogenization result of PDEs system with Markov chains when there exists the classical solution under sufficiently regular assumptions.
(1.3) As we know, it is very necessary that some structural and regular assumptions are imposed on the tensorial-valued operator (A x,u)) and the inhomogeneity (B x,u,Du)).
Here I show that, given regular assumptions on the primitives of the design problem, a weaker condition that includes the case of non-smooth valuations is sufficient and necessary for the uniqueness of Groves mechanisms among all efficient, dominant strategy mechanisms.
To this end, it is necessary to impose some suitable minimal regular assumptions on the coefficients (A x)) and geometric restriction on the boundary of the domain under the assumption that the given variable exponent (p(x)) has log-Hölder continuity.
To the best knowledge of the authors of this paper, this is the first time to consider the regularity in the category of the Lorentz spaces with variable exponents for the weak solution of variational inequalities (1.1) under minimal regular assumptions on the leading coefficients and the boundary of domain.
Furthermore, in the context of mixture models, the regular assumptions for the AIC and BIC are no longer valid [ 42, 43].
The frequent experimental observation of a semi-regular system of patches, centered on the site of a small injection of tracer, lead us to propose an intuitive model of projections onto a regular grid (assumption 2) that shifts smoothly across the cortical surface.
A relevant result is that even when the assumption of regular spatial structure of interaction, all the persistent regimes obtained in the global interaction case are also reached.
