Adjoint methods for obstacle problems and weakly coupled systems of PDE

The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton--Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived

Nondivergence form degenerate linear parabolic equations on the upper half space

We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in {\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in [14]. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.Comment: 40 pages, minor revision. Added a weighted parabolic embedding result and a local boundary estimat

Cooperative Group In Current Vietnamese Commercial Environment

Purpose: The article researches the compatibility between the cooperative group and the current commercial environment in Vietnam to propose for improvement to the laws and for promotion of development of the cooperative group.   Theoretical framework: Recent literature points out that the cooperative group model is quite suitable for the production and business psychology of citizens in Vietnam. In the current stage of development, however, from the influence and impact of international integration and the increasingly fierce competitive pressure of various types of enterprises, the cooperative group have encountered many difficulties to survive. On the other hand, in the context of judicial reform in Vietnam, the issue of complete law system and expanding the freedom to business has always been paid special attention by the Party and Vietnam State.   Design/methodology/approach: The authors have combined traditional research methods of legal science such as legal analysis method, legal efficiency assessment method and legal comparison method.   Findings: According to research, the cooperative group is well adapted to Vietnam's commercial environment because it is compatible with investor psychology and meets the conditions and circumstances of nature, economy and society. However, since Vietnam's deep integration with the rest of the world, the cooperative group model has faced numerous difficulties and challenges, and if suitable solutions are not found, it is very likely that this collective economy will become increasingly difficult. As a result, the article suggests some important solutions, such as offering cooperative group legal status and requiring more preferential lending policies with long loan terms, simple and quick loan procedures, and easy team cooperation.   Research, Practical & Social implications: This research may support many develop future research in Vietnam.   Originality/value: The study of the freedom to business become more and more urgent and very important in Vietnam