In this paper, we study the regularity of solutions for two–dimensional Cahn–Hilliard equation with non–constant mobility. Basing on the L p type estimates and Schauder type estimates, we prove the global existence of classical solutions. [ABSTRACT FROM AUTHOR]
A right R–module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R–homomorphism f : m → E can be extended to an R–homomorphism f' : R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R–module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which asserts that every injective right R–module over any left perfect right self–injective ring R is the injective hull of a projective submodule. [ABSTRACT FROM AUTHOR]
The article offers information on how to determine the mean value formula for circular mathematical sector's harmonic functions. It presents the solution to delineate the mean value formula's validity for homogeneous boundary conditions. In addition, it cites the statement of the problems along with its domain, boundary condition imposition, and formula. It highlights the computational method and formula in finding the Euler gamma-function. Furthermore, the use of coefficients in search for the mean value formula's solution is also suggested.
DIFFERENTIAL equations, SMOOTHNESS of functions, COMPLEX variables, MATHEMATICAL formulas, ALGEBRAIC functions, MATHEMATICAL analysis, EQUATIONS, FUNCTIONAL identities, PFAFFIAN problem
Abstract
The article offers information on how to determine and classify the ordinary differential equations' undetermined mathematical system. It cites the formulation for the autonomous differential equations as well as the smooth functions. However, it entails the complexness of the undetermined systems in terms of mathematical objects and differential equation. It connotes the usability of the Pfaff systems for nondegenerate linear transformation and coordinate changes. Furthermore, the solution in finding the equivalence of the undetermined systems as well as theorems are also provided.