Among his topics are Bellman's equations with constant "coefficients" in the whole space, finite-different equations of elliptic type
, a priori estimates in Ca for solutions of linear and nonlinear equations, elements of the C2+a-theory of fully nonlinear elliptic and parabolic equations, and unique and existence of extremal viscosity solutions for parabolic equations.
Most important geometric structures of interest for (5) are periodic points, which are typically of hyperbolic or elliptic type; the invariant manifolds associated with hyperbolic periodic points; KAM invariant curves (around the elliptic fixed point or around elliptic periodic orbits); and cantori, which are remnant sets of Cantor type of destroyed invariant circles.
In the case-1 < a < 1 characteristic roots are complex conjugate numbers lying on the unit circle, which means that the zero equilibrium is nonhyperbolic of the elliptic type; see [10, 19].
When a > 1 characteristic roots are complex conjugate numbers lying on the unit circle, which means that the non-zero equilibrium solutions are nonhyperbolic of the elliptic type see [10, 19].
We consider the case where -1 < a < 1 in which case [E.sub.0] 0 is nonhyperbolic equilibrium of elliptic type.
As we mentioned earlier, when a > 1 the characteristic roots are complex conjugate numbers lying on the unit circle, which means that the non-zero equilibrium is nonhyperbolic of the elliptic type and so KAM theory is a natural tool to be applied.
The system of equations to be solved is of elliptic type, needing a boundary condition along the whole contour of the solution domain.
An iterative procedure to solve the elliptic type MSE is introduced.
Quraishi, Evaluation of Certain Elliptic Type
Single, Double Integrals of Ramanujan and Erdelyi, J.
They present some basic methods for obtaining various a priori estimates for second-order partial differential equations of the elliptic type
, with a particular emphasis on maximal principles, Harnack inequalities, and their applications.
Advantages are high efficiency performance of elliptic type
designs and the bi-directional capability.
From the perspective of modern research in the field, Radulescu and Repovs introduce graduate students and researchers to the theory of nonlinear partial differential equations with variable exponents, in particular, those of elliptic types
. They look at the most important variational methods for those elliptic partial differential equations that are described by non-homogeneous differential operators and that contain one or more power-type non-linearities with variable exponents.