Applied analysis including nonlinear analysis and variational analysis as well as optimization are rapidly growing areas of mathematics with numerous applications to optimization, control theory, economics, engineering, and other disciplines. The main objective of this special issue is to discuss recent results in nonlinear analysis, optimization, optimal control and their applications to elliptic and sub-elliptic partial differential equations, particularly to optimization and applications to medicine. Seven contributed papers in this special issue contain most recent results in this research area.
The paper by Y.R. Jiang, N.J. Huang and J.C. Yao introduced and studied a control system governed by a semilinear nonlocal fractional evolution inclusion with Clarke subdifferential and its optimal control. The authors showd an existence theorem of the mild solution for the presented control system by employing the fixed point theorem of a condensing multi-valued map and provide a result concerned with the existence of an optimal control to the presented control system. An example was provided to support their results.
The paper by Lu-Chuan Ceng, D. R. Sahu, Ching-Feng Wen and Ngai-Ching Wong studied the generalized -D-gap function for an equilibrium problem. They gave sufficient conditions under which a critical point of the generalized
-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized
-D-gap function, the authors constructed an error bound for the equilibrium problem. Concrete examples were provided to suppot the results.
In the paper ‘Sub-Riemannian structure in a principal bundle and their Popp measures’ by Wolfram Bauer, Kenro Furutani and Chisato Iwasaki, they discussed a relation between the Popp’s measures for 2-step sub-Riemannian structures defined on the total space of a principal bundle and its base manifold. In some examples we determine the Popp measure explicitly. This has important in optimization and control theory.
The paper by Liguang Liu, Der-Chen Chang, Xing Fu and Dachun Yang on ‘Endpoint boundedness of commutators on spaces of homogeneous type’ studied regularity properties of the commutator [b, T] where T is a sublinear operators that include the well-known Calderón-Zygmund operator and b is a non-constant -function. Here
is a space of homogeneous type in the sense of Coifman and Weiss. In this paper, they proved the commutator [b, T] is bounded from a Hardy-type subspace
of
to
, where
denotes the Lebesgue space of integrable functions. This plays an important role in both harmonic analysis and partial differential equations.
In the paper ‘Singular degenerate operators’ by Der-Chen Chang, M. Hedayat Mahmoudi and Bert-Wolfgang Schulze, the authors studied some simplified and more general method for constructing parametrices for pseudo-differential operators on manifolds with higher singular spaces. They outlined ideas on establishing an algebra of such operators where the order of composition is equal to sum of orders, and the symbols behave multiplicatively, except for a translation in the complex Mellin covariable when the highest singularity This provides deep insight of theory of pseudo-differential operators on stratified spaces of any singularity order
, with
indicating smoothness that has wide impact on study of elliptic boundary value problems.
The study of Chandrasekhar operator originates from S. Chandrasekhar’s pioneer work on the theory of radiative transfer in 1940’s. The paper by Sheng-Ya Feng on ‘Variational analysis for the heat kernel of Chandrasekhar-type operators’ generalized the operator to cases with generic quadratic potentials. Using Hamilton-Jacobi theory, geodesics were calculated explicitly and the exact heat kernels are given in terms of integral representation. Results in this paper share great benefits with many other areas of mathematics, including harmonic analysis, partial differential equations, variation calculus, Brownian motion, boundary element analysis, etc.
The paper by Yun Shi and Wei Wang on ‘The Szego kernel for k-CF functions on the Quaternionic Heisenberg group’ studied the tangential k-Cauchy-Fueter operator and the k-CF functions on the quaternionic Heisenberg group. These are quaternionic counterparts of the tangential CR operator and CR functions on the Heisenberg group in the theory of sub-ellptic operators and sub-Riemannian geomrtry. The authors used the group Fourier transform on the quaternionic Heisenberg group to analyze the operator associated the tangential k-Cauchy-Fueter operator and to construct its kernel, from which they obtained the Szegö kernel of the orthonormal projection from the space of
functions to the space of
integrable k-CF functions on the quaternionic Heisenberg group. This provides an very important example in function theory on sub-Riemannian manifolds of step 2.
We sincerely thank the authors of this special issue. Without their contributions, this special issue would not come out successfully.
Guest Editors:
Der-Chen Chang
Robert P. Gilbert
Yongzhi Steve Xu
Jen-Chih Yao