scholarly journals Weighted Morrey Regularity for Linear Degenerate Elliptic Dirichlet Problems on Hörmander's Vector Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 301-315 ◽  
Author(s):  
Xuewei Cui ◽  
Pengcheng Niu
Keyword(s):  
Vector Fields ◽  
Dirichlet Problems ◽  
Degenerate Elliptic ◽  
Morrey Regularity ◽  
Linear Degenerate

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

scholarly journals On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Olha P. Kupenko ◽  
Rosanna Manzo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Direct Method ◽  
Dirichlet Boundary Conditions ◽  
Elliptic Variational Inequalities ◽  
The Matrix ◽  
Degenerate Elliptic ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Linear Degenerate

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.

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