Quasiclassical asymptotics of eigenvalues op elliptic systems of differential operators on compact manifolds

1992 ◽  
Vol 51 (2) ◽  
pp. 210-212
Author(s):  
K. Kh. Boimatov
Keyword(s):  
Differential Operators ◽  
Elliptic Systems ◽  
Compact Manifolds ◽  
Asymptotics Of Eigenvalues ◽  
Quasiclassical Asymptotics

scholarly journals Unique continuation property of solutions to general second order elliptic systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Naofumi Honda ◽  
Ching-Lung Lin ◽  
Gen Nakamura ◽  
Satoshi Sasayama
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Unique Continuation ◽  
Elliptic Systems ◽  
General System ◽  
Second Order ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
Constant Coefficients ◽  
Continuation Property

Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

2021 ◽  
Author(s):  
Anna Kh. Balci ◽  
Andrea Cianchi ◽  
Lars Diening ◽  
Vladimir Maz’ya
Keyword(s):  
Differential Inequality ◽  
Differential Operators ◽  
Elliptic Systems ◽  
Second Order ◽  
Nonlinear Elliptic Systems ◽  
Nonlinear Elliptic ◽  
Regularity Properties ◽  
Partial Differential Operators ◽  
Global Estimates ◽  
Special Case

AbstractA sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $${\mathbb R^n}$$ R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

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