Generalized G-convergence for quasilinear elliptic differential operators

2008 ◽  
Vol 68 (2) ◽  
pp. 304-314
Author(s):  
Josef Malík
Keyword(s):  
Differential Operators ◽  
Elliptic Differential Operators ◽  
Quasilinear Elliptic

scholarly journals Adams inequalities with exact growth condition : on Rn and the Heisenberg group

2020 ◽  
Author(s):  
◽  
Liuyu Qin
Keyword(s):  
Heisenberg Group ◽  
Growth Condition ◽  
Differential Operators ◽  
Riesz Potential ◽  
Type Inequality ◽  
Integral Operators ◽  
Ground State Solution ◽  
Elliptic Differential Operators ◽  
Exact Growth Condition ◽  
Quasilinear Elliptic

In this thesis we prove sharp Adams inequality with exact growth condition for the Riesz potential as well as the more general strictly Riesz-like potentials on R[superscript n]. Then we derive the Moser-Trudinger type inequality with exact growth condition for fractional Laplacians with arbitrary 0 [less than] [alpha] [less than] n, higher order gradients and homogeneous elliptic differential operators. Next we give an application to a quasilinear elliptic equation, and prove the existence of ground state solution of this equation. Lastly, we extend our result to the Heisenberg group. By applying the same technique used in R[superscript n], we derive a sharp Adams inequality with critical growth condition on H[superscript n] for integral operators whose kernels are strictly Riesz-like on H[superscript n]. As a consequence we then derive the corresponding sharp Moser-Trudinger inequalities with exact growth condition for the powers of sublaplacian -L[subscript 0] [superscript alpha/2] when [alpha] is an even integer, and for the subgradient [del] H[subscript n].

scholarly journals Noncoercive quasilinear elliptic operators with singular lower order terms

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Obstacle Problem ◽  
Differential Operators ◽  
Existence Of A Solution ◽  
Marcinkiewicz Spaces ◽  
Higher Integrability ◽  
Elliptic Differential Operators ◽  
Quasilinear Elliptic ◽  
Lower Order Terms

AbstractWe consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

scholarly journals The approximation of the solution of wave problems by spectral expansions connected with elliptic differential operators

2021 ◽  
Vol 86 (2) ◽  
pp. 101-106
Author(s):  
Abdulkasim Akhmedov ◽  
Mohd Zuki Salleh ◽  
Abdumalik Rakhimov
Keyword(s):  
Wave Propagation ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Elastic Membrane ◽  
Conductive Material ◽  
Spectral Expansions ◽  
Vibration Process ◽  
Thermally Conductive ◽  
Elliptic Differential Operators ◽  
Wave Problems

In this research, we investigate the spectral expansions connected with elliptic differential operators in the space of singular distributions, which describes the vibration process made of thin elastic membrane stretched tightly over a circular frame. The sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material.

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