On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

1999 ◽  
Vol 6 (2) ◽  
pp. 207-225 ◽  
Author(s):  
M. García-Huidobro ◽  
V.K. Le ◽  
R. Manásevich ◽  
K. Schmitt
Keyword(s):  
Sobolev Space ◽  
Differential Operators ◽  
Principal Eigenvalues ◽  
Space Setting ◽  
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].

Elliptic problems for a Douglis–Nirenberg system over ℝn and over an exterior subregion

2016 ◽  
Vol 146 (3) ◽  
pp. 579-594
Author(s):  
M. Faierman
Keyword(s):  
Sobolev Space ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Complex Plane ◽  
Spectral Parameter ◽  
Differential Operators ◽  
Elliptic Problems ◽  
Fredholm Theory ◽  
Space Setting

We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

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.

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