The homogeneous Dirichlet problem

2019 ◽  
pp. 27-46
Author(s):  
Francisco-Javier Sayas ◽  
Thomas S. Brown ◽  
Matthew E. Hassell
Keyword(s):  
Dirichlet Problem ◽  
Homogeneous Dirichlet Problem

scholarly journals Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

2020 ◽  
Vol 6 (2) ◽  
pp. 198-209
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Dirichlet Problem ◽  
Variational Approach ◽  
Type Inequality ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Hardy Type Inequality ◽  
Principal Curve ◽  
Hardy Type ◽  
Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

scholarly journals ON A DIRICHLET PROBLEM WITH p-LAPLACIAN AND SET-VALUED NONLINEARITY

2011 ◽  
Vol 86 (1) ◽  
pp. 83-89 ◽  
Author(s):  
S. A. MARANO
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Banach Spaces ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Type Theorem ◽  
Special Cases ◽  
Homogeneous Dirichlet Problem ◽  
Point Type

AbstractThe existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

scholarly journals Existence and uniqueness of solutions for a semilinear elliptic system

2005 ◽  
Vol 2005 (10) ◽  
pp. 1507-1523 ◽  
Author(s):  
Robert Dalmasso
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Nonlinear Elliptic Equations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Homogeneous Dirichlet Problem

We consider the existence, the nonexistence, and the uniqueness of solutions of some special systems of nonlinear elliptic equations with boundary conditions. In a particular case, the system reduces to the homogeneous Dirichlet problem for the biharmonic equationΔ2u=|u|pin a ball withp>0.

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