SYSTEMS OF ELLIPTIC EQUATIONS IN DIVERGENCE FORM: A PRIORI BOUNDS FOR SOLUTIONS OF DIRICHLET PROBLEM

2019 ◽  
Vol 113 (2) ◽  
pp. 221-232 ◽  
Author(s):  
Patrizia Di Gironimo
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
Divergence Form ◽  
A Priori Bounds ◽  
Systems Of Elliptic Equations

scholarly journals A Priori Bounds in and in for Solutions of Elliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Variational Problems ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bounds ◽  
A Priori Bound

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

scholarly journals A-priori bounds and existence for solutions of weighted elliptic equations with a convection term

2017 ◽  
Vol 6 (4) ◽  
pp. 427-445 ◽  
Author(s):  
Ky Ho ◽  
Inbo Sim
Keyword(s):  
Elliptic Equations ◽  
Neumann Boundary Condition ◽  
Existence Of Solutions ◽  
A Priori ◽  
Neumann Boundary ◽  
Variable Exponents ◽  
Convection Term ◽  
Pseudomonotone Operators ◽  
A Priori Bounds ◽  
Localization Method

AbstractWe investigate weighted elliptic equations containing a convection term with variable exponents that are subject to Dirichlet or Neumann boundary condition. By employing the De Giorgi iteration and a localization method, we give a-priori bounds for solutions to these problems. The existence of solutions is also established using Brezis’ theorem for pseudomonotone operators.

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