Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms

2009 ◽  
Vol 189 (2) ◽  
pp. 335-356 ◽  
Author(s):  
Vitali Liskevich ◽  
Igor I. Skrypnik
Keyword(s):  
Harnack Inequality ◽  
Lower Order ◽  
Elliptic Equations ◽  
Growth Conditions ◽  
Nonstandard Growth ◽  
Continuity Of Solutions ◽  
Nonstandard Growth Conditions ◽  
Lower Order Terms

On the continuity of solutions of the equations of a porous medium and the fast diffusion with weighted and singular lower-order terms

2021 ◽  
Vol 18 (1) ◽  
pp. 104-139
Author(s):  
Yevhen Zozulia
Keyword(s):  
Porous Medium ◽  
Parabolic Equation ◽  
Harnack Inequality ◽  
Lower Order ◽  
Riesz Potential ◽  
Fast Diffusion ◽  
Right Hand ◽  
Continuity Of Solutions ◽  
The Right ◽  
Lower Order Terms

For the parabolic equation $$ \ v\left(x \right)u_{t} -{div({\omega(x)u^{m-1}}} \nabla u) = f(x,t)\: ,\; u\geq{0}\:,\; m\neq{1} $$ we prove the continuity and the Harnack inequality for generalized k solutions, by using the weighted Riesz potential on the right-hand side of the equation.

scholarly journals On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids

2017 ◽  
Vol 20 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Nam Q. Le
Keyword(s):  
Elliptic Equation ◽  
Harnack Inequality ◽  
Lower Order ◽  
Elliptic Equations ◽  
Singular Elliptic Equation ◽  
Large Gradient ◽  
Singular Elliptic Equations ◽  
Monge Ampere ◽  
Lower Order Terms ◽  
Linear Degenerate

We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge–Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.

Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity

2016 ◽  
Vol 2016 (715) ◽  
pp. 1-38 ◽  
Author(s):  
Sun-Sig Byun ◽  
Jihoon Ok ◽  
Seungjin Ryu
Keyword(s):  
Weak Solution ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Growth Conditions ◽  
Divergence Form ◽  
Gradient Estimates ◽  
Nonlinear Elliptic Problem ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic ◽  
Nonstandard Growth Conditions

AbstractWe consider a nonlinear elliptic problem in divergence form, with nonstandard growth conditions, on a bounded domain. We obtain the global Calderón–Zygmund type gradient estimates for the weak solution of such a problem in the setting of Lebesgue and Sobolev spaces with variable

scholarly journals Existence results for nonlinear degenerate elliptic equations with lower order terms

2020 ◽  
Vol 10 (1) ◽  
pp. 301-310
Author(s):  
Weilin Zou ◽  
Xinxin Li
Keyword(s):  
Lower Order ◽  
Elliptic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Initial Boundary Value ◽  
Lower Order Terms ◽  
Nonlinear Degenerate

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

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