We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.
We introduce a new boundary Harnack principle in Lipschitz domains for equations with right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle f
or divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
We investigate the Boundary Harnack Principle in Holder domains of exponent $alpha>0$ by the analytical method developed in our previous work A short proof of Boundary Harnack Principle.
This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perrons method. If the equation is uniformly e
lliptic in the sense of cite{SS}, we prove the discontinuous viscosity solution is Holder continuous and thus it is a viscosity solution.
The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type begin{align*} begin{cases} mathrm
{div}(| abla u|^{p-2} abla u) &= f+ mathrm{div}(|mathbf{F}|^{p-2}mathbf{F}) qquad text{in} Omega, hspace{1.2cm} u &= g hspace{3.1cm} text{on} partial Omega, end{cases} end{align*} in Lorentz space, with given data $mathbf{F} in L^p(Omega;mathbb{R}^n)$, $f in L^{frac{p}{p-1}}(Omega)$, $g in W^{1,p}(Omega)$ for $p>1$ and $Omega subset mathbb{R}^n$ ($n ge 2$) satisfying a Reifenberg flat domain condition or a $p$-capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-$lambda$ approach technique proposed in our preceding works~cite{MPT2018,PNCCM,PNJDE,PNCRM}, to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.
Mark Allen
,Dennis Kriventsov
,Henrik Shahgholian
.
(2020)
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"The Inhomogeneous Boundary Harnack Principle for Fully Nonlinear and p-Laplace equations"
.
Dennis Kriventsov
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