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Register a new userThe Inhomogeneous Boundary Harnack Principle for Fully Nonlinear and p-Laplace equations
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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.
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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 for 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 elliptic 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.
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