Related papers: A new boundary Harnack principle (equations with r…

We establish a Harnack inequality of fractional Laplace equations without imposing sign condition on the coefficient of zero order term via the Moser's iteration and John-Nirenberg inequality.

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $\Omega\cap B_1$, $u_1=u_2=0$ in…

We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary Harnack inequalities on…

We provide some versions of the Zaremba-Hopf-Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains.

We prove the Harnack inequality for general nonlocal elliptic equations with zero order terms. As an application we prove the existence of the principal eigenvalue in general domains. Furthermore, we study the eigenvalue problem associated…

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary…

We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…

In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum…

For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. In this paper, we…

We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate,…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the…

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient…

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free…

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \]…

For a class of one-dimensional linear elliptic fourth-order equations with homogeneous Dirichlet boundary conditions it is shown that a non-positive and non-vanishing right-hand side gives rise to a negative solution. A similar result is…

Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in…

This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $\Omega \subset \mathbb{R}^N$ with a…

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants…