Around the normal derivative lemma

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum…

On Phragmén-Lindelöf principle for Non-divergence Type Elliptic Equation and Mixed Boundary conditions

Paper dedicated to qualitative study of the solution of the Zaremba type problem in Lipschitz domain with respect to the elliptic equation in non-divergent form. Main result is Landis type Growth Lemma in spherical layer for Mixed Boundary Value Problem in the class of "admissible domain". Based on the Growth Lemma Phragmén-Lindelöf theorem is proved at junction point of Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.

Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients

For functions from the Sobolev space $H^s(Ω)$, 1/2<s<3/2, definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain $Ω$, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness are given. Some new facts about trace operator estimates, Sobolev spaces characterisations, and solution regularity of PDEs …

分布意义下法向导数的表示公式

A representation formula for the distributional normal derivative

本文针对带非负位势和有限 Borel 测度右端的 Poisson 方程，建立了分布意义下法向导数的积分表示公式，并由此证明非负 Hopf 位势情形下 Hopf 引理在边界上几乎处处成立。

Birkhoff Normal Form for the Derivative Nonlinear Schrödinger Equation

This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.

An inequality for the normal derivative of the Lane-Emden ground state

We consider Lane-Emden ground states with polytropic index $0\leq q-1\leq 1$, that is, minimizers of the Dirichlet integral among $L^q$-normalized functions. Our main result is a sharp lower bound on the $L^2$-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets $Ω\subset\mathbb{R}^d$, without assuming convexity.