# Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

• Volume: 010, Issue: 1, page 73-104
• ISSN: 1435-9855

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## Abstract

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We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^{N}$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_{1}$ such the positivity principle holds for $\lambda \in \left({\lambda }_{-1},{\lambda }_{1}\right)$ and the anti-maximum principle holds if $\lambda \in \left({\lambda }_{1},{\lambda }_{1}+\delta \right)$ or $\lambda \in \left({\lambda }_{-1}-ϵ,{\lambda }_{-1}\right)$. A similar, but more complicated result holds if $q\equiv 0$. This is due to the fact that ${\lambda }_{0}=0$ becomes an eigenvalue in this case and that ${\lambda }_{1}\left(\sigma \right)$ as a function of $\sigma$ connects to ${\lambda }_{-1}\left(\sigma \right)$ when the mean value of $\sigma$ crosses the value ${\sigma }_{0}=-|D|/|\partial D|$. In dimension $N=1$ we determine the optimal $\lambda$-interval such that the anti-maximum principles holds uniformly for all right-hand sides $f,g\ge 0$. Finally, we apply our result to the problem $-\Delta u+q\left(x\right)u=\alpha u+f$ in $D$, $\partial u/\partial n=\beta u+g$ on $\partial D$ with constant coefficients $\alpha ,\beta \in ℝ$.

## How to cite

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Bandle, Catherine, von Below, Joachim, and Reichel, Wolfgang. "Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions." Journal of the European Mathematical Society 010.1 (2008): 73-104. <http://eudml.org/doc/277364>.

@article{Bandle2008,
abstract = {We consider linear elliptic equations $-\Delta u+q(x)u=\lambda u+f$ in bounded Lipschitz domains $D\subset \mathbb \{R\}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma (x)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma (x)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and $\int _D q(x)dx>0$ then there exist two eigenvalues $\lambda _\{-1\}$, $\lambda _1$ such the positivity principle holds for $\lambda \in (\lambda _\{-1\},\lambda _1)$ and the anti-maximum principle holds if $\lambda \in (\lambda _1,\lambda _1+\delta )$ or $\lambda \in (\lambda _\{-1\}-\epsilon ,\lambda _\{-1\})$. A similar, but more complicated result holds if $q\equiv 0$. This is due to the fact that $\lambda _0=0$ becomes an eigenvalue in this case and that $\lambda _1(\sigma )$ as a function of $\sigma$ connects to $\lambda _\{-1\}(\sigma )$ when the mean value of $\sigma$ crosses the value $\sigma _0=-|D|/|\partial D|$. In dimension $N=1$ we determine the optimal $\lambda$-interval such that the anti-maximum principles holds uniformly for all right-hand sides $f,g\ge 0$. Finally, we apply our result to the problem $-\Delta u+q(x)u=\alpha u+f$ in $D$, $\partial u/\partial n=\beta u+g$ on $\partial D$ with constant coefficients $\alpha ,\beta \in \mathbb \{R\}$.},
author = {Bandle, Catherine, von Below, Joachim, Reichel, Wolfgang},
journal = {Journal of the European Mathematical Society},
keywords = {positivity principle; anti-maximum principle; eigenvalues; Harnack inequality; positivity principle; anti-maximum principle; eigenvalues; Harnack inequality},
language = {eng},
number = {1},
pages = {73-104},
publisher = {European Mathematical Society Publishing House},
title = {Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions},
url = {http://eudml.org/doc/277364},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Bandle, Catherine
AU - von Below, Joachim
AU - Reichel, Wolfgang
TI - Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 1
SP - 73
EP - 104
AB - We consider linear elliptic equations $-\Delta u+q(x)u=\lambda u+f$ in bounded Lipschitz domains $D\subset \mathbb {R}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma (x)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma (x)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and $\int _D q(x)dx>0$ then there exist two eigenvalues $\lambda _{-1}$, $\lambda _1$ such the positivity principle holds for $\lambda \in (\lambda _{-1},\lambda _1)$ and the anti-maximum principle holds if $\lambda \in (\lambda _1,\lambda _1+\delta )$ or $\lambda \in (\lambda _{-1}-\epsilon ,\lambda _{-1})$. A similar, but more complicated result holds if $q\equiv 0$. This is due to the fact that $\lambda _0=0$ becomes an eigenvalue in this case and that $\lambda _1(\sigma )$ as a function of $\sigma$ connects to $\lambda _{-1}(\sigma )$ when the mean value of $\sigma$ crosses the value $\sigma _0=-|D|/|\partial D|$. In dimension $N=1$ we determine the optimal $\lambda$-interval such that the anti-maximum principles holds uniformly for all right-hand sides $f,g\ge 0$. Finally, we apply our result to the problem $-\Delta u+q(x)u=\alpha u+f$ in $D$, $\partial u/\partial n=\beta u+g$ on $\partial D$ with constant coefficients $\alpha ,\beta \in \mathbb {R}$.
LA - eng
KW - positivity principle; anti-maximum principle; eigenvalues; Harnack inequality; positivity principle; anti-maximum principle; eigenvalues; Harnack inequality
UR - http://eudml.org/doc/277364
ER -

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