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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strong comparison principle for solutions of quasilinear equations


Authors: M. Lucia and S. Prashanth
Journal: Proc. Amer. Math. Soc. 132 (2004), 1005-1011
MSC (2000): Primary 35B50
Published electronically: November 19, 2003
MathSciNet review: 2045415
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Abstract: Let $\Omega \subset \mathbb{R} ^N$, $N \geq 1$, be a bounded smooth connected open set and $\mathbf{a} : \Omega \times \mathbb{R} ^N \to \mathbb{R} ^N$ be a map satisfying the hypotheses (H1)-(H4) below. Let $f_1,f_2 \in \mathrm{L}_{loc}^{1} (\Omega)$ with $f_2 \geq f_1$, $f_1 \not\equiv f_2$ in $\Omega$ and $u_1, u_2 \in \mathcal{C}^{1,\theta} (\overline \Omega)$ with $\theta \in (0,1]$ be two weak solutions of

\begin{displaymath}(P_i)\quad -\mathrm{div} (\mathbf{a}(x,\nabla u_i)) = f_i \quad \hbox{in } \Omega, \,\quad i=1,2.\end{displaymath}

Suppose that $u_2 \geq u_1$ in $\Omega$. Then we show that $u_2 > u_1$ in $\Omega$ under the following assumptions: either $u_2>u_1$ on $\partial \Omega$, or $u_1=u_2=0$ on $\partial \Omega$ and $u_1 \geq 0$ in $\Omega$. We also show a measure-theoretic version of the Strong Comparison Principle.


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Additional Information

M. Lucia
Affiliation: Department Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854
Email: mlucia@math.rutgers.edu

S. Prashanth
Affiliation: TIFR Center, IISc. Campus, Post Box No. 1234, Bangalore 560012, India
Email: pras@math.tifrbng.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07285-X
PII: S 0002-9939(03)07285-X
Received by editor(s): August 20, 2002
Published electronically: November 19, 2003
Additional Notes: The first author was supported by Swiss National Foundation, Contract No. 8220-64676
The second author was supported by Indo-French Project (IFCPAR) No. 1901-2
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society