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

DOI:
https://doi.org/10.1090/S0002-9939-03-07285-X

Published electronically:
November 19, 2003

MathSciNet review:
2045415

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Abstract | References | Similar Articles | Additional Information

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

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