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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Strong comparison principle for solutions of quasilinear equations
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by M. Lucia and S. Prashanth PDF
Proc. Amer. Math. Soc. 132 (2004), 1005-1011 Request permission

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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1005-1011
  • MSC (2000): Primary 35B50
  • DOI: https://doi.org/10.1090/S0002-9939-03-07285-X
  • MathSciNet review: 2045415