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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Solvability of quasilinear elliptic equations with nonlinear boundary conditions

Author: Gary M. Lieberman
Journal: Trans. Amer. Math. Soc. 273 (1982), 753-765
MSC: Primary 35J65
MathSciNet review: 667172
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Abstract: On an $ n$-dimensional domain $ \Omega $, we consider the boundary value problem

$\displaystyle (\ast)\quad Qu = 0\;{\text{in}}\Omega {\text{,}}\quad Nu = 0\;{\text{on}}\;\partial \Omega $

where $ Q$ is a quasilinear elliptic second-order differential operator and $ N$ is a nonlinear first order differential operator satisfying an Agmon-Douglis-Nirenberg consistency condition. If the coefficients of $ Q$ and $ N$ satisfy additional hypotheses (such as sufficient smoothness), Fiorenza was able to reduce the solvability of $ (\ast)$ to the establishment of a priori bounds for solutions of a related family of boundary value problems. We simplify Fiorenza's argument, obtaining the reduction under more general hypotheses and requiring a priori bounds only for solutions of $ Qu = f$, $ Nu = g$ where $ f$ and $ g$ range over suitable function spaces. As an example, classical solutions of the capillary problem are shown to exist without using the method of elliptic regularization.

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Keywords: Quasilinear elliptic equations, nonlinear boundary conditions, capillary problem, fixed point theorem
Article copyright: © Copyright 1982 American Mathematical Society