Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

Franco Obersnel; Pierpaolo Omari

doi:10.3934/dcds.2013.33.305

Article Contents

2013, Volume 33, Issue 1: 305-320. Doi: 10.3934/dcds.2013.33.305

This issue Previous Article Generalized linear differential equations in a Banach space: Continuous dependence on a parameter Next Article Infinitely many solutions for some singular elliptic problems

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

Franco Obersnel^1, and
Pierpaolo Omari^1,

1.
Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste

Received: July 31, 2011

Published: December 2012

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Abstract

We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = f(t,u) \quad \hbox{ in } {]-r,r[}, \qquad u(-r)=a, \, u(r) = b. \end{equation*} We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.

Keywords:

Quasilinear ordinary differential equation,
capillarity equation,
Dirichlet problem,
bounded variation solution,
classical solution,
existence,
regularity,
boundary behaviour.

Mathematics Subject Classification: Primary: 34B15, 35J93; Secondary: 49Q20, 76D45.

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