An <mml:math alttext="$H$"> <mml:mi>H</mml:mi> </mml:math>-system for a revolution surface without boundary

• Format: Article
• Language: English
• Published: Hindawi Limited 2006-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/93163

<p>We study the existence of solutions an <mml:math alttext="$H$"> <mml:mi>H</mml:mi> </mml:math>-system for a revolution surface without boundary for <mml:math alttext="$H$"> <mml:mi>H</mml:mi> </mml:math> depending on the radius <mml:math alttext="$f$"> <mml:mi>f</mml:mi> </mml:math>. Under suitable conditions we prove that the existence of a solution is equivalent to the solvability of a scalar equation <mml:math alttext="$N(a)={L}/{sqrt{2}}$"> <mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow></mml:mrow> </mml:math>, where <mml:math alttext="$N:mathcal{A}subset mathbb{R}^+ ightarrow mathbb{R}$"> <mml:mi>N</mml:mi><mml:mo>:</mml:mo><mml:mi>&#x1D49C;</mml:mi><mml:mo>&#x2282;</mml:mo><mml:msup> <mml:mi>&#x211D;</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>&#x2192;</mml:mo><mml:mi>&#x211D;</mml:mi> </mml:math> is a function depending on <mml:math alttext="$H$"> <mml:mi>H</mml:mi> </mml:math>. Moreover, using the method of upper and lower solutions we prove existence results for some particular examples. In particular, applying a diagonal argument we prove the existence of unbounded surfaces with prescribed <mml:math alttext="$H$"> <mml:mi>H</mml:mi> </mml:math>.</p>