A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight

• Main Authors: Hongya Gao, Jinjing Qiao
• Format: Article
• Language: English
• Published: SpringerOpen 2011-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Caccioppoli-type estimate; very weak solution; obstacle problem; Mucken-houpt weight; <inline-formula><m:math name="1029-242X-2011-58-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="script">A</m:mi> </m:math> </inline-formula>-harmonic equation
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2011/1/58
• ISSN: 1025-5834; 1029-242X

<p>Abstract</p> <p>This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the <inline-formula><m:math xmlns:m="http://www.w3.org/1998/Math/MathML" name="1029-242X-2011-58-i1"><m:mi mathvariant="script">A</m:mi></m:math> </inline-formula>-harmonic equation <inline-formula><m:math name="1029-242X-2011-58-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">div</m:mtext> </m:mstyle> <m:mi mathvariant="script">A</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mo class="MathClass-op">&#8711;</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> </m:math> </inline-formula> with <inline-formula><m:math name="1029-242X-2011-58-i3" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo class="MathClass-rel">|</m:mo> <m:mi mathvariant="script">A</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>&#958;</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">|</m:mo> <m:mo class="MathClass-rel">&#8776;</m:mo> <m:mi>w</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">|</m:mo> <m:mi>&#958;</m:mi> <m:msup> <m:mrow> <m:mo class="MathClass-rel">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> </inline-formula>, where 1 &lt; <it>p </it>&lt; &#8734; and <it>w</it>(<it>x</it>) be a Muckenhoupt <it>A</it>₁weight.</p> <p><b>Mathematics Subject Classification (2000) </b>35J50, 35J60</p>