Type: Article
Publication Date: 1988-01-01
Citations: 22
DOI: https://doi.org/10.1090/s0002-9947-1988-0965751-8
We study the Wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis x comma nabla u right-parenthesis dot nabla u almost-equals StartAbsoluteValue nabla u EndAbsoluteValue Superscript p"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>≈<!-- ≈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A(x,\,\nabla u) \cdot \nabla u \approx |\nabla u{|^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Local solutions are continuous at Wiener points of the obstacle function; if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the converse is also shown to be true. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then a characterization of the thinness of a set at a point is given in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-superharmonic functions.