Beschreibung:
<p>A theorem about the separation of sub- and superfunctions <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon">
<mml:semantics>
<mml:mi>υ<!-- υ --></mml:mi>
<mml:annotation encoding="application/x-tex">\upsilon</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w">
<mml:semantics>
<mml:mi>w</mml:mi>
<mml:annotation encoding="application/x-tex">w</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> by solutions of an ordinary differential equation of second order is proved, where <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon greater-than-or-slanted-equals w">
<mml:semantics>
<mml:mrow>
<mml:mi>υ<!-- υ --></mml:mi>
<mml:mo>⩾<!-- ⩾ --></mml:mo>
<mml:mi>w</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\upsilon \geqslant w</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> throughout the given interval. Examples show that the condition imposed on the right side <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f">
<mml:semantics>
<mml:mi>f</mml:mi>
<mml:annotation encoding="application/x-tex">f</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of the equation is sharp, and that an analogous theorem is not true for Laplace’s equation, even in the case <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f identical-to 0 comma upsilon">
<mml:semantics>
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mo>≡<!-- ≡ --></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>υ<!-- υ --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">f \equiv 0,\upsilon</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> sub- and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w">
<mml:semantics>
<mml:mi>w</mml:mi>
<mml:annotation encoding="application/x-tex">w</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> superharmonic.</p>