On solutions of elliptic equations that decay rapidly on paths

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an elliptic differential operator on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {R}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant coefficients. It is known that if <italic>u</italic> is a solution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis upper D right-parenthesis u equals 0"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">P(D)u = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on an unbounded domain and if <italic>u</italic> decays uniformly and sufficiently rapidly, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u equals 0"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">u = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this note it is shown that the same conclusion holds if <italic>u</italic> decays rapidly, but not a priori uniformly, on a sufficiently large set of unbounded paths.

Locations

• Proceedings of the American Mathematical Society - View - PDF