Unique continuation for Δ+𝑣 and the C. Fefferman-Phong class

Abstract

We show that the strong unique continuation property holds for the inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue normal upper Delta u EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue upsilon EndAbsoluteValue StartAbsoluteValue u EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>υ<!-- υ --></mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\left | {\Delta u} \right | \leq \left | \upsilon \right |\left | u \right |</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the potential <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>υ<!-- υ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\upsilon (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies the C. Fefferman-Phong condition in a certain range of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> values. We also deal with the situation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishing at infinity. These are all consequences of appropriate Carleman inequalities.

Locations

• Transactions of the American Mathematical Society - View - PDF