Tunneling estimates and approximate controllability for hypoelliptic equations

Abstract

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a compact manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> assuming: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</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="left-parenthesis i i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(ii)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the analyticity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the coefficients of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The first result is the tunneling estimate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar phi double-vertical-bar Subscript upper L squared left-parenthesis omega right-parenthesis Baseline greater-than-or-equal-to upper C e Superscript minus c lamda Super Superscript StartFraction k Over 2 EndFraction"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi> φ </mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo> ≥ </mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mi> λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for normalized eigenfunctions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a nonempty open set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega subset-of script upper M"> <mml:semantics> <mml:mrow> <mml:mi> ω </mml:mi> <mml:mo> ⊂ </mml:mo>

Topics & Concepts