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• W2024430140 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a subdomain of a bounded domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</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=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> . The conductivity coefficient of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k not-equals 1> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>k ne 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the conductivity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega minus upper D> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mi class=MJX-variant mathvariant=normal>∖<!-- ∖ --></mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega backslash D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equal to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For a given current density <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we compute the resulting potential <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding=application/x-tex>u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and denote by <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> the value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding=application/x-tex>u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The general inverse problem is to estimate the location of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the known measurements of the voltage <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>. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a family of domains for which the Hausdorff distance <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper D comma upper D Subscript h Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>d(D,{D_h})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equal to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis h right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(h)</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=h> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=application/x-tex>h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> small), then the corresponding measurements <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis h right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(h)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> close to <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>. This paper is concerned with proving the inverse, that is, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper D comma upper D Subscript h Baseline right-parenthesis less-than-or-equal-to StartFraction 1 Over c EndFraction double-vertical-bar f Subscript h Baseline minus f double-vertical-bar> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>c</mml:mi> </mml:mfrac> <mml:mrow> <mml:mo symmetric=true>‖</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>f</mml:mi> <mml:mo symmetric=true>‖</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>d(D,{D_h}) leq frac {1}{c}left | {f_h} - fright |</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=c greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>c > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; the domains <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</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=upper D Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are assumed to be piecewise smooth. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 3> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we assume in proving the above result, that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript h Baseline superset-of upper D> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊃<!-- ⊃ --></mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_h} supset D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript h Baseline subset-of upper D> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_h} subset D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) for all small <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=application/x-tex>h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals 2> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> this monotonicity condition is dropped, provided <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is appropriately chosen. The above stability estimate provides quantitative information on the location of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by means of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ." @default.
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• W2024430140 title "Stability for an inverse problem in potential theory" @default.
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