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• W1618098293 abstract "We consider the Dirichlet problem <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout Enlarged left-brace 1st Row 1st Column script upper L u 2nd Column a m p semicolon equals 0 3rd Column a m p semicolon 4th Column a m p semicolon in upper D comma 2nd Row 1st Column u 2nd Column a m p semicolon equals g 3rd Column a m p semicolon 4th Column a m p semicolon on partial-differential upper D EndLayout> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=right left right left right left right left right left right left rowspacing=3pt columnspacing=0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mrow> <mml:mtext>in </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>D</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mrow> <mml:mtext>on </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>left { begin {aligned} mathcal {L} u & = 0 &&text {in $D$}, u &= g &&text {on $partial D$} end {aligned} right .</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> for two second-order elliptic operators <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L Subscript k Baseline u equals sigma-summation Underscript i comma j equals 1 Overscript n Endscripts a Subscript k Superscript i comma j Baseline left-parenthesis x right-parenthesis partial-differential Subscript i j Baseline u left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <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>mathcal {L}_k u=sum _{i,j=1}^na_k^{i,j}(x),partial _{ij} u(x)</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=k equals 0 comma 1> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>k=0,1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in a bounded Lipschitz domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D subset-of double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <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>Dsubset mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript k Superscript i comma j> <mml:semantics> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=application/x-tex>a_k^{i,j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L 0> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>{mathcal {L}}_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is regular in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript p Baseline left-parenthesis partial-differential upper D comma d sigma right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^p(partial D, dsigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <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>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than p greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>1>p>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar upper N u double-vertical-bar Subscript upper L Sub Superscript p Baseline less-than-or-equal-to upper C double-vertical-bar g double-vertical-bar Subscript upper L Sub Superscript p> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>N</mml:mi> <mml:mi>u</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:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>g</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:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>|Nu|_{L^p}le C,|g|_{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all continuous boundary data <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>. Here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=application/x-tex>sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the surface measure on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential upper D> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial 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 N u> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon Superscript i comma j Baseline left-parenthesis x right-parenthesis equals a 1 Superscript i comma j Baseline left-parenthesis x right-parenthesis minus a 0 Superscript i comma j Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>ε<!-- ε --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msubsup> <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>varepsilon ^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that will assure the perturbed operator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L 1> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {L}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be regular in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript q Baseline left-parenthesis partial-differential upper D comma d sigma right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^q(partial D,dsigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</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=1 greater-than q greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>1>q>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1618098293 date "2002-10-01" @default.
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• W1618098293 title "The 𝐿^{𝑝} Dirichlet problem and nondivergence harmonic measure" @default.
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