2011/11/07 by J. Taylor, Justin L. Taylor, K. Ott +3 · 1 citation
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Boundary (topology) #Bounded function #Domain (mathematical analysis) #Lipschitz continuity #Lipschitz domain #Nonlinear Partial Differential Equations #Upper and lower bounds #math.AP #msc:35J05 #msc:35J25
paper · pdf · doi:10.1090/s0002-9947-2012-05711-4
published as Trans. Amer. Math. Soc., 365 (2013), 2895-2930 · 36 pages
arxiv created 2011/11/07 · openalex publication_date 2012/12/13 · arxiv updated 2013/05/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of bold upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">Ω ⊂ \mathbf Rn</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="n greater-than-or-equal-to 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> , with boundary that is decomposed as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential normal upper Omega equals upper D union upper N"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>=</mml:mo> <mml:mi>D</mml:mi> <mml:mo> ∪ </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">∂ Ω =D∪ N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <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 N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> disjoint. We let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">Λ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the boundary 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> (relative to <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 </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">∂ Ω</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) and impose conditions on the dimension and shape of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">Λ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</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"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Under these geometric criteria, we show that there exists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p 0 greater-than 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p0>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depending on the 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 </mml:mi> <mml:annotation encoding="application/x-tex">Ω</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for <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> in the interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma p 0 right-parenthesis">