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<title>Dirichlet-to-Neumann-Operator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Dirichlet-to-Neumann-Operator</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Dirichlet-to-Neumann-Operator</b> (auch <b>Poincaré-Steklow-Operator</b> genannt) ist in der Theorie der <a href="Elliptische_partielle_Differentialgleichung" title="Elliptische partielle Differentialgleichung">elliptischen partiellen Differentialgleichungen</a> ein <a href="Elliptischer_Operator" class="mw-redirect" title="Elliptischer Operator">elliptischer</a>, <a href="Selbstadjungiert" class="mw-redirect" title="Selbstadjungiert">selbstadjungierter</a> <a href="Pseudodifferentialoperator" title="Pseudodifferentialoperator">Pseudodifferentialoperator</a> der Ordnung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>, der die <a href="Dirichlet-Randbedingung" title="Dirichlet-Randbedingung">Dirichlet-Randbedingungen</a> auf die <a href="Neumann-Randbedingung" title="Neumann-Randbedingung">Neumann-Randbedingungen</a> abbildet. Im einfachen Fall bildet der Operator eine auf dem Rand einer <a href="Kompakter_Raum" title="Kompakter Raum">kompakten</a>, <a href="Glatte_Mannigfaltigkeit" class="mw-redirect" title="Glatte Mannigfaltigkeit">glatten Mannigfaltigkeit</a> glatte Funktion auf die <a href="Tangentialraum#Richtungen_der_Tangentialvektoren" title="Tangentialraum">äußere</a> <a href="Richtungsableitung#Normalenableitung_auf_Gebieten" title="Richtungsableitung">Normalenableitung</a> der <a href="Harmonische_Funktion" title="Harmonische Funktion">harmonischen</a> Erweiterung ab.

</p><p>Der Operator taucht in diversen <a href="Inverses_Problem" title="Inverses Problem">inversen Problemen</a> auf. Die Eigenwerte des Operators nennt man Steklow-Eigenwerte (nach <a href="Wladimir_Andrejewitsch_Steklow" title="Wladimir Andrejewitsch Steklow">Wladimir Andrejewitsch Steklow</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> eine glatte, kompakte Mannigfaltigkeit der Dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> mit Rand <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16feddaad462c2a1c9efdaeee062a0484a023fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }" loading="lazy"></span>. Für eine Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{\infty }(\partial \Omega )}">

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<annotation encoding="application/x-tex">{\displaystyle f\in C^{\infty }(\partial \Omega )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47a741e818d008af75a0ac57b0209d4c34ced1cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.598ex; height:2.843ex;" alt="{\displaystyle f\in C^{\infty }(\partial \Omega )}" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {f}}\in C^{\infty }({\overline {\Omega }})}">

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<annotation encoding="application/x-tex">{\displaystyle {\widetilde {f}}\in C^{\infty }({\overline {\Omega }})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/442e49d6922af6909504e2b3b40495f123fb31b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.815ex; height:3.509ex;" alt="{\displaystyle {\widetilde {f}}\in C^{\infty }({\overline {\Omega }})}" loading="lazy"></span> die <i>harmonische Erweiterung</i>, das heißt, es gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widetilde {f}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta {\widetilde {f}}=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c364bbc9cd778043e2ff1f005491b70e57752484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.896ex; height:3.009ex;" alt="{\displaystyle \Delta {\widetilde {f}}=0}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {f}}\mid _{\partial \Omega }=f}">

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<annotation encoding="application/x-tex">{\displaystyle {\widetilde {f}}\mid _{\partial \Omega }=f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d6ca92834145581b0545047cfcf52d9bd2f17d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.074ex; height:3.176ex;" alt="{\displaystyle {\widetilde {f}}\mid _{\partial \Omega }=f}" loading="lazy"></span>.

</p><p>Der <b>Dirichlet-to-Neumann-Operator</b> ist der Operator

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}:C^{\infty }(\partial \Omega )\to C^{\infty }(\partial \Omega )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a04df49761b96a050a14957741de4851209c4164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.3ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}:C^{\infty }(\partial \Omega )\to C^{\infty }(\partial \Omega )}" loading="lazy"></span>,</dd></dl>

<p>definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}f=\partial _{v}({\widetilde {f}})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a25e87d04f5c2effd7ff90ceff9d3a0d5390e6ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.941ex; height:3.176ex;" alt="{\displaystyle {\mathcal {D}}f=\partial _{v}({\widetilde {f}})}" loading="lazy"></span>,</dd></dl>

<p>wobei

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<annotation encoding="application/x-tex">{\displaystyle \partial _{v}({\widetilde {f}})=\langle \nabla ({\widetilde {f}})\mid _{\partial \Omega },v\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca683e619621803162b4aab4d133066d20bdbedd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.929ex; height:3.176ex;" alt="{\displaystyle \partial _{v}({\widetilde {f}})=\langle \nabla ({\widetilde {f}})\mid _{\partial \Omega },v\rangle }" loading="lazy"></span></dd></dl>

<p>die äußere <a href="Normalenableitung" class="mw-redirect" title="Normalenableitung">Normalenableitung</a> ist.

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<div class="mw-heading mw-heading3"><h3 id="Allgemeine_Form">Allgemeine Form</h3></div>

<p>Ersetzt man die Bedingung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widetilde {f}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta {\widetilde {f}}=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c364bbc9cd778043e2ff1f005491b70e57752484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.896ex; height:3.009ex;" alt="{\displaystyle \Delta {\widetilde {f}}=0}" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widetilde {f}}=\lambda {\widetilde {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta {\widetilde {f}}=\lambda {\widetilde {f}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0887afe237ddbbb002efabf60e4cf67c2c0bfad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.788ex; height:3.009ex;" alt="{\displaystyle \Delta {\widetilde {f}}=\lambda {\widetilde {f}}}" loading="lazy"></span>, dann erhält man eine allgemeinere Form des <i>Dirichlet-to-Neumann-Operator</i>s, welche mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{\lambda }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{\lambda }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c197df9d88fca24c9a164badaa4f09c57abc2899.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{\lambda }}" loading="lazy"></span> notiert wird.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Michael_E._Taylor" title="Michael E. Taylor">Michael E. Taylor</a>: <i>Partial Differential Equations II: Qualitative Studies of Linear Equations.</i> Springer-Verlag, New York 1996, ISBN 978-1-4757-4187-2, S. 41</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Alexandre Girouard, Mikhail Karpukhin, Michael Levitin und Iosif Polterovich: <cite style="font-style:italic">The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript</cite>. Hrsg.: arXiv. 2021.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dirichlet-to-Neumann-Operator&rft.au=Alexandre+Girouard%2C+Mikhail+Karpukhin%2C+Michael+Levitin+und+Iosif+Polterovich&rft.btitle=The+Dirichlet-to-Neumann+map%2C+the+boundary+Laplacian%2C+and+H%C3%B6rmander%27s+rediscovered+manuscript&rft.date=2021&rft.genre=book" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">W. Arendt, A. F. M. ter Elst, J. B. Kennedy und M. Sauter: <cite style="font-style:italic">The Dirichlet-to-Neumann operator via hidden compactness</cite>. In: <cite style="font-style:italic">J. Funct. Anal.</cite> <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>266</span>, 2014, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1757 ––1786</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dirichlet-to-Neumann-Operator&rft.atitle=The+Dirichlet-to-Neumann+operator+via+hidden+compactness&rft.au=W.+Arendt%2C+A.+F.+M.+ter+Elst%2C+J.+B.+Kennedy+und+M.+Sauter&rft.btitle=J.+Funct.+Anal.&rft.date=2014&rft.genre=book&rft.pages=1757+--1786&rft.volume=266" style="display:none"> </span></span>

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