GermanWikipedia - /page/entry.mu

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<title>Neumann-Randbedingung</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Neumann-Randbedingung</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>Eine <b>Neumann-Randbedingung</b> (nach <a href="Carl_Gottfried_Neumann" title="Carl Gottfried Neumann">Carl Gottfried Neumann</a>) bezeichnet im Zusammenhang mit <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a> (genauer: <a href="Randwertproblem" title="Randwertproblem">Randwertproblemen</a>) Werte, die auf dem <a href="Rand_(Topologie)" title="Rand (Topologie)">Rand</a> des Definitionsbereichs für die <a href="Richtungsableitung" title="Richtungsableitung">Normalableitung</a> der Lösung vorgegeben werden. Bei Neumann-Randwertproblemen werden nicht Funktionswerte, sondern Ableitungswerte vorgegeben. Weitere Randbedingungen sind beispielsweise <a href="Dirichlet-Randbedingung" title="Dirichlet-Randbedingung">Dirichlet-Randbedingungen</a> (bei denen die Funktionswerte auf dem Rand vorgegeben sind) oder <a href="Randwertproblem#Schiefe_Randbedingung" title="Randwertproblem">schiefe Randbedingungen</a>.

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<div class="mw-heading mw-heading2"><h2 id="Gewöhnliche_Differentialgleichung"><span id="Gew.C3.B6hnliche_Differentialgleichung"></span>Gewöhnliche Differentialgleichung</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Das_Neumannproblem">Das Neumannproblem</h3></div>

<p>Im Falle einer <a href="Gew%C3%B6hnliche_Differentialgleichung" title="Gewöhnliche Differentialgleichung">gewöhnlichen Differentialgleichung</a> ist der Definitionsbereich der Funktion ein abgeschlossenes Intervall. Folglich besteht der Rand des Definitionsbereiches nur aus dem rechten und dem linken Intervallende. Aufgrund der Freiheit in gewöhnlichen Differentialgleichungen sind Neumann-Randbedingungen nur für Gleichungen von zweiter oder höherer Ordnung sinnvoll. In diesem Fall sieht ein Neumannproblem, d. h. eine Differentialgleichung mit Neumann-Randbedingung folgendermaßen aus:

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<annotation encoding="application/x-tex">{\displaystyle y\in C^{2}(a,b)\cap C^{1}[a,b]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3075d38825817a2d55f461f3f81cacaadd117fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.908ex; height:3.176ex;" alt="{\displaystyle y\in C^{2}(a,b)\cap C^{1}[a,b]}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle y^{\prime \prime }=f(x,y,y')}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b51f164c840a8dbd5652488e90c0f9ca001895d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.882ex; height:3.009ex;" alt="{\displaystyle y^{\prime \prime }=f(x,y,y')}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle y^{\prime }(a)=\alpha ,\quad y'(b)=\beta }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8cf2be2f0af74e0e378e73af99631f54e99dca4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.909ex; height:3.009ex;" alt="{\displaystyle y^{\prime }(a)=\alpha ,\quad y'(b)=\beta }" loading="lazy"></span></dd></dl>

<p>Hierbei ist die rechte Seite <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> der Differentialgleichung eine vorgeschriebene 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 \alpha }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" 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 \beta }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> sind vorgeschriebene <a href="Reelle_Zahl" title="Reelle Zahl">reelle Zahlen</a> für die Werte der ersten Ableitung einer Lösung an den Intervallenden. Schließlich wird eine Lösung <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 y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> aus der angegebenen <a href="Regularit%C3%A4tsklasse" class="mw-redirect" title="Regularitätsklasse">Regularitätsklasse</a> gesucht.

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<div class="mw-heading mw-heading3"><h3 id="Beispiel_für_eine_gewöhnliche_Differentialgleichung"><span id="Beispiel_f.C3.BCr_eine_gew.C3.B6hnliche_Differentialgleichung"></span>Beispiel für eine gewöhnliche Differentialgleichung</h3></div>

<p>Wir wählen als unser Intervall <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 [0,\pi ]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e2a912eda6ef1afe46a81b518fe9da64a832751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}" loading="lazy"></span> und betrachten das folgende Problem:

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<annotation encoding="application/x-tex">{\displaystyle y\in C^{2}(0,\pi )\cap C^{1}[0,\pi ]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7df53d8f96c1d4385895c005f3c925f57db5bb48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.443ex; height:3.176ex;" alt="{\displaystyle y\in C^{2}(0,\pi )\cap C^{1}[0,\pi ]}" loading="lazy"></span></dd>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc7293b16d52f2a42cf102e32153c63e341ef7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle y''=-y}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle y'(0)=0,\quad y'(\pi )=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84ec03d20d1a3666315f61a33276ac26cca5941f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.682ex; height:3.009ex;" alt="{\displaystyle y'(0)=0,\quad y'(\pi )=0}" loading="lazy"></span></dd></dl>

<p>Mit der Theorie der linearen gewöhnlichen Differentialgleichungen mit konstanten Koeffizienten erhalten wir zunächst als allgemeine Lösung der Differentialgleichung:

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<annotation encoding="application/x-tex">{\displaystyle y(x)=C\cos x+D\sin x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f96bddb5bf7b0f604a7b03fc48f9d54f26154124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.098ex; height:2.843ex;" alt="{\displaystyle y(x)=C\cos x+D\sin x}" loading="lazy"></span></dd></dl>

<p>mit der Ableitung

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<annotation encoding="application/x-tex">{\displaystyle y'(x)=-C\sin x+D\cos x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a77772221ed50e0712334c325424f811fa5e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.596ex; height:3.009ex;" alt="{\displaystyle y'(x)=-C\sin x+D\cos x}" loading="lazy"></span></dd></dl>

<p>und zwei frei wählbaren reellen Konstanten <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 C}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" 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 D}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>. Wir benutzen die Randbedingungen, um diese Konstanten zu fixieren. Dabei erhalten wir ein <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">lineares Gleichungssystem</a> in den Unbekannten <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 C}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" 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 D}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle D=0,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eb59a951bb34fd3ffd06c8ef29d8c7f9f7e234e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.832ex; height:2.509ex;" alt="{\displaystyle D=0,}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle -D=0.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6dc8ea310fea63a4d2354621e90dc261cea021b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.64ex; height:2.343ex;" alt="{\displaystyle -D=0.}" loading="lazy"></span></dd></dl>

<p>Bemerkenswerterweise ist dieses System nicht eindeutig lösbar, aber es ist für beliebiges reelles <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 C}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> eine Lösung gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle y(x)=C\cos x.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7a426be1660465ace158119e1510b94194ff0e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.021ex; height:2.843ex;" alt="{\displaystyle y(x)=C\cos x.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Partielle_Differentialgleichungen">Partielle Differentialgleichungen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Das_Neumannproblem_2">Das Neumannproblem</h3></div>

<p>Bei einer <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen Differentialgleichung</a> ist die alleinige Angabe von Neumann-Randbedingungen nur für elliptische Gleichungen auf einem <a href="Beschr%C3%A4nktheit" class="mw-redirect" title="Beschränktheit">beschränkten</a> <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> <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 \Omega \subset \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79774d994aac0be34ef390915fed12cbce816f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{n}}" loading="lazy"></span> sinnvoll, da die anderen Typen auch Vorgaben der Anfangswerte benötigen. Dabei werden Neumann-Randbedingungen auf dem Rand des Gebietes <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> vorgeschrieben. Es wird also die Ableitung der Lösung in Richtung der äußeren Normalen vorgeschrieben. Damit die Ableitung in Richtung der äußeren Normalen an das Gebiet sinnvoll ist, muss dabei notwendig vorausgesetzt werden, dass es sich um einen <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 C^{1}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"></span>-Rand handelt.

</p><p>Wir definieren hier das Neumannproblem für eine quasilineare partielle Differentialgleichung:

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<annotation encoding="application/x-tex">{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecdb18779badb99124c043620fce36dfd32c7f59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.547ex; height:3.509ex;" alt="{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8738d6fb5445837369899b2c228d48d5bbad420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.049ex; height:7.176ex;" alt="{\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9b5ad241788fb5bff23e1d531a23690fd392a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.822ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega }" loading="lazy"></span></dd></dl>

<p>Hierbei stellt die 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 g\colon \partial \Omega \rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle g\colon \partial \Omega \rightarrow \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65712b977e11bc14b8b61829871428678935f4f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.438ex; height:2.509ex;" alt="{\displaystyle g\colon \partial \Omega \rightarrow \mathbb {R} }" loading="lazy"></span> die vorgeschriebene Ableitung in Richtung der äußeren Normalen <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 \mathbf {n} (x)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8730e1c4ba540206433c7e9fe82542c2f7ebc59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.624ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} (x)}" loading="lazy"></span> an <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 }">

</mstyle>

</mrow>

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

</semantics>

</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> von unserer Lösung dar. Allein die Frage nach der Lösbarkeit eines solchen Problemes ist sehr anspruchsvoll und steht im Mittelpunkt der aktuellen Forschung. Es ist auch sehr schwierig, eine allgemeingültige Lösungsmethode anzugeben.

</p>

<div class="mw-heading mw-heading3"><h3 id="Ermittlung_notwendiger_Bedingungen">Ermittlung notwendiger Bedingungen</h3></div>

<p>Es ist jedoch zu beachten, dass allein die Gültigkeit des <a href="Gau%C3%9Fscher_Integralsatz" title="Gaußscher Integralsatz">gaußschen Integralsatzes</a> eine weitere (notwendige) Bedingung an die Daten und an Lösungen unseres Neumannproblems darstellt. Wir haben hierzu lediglich den gaußschen Integralsatz auf das Vektorfeld <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(x)=\nabla u(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\nabla u(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac8536faa4a06ef651aa6bbc6186b7b87692e565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.921ex; height:2.843ex;" alt="{\displaystyle f(x)=\nabla u(x)}" loading="lazy"></span> anzuwenden.

</p><p>Wenn wir beispielsweise eine Lösung eines einfachen linearen Neumannproblems mit dem <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> <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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span> betrachten:

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecdb18779badb99124c043620fce36dfd32c7f59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.547ex; height:3.509ex;" alt="{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}" loading="lazy"></span></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta u(x)=f(x),\qquad x\in \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56d920775ffc48ac59a524a0932cbe3af53add40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.448ex; height:2.843ex;" alt="{\displaystyle \Delta u(x)=f(x),\qquad x\in \Omega }" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega ,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e1619e16933c70e67181d7d7ef5c9ec706332d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.469ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega ,}" loading="lazy"></span></dd></dl>

<p>so erhalten wir unter Anwendung des gaußschen Integralsatzes die Bedingung an die Daten <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msup>

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</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f(x)\,d^{n}x=\int _{\Omega }\operatorname {div} \nabla u(x)\,d^{n}x=\oint _{\partial \Omega }\nabla u(x)\cdot \mathbf {n} (x)\,d^{n-1}\sigma (x)=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d6de36e1331d430c197c796b12f10ddc833fb31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:82.631ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f(x)\,d^{n}x=\int _{\Omega }\operatorname {div} \nabla u(x)\,d^{n}x=\oint _{\partial \Omega }\nabla u(x)\cdot \mathbf {n} (x)\,d^{n-1}\sigma (x)=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x).}" loading="lazy"></span></dd></dl>

<p>Folglich ist die Gültigkeit der Gleichung

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f(x)\,d^{n}x=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25416b4734293d1169f8db4dcfaaf369627f78d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.446ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f(x)\,d^{n}x=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x)}" loading="lazy"></span></dd></dl>

<p>notwendig für die Lösbarkeit dieses Neumannproblems. Bei anderen Problem ist es gegebenenfalls hilfreich geeignete andere Vektorfelder zu betrachten.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_für_eine_partielle_Differentialgleichung"><span id="Beispiel_f.C3.BCr_eine_partielle_Differentialgleichung"></span>Beispiel für eine partielle Differentialgleichung</h3></div>

<p>Wir betrachten in diesem Beispiel auf dem Gebiet <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 \Omega =(0,\pi )^{n}=\{x=(x_{1},...,x_{n})\in \mathbb {R} ^{n}\,:\,0<x_{i}<\pi ,\quad i=1,\dots ,n\},}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =(0,\pi )^{n}=\{x=(x_{1},...,x_{n})\in \mathbb {R} ^{n}\,:\,0<x_{i}<\pi ,\quad i=1,\dots ,n\},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f9a9852db9a105ae36337936ad30c9cf6b3d1b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.004ex; height:2.843ex;" alt="{\displaystyle \Omega =(0,\pi )^{n}=\{x=(x_{1},...,x_{n})\in \mathbb {R} ^{n}\,:\,0<x_{i}<\pi ,\quad i=1,\dots ,n\},}" loading="lazy"></span> mit dem regulären Rand

</p>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta \Omega =\{x=(x_{1},...,x_{n})\in \partial \Omega \,:\,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae0ef064ad881740ef6bacba8e860fb69cb53dc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.23ex; height:2.843ex;" alt="{\displaystyle \delta \Omega =\{x=(x_{1},...,x_{n})\in \partial \Omega \,:\,}" loading="lazy"></span> für genau ein <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 i_{0}\in \{1,...,n\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i_{0}\in \{1,...,n\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d063b8fc86fc5a77fd42ed426d6961efc13b3c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.749ex; height:2.843ex;" alt="{\displaystyle i_{0}\in \{1,...,n\}}" loading="lazy"></span> 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 x_{i_{0}}\in \{0,\pi \}\}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i_{0}}\in \{0,\pi \}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f6ecec4d5591ea487da20754678bb7d425a4c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.817ex; height:3.009ex;" alt="{\displaystyle x_{i_{0}}\in \{0,\pi \}\}}" loading="lazy"></span></dd></dl>

<p>das folgende Randwertproblem:

</p>

<msup>

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</msup>

<msup>

</mrow>

</msup>

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in C^{2}(\Omega )\cap C^{0}({\overline {\Omega }})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b4b8160e96bf2809ab1669471d93a0e6de3d13b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.547ex; height:3.509ex;" alt="{\displaystyle u\in C^{2}(\Omega )\cap C^{0}({\overline {\Omega }})}" loading="lazy"></span></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta u(x)=-nu(x),\qquad x\in \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc580b10302316e27f13b87f3161741ecfc9c41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.702ex; height:2.843ex;" alt="{\displaystyle \Delta u(x)=-nu(x),\qquad x\in \Omega }" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=0,\qquad x\in \partial \Omega .}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e7a69afb686fd890cfa0efefeb6cfc81c4b91f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.376ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=0,\qquad x\in \partial \Omega .}" loading="lazy"></span></dd></dl>

<p>Hierbei bezeichnet <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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span> den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a>. Zunächst stellen wir fest, dass <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 u\equiv 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\equiv 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a574154028b4815f68459f09d85e82f0e983ff9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u\equiv 0}" loading="lazy"></span> eine Lösung des Problems ist. Um weitere Lösungen zu finden, können wir rein formal dem Beispiel zu <a href="Dirichlet-Randbedingung" title="Dirichlet-Randbedingung">Dirichlet-Randbedingungen</a> partieller Differentialgleichungen folgen, und erhalten nach einem Produktansatz:

</p>

</mrow>

</mrow>

</munderover>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x)=D\prod _{k=1}^{n}\cos(x_{k}).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b4e469d66066e835571b920bf0e6a03003b046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.221ex; height:6.843ex;" alt="{\displaystyle u(x)=D\prod _{k=1}^{n}\cos(x_{k}).}" loading="lazy"></span></dd></dl>

<p>Wir müssen aber beachten, dass wir hier eigentlich nicht die Nullstellenfreiheit von <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 u}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> fordern können, da die Cosinusfunktion bekanntermaßen eine <a href="Nullstelle" title="Nullstelle">Nullstelle</a> bei <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 {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy"></span> hat. Das bedeutet, dass wir nicht wissen, ob unsere formale Lösung auch wirklich Lösung unseres Neumannproblems ist. Wenn wir dies aber einsetzen, stellen wir fest, dass wir Glück haben und unser <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 u}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> tatsächlich Lösung unseres Problems ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung_für_partielle_Differentialgleichungen"><span id="Verallgemeinerung_f.C3.BCr_partielle_Differentialgleichungen"></span>Verallgemeinerung für partielle Differentialgleichungen</h2></div>

<p>Häufig ist es ratsam, allgemeinere Randwertprobleme wie

</p>

<msup>

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<msup>

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</msup>

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecdb18779badb99124c043620fce36dfd32c7f59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.547ex; height:3.509ex;" alt="{\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}" loading="lazy"></span></dd>

</mrow>

</mrow>

</munderover>

<msub>

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</msub>

<mfrac>

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<msub>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8738d6fb5445837369899b2c228d48d5bbad420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.049ex; height:7.176ex;" alt="{\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {\nu } }}=g(x),\qquad x\in \partial \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a3a120d2ee3303df0f610d5ca87dfd6a5a5da44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.822ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial u(x)}{\partial \mathbf {\nu } }}=g(x),\qquad x\in \partial \Omega }" loading="lazy"></span></dd></dl>

<p>zu betrachten. In diesem Fall 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 {\tfrac {\partial }{\partial \mathbf {\nu } }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c29eb14d58664ce5bb5dd1a5b52bd6e37032a86f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.639ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial }{\partial \mathbf {\nu } }}}" loading="lazy"></span> eine <a href="Richtungsableitung" title="Richtungsableitung">Richtungsableitung</a> in eine äußere Richtung. 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 \mathbf {\nu } (x)\cdot \mathbf {n} (x)>0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\nu } (x)\cdot \mathbf {n} (x)>0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ca968fe9572780a51c6c6ab51f2ad64e2ce2eae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.935ex; height:2.843ex;" alt="{\displaystyle \mathbf {\nu } (x)\cdot \mathbf {n} (x)>0}" loading="lazy"></span> für alle <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 x\in \partial \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3d1eeef2325457e76e35ef35facf8979ea17b65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.166ex; height:2.176ex;" alt="{\displaystyle x\in \partial \Omega }" loading="lazy"></span>. Wir beachten aber, dass der Richtungsvektor <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 \mathbf {\nu } (x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\nu } (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/842829dcb9919bc4122f556937893345efba44fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.371ex; height:2.843ex;" alt="{\displaystyle \mathbf {\nu } (x)}" loading="lazy"></span> ein Datum des Problems ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Dirichlet-to-Neumann-Operator">Dirichlet-to-Neumann-Operator</h2></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Dirichlet-to-Neumann-Operator" title="Dirichlet-to-Neumann-Operator">Dirichlet-to-Neumann-Operator</a></i></div>

<p>Der <a href="Dirichlet-to-Neumann-Operator" title="Dirichlet-to-Neumann-Operator">Dirichlet-to-Neumann-Operator</a> ist ein elliptischer, selbstadjungierter <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}">

</mstyle>

</mrow>

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

</semantics>

</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 Dirichlet-Randbedingungen auf die Neumann-Randbedingungen abbildet.

</p>

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

<ul><li>D. Gilbarg, N.S. Trudinger: <i>Partial Differential Equations of Second Order</i>, Springer-Verlag, Berlin 1998, ISBN 3-540-41160-7.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-12-26" href="https://de.wikipedia.org/wiki/?title=Neumann-Randbedingung&oldid=251611874">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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