@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-XCJD281P-M
  skos:prefLabel "Lamé function"@en, "harmonique ellipsoïdale"@fr ;
  a skos:Concept ;
  skos:broader psr:-H7WG59FS-L .

psr:-XJ77SGF5-S
  skos:prefLabel "Laplace's equation"@en, "équation de Laplace"@fr ;
  a skos:Concept ;
  skos:broader psr:-H7WG59FS-L .

psr:-X5ZLLBMK-J
  skos:prefLabel "spherical harmonic"@en, "harmonique sphérique"@fr ;
  a skos:Concept ;
  skos:broader psr:-H7WG59FS-L .

psr:-NV3MNMHN-6
  skos:prefLabel "harmonic measure"@en, "mesure harmonique"@fr ;
  a skos:Concept ;
  skos:related psr:-H7WG59FS-L .

psr:-H7WG59FS-L
  skos:narrower psr:-X5ZLLBMK-J, psr:-SX35HXWB-P, psr:-XJ77SGF5-S, psr:-XCJD281P-M ;
  a skos:Concept ;
  skos:prefLabel "fonction harmonique"@fr, "harmonic function"@en ;
  skos:definition """En mathématiques, une fonction harmonique est une fonction qui satisfait l'équation de Laplace. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_harmonique">https://fr.wikipedia.org/wiki/Fonction_harmonique</a>)"""@fr, """In mathematics, mathematical physics and the theory of stochastic processes, a <b>harmonic function</b> is a twice continuously differentiable function <span class="mwe-math-element"><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:U\\	o \\\\mathbb {R} ,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>:</mo>
<br/>        <mi>U</mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f:U\\	o \\\\mathbb {R} ,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12267b5b611997fb31b8f88a0f8964adb3bbe3d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:10.937ex; height:2.509ex;" alt="{\\\\displaystyle f:U\\	o \\\\mathbb {R} ,}"></span> where <span class="texhtml mvar" style="font-style:italic;">U</span> is an open subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {R} ^{n},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="\\\\mathbb {R} ^{n},"></span> that satisfies Laplace's equation, that is,
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\rac {\\\\partial ^{2}f}{\\\\partial x_{1}^{2}}}+{\\rac {\\\\partial ^{2}f}{\\\\partial x_{2}^{2}}}+\\\\cdots +{\\rac {\\\\partial ^{2}f}{\\\\partial x_{n}^{2}}}=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>f</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <msubsup>
<br/>                <mi>x</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msubsup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>f</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <msubsup>
<br/>                <mi>x</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msubsup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>f</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <msubsup>
<br/>                <mi>x</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msubsup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {\\\\partial ^{2}f}{\\\\partial x_{1}^{2}}}+{\\rac {\\\\partial ^{2}f}{\\\\partial x_{2}^{2}}}+\\\\cdots +{\\rac {\\\\partial ^{2}f}{\\\\partial x_{n}^{2}}}=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25157cd4c4b88b223f36a2885d56d0d10b753327" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:29.284ex; height:6.676ex;" alt=" \\rac{\\\\partial^2f}{\\\\partial x_1^2} + \\rac{\\\\partial^2f}{\\\\partial x_2^2} + \\\\cdots + \\rac{\\\\partial^2f}{\\\\partial x_n^2} = 0"></span></dd></dl>
<br/>everywhere on <span class="texhtml mvar" style="font-style:italic;">U</span>. This is usually written as
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\
abla ^{2}f=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>f</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}f=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d96e48b4ac582542a70e54bf299226aaa0e812" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.53ex; height:3.009ex;" alt=" \\
abla^2 f = 0 "></span></dd></dl>
<br/>or
<br/>
<br/><dl><dd><span class="mwe-math-element"><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 f=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Δ<!-- Δ --></mi>
<br/>        <mi>f</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta f=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524f4acc990d8c061ca122776ebc3b0a48f8acbb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.475ex; height:2.509ex;" alt="{\\\\displaystyle \\\\Delta f=0}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Harmonic_function">https://en.wikipedia.org/wiki/Harmonic_function</a>)"""@en ;
  skos:broader psr:-FH1H1FB9-1 ;
  skos:related psr:-NV3MNMHN-6 ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_harmonique>, <https://en.wikipedia.org/wiki/Harmonic_function> ;
  skos:inScheme psr: .

psr: a skos:ConceptScheme .
psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-H7WG59FS-L .

psr:-SX35HXWB-P
  skos:prefLabel "Harnack's inequality"@en, "inégalité de Harnack"@fr ;
  a skos:Concept ;
  skos:broader psr:-H7WG59FS-L .