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Difference between revisions of "Degenerate elliptic equation"

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A partial differential equation
 
A partial differential equation
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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\label{eq1}
 
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F(x,Du) = 0
where the real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307602.png" /> satisfies the condition
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\end{equation}
 
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where the real-valued function $F(x,q)$ satisfies the condition
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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\begin{equation}
 
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\label{eq2}
for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307604.png" />, and there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307605.png" /> for which (2) becomes an equality. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307606.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307607.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307608.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d0307609.png" /> is the unknown function; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076010.png" /> is a multi-index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076011.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076012.png" /> is a vector with components
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\sum_{\abs{\alpha} = m} \frac{\partial F(x,Du)}{\partial q_\alpha} \xi^\alpha \geq 0
 
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\end{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076013.png" /></td> </tr></table>
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for all real $\xi$, and there exists a $\xi \neq 0$ for which \ref{eq2} becomes an equality. Here, $x$ is an $n$-dimensional vector $(x_1,\ldots,x_n)$; $u$ is the unknown function; $\alpha$ is a [[Multi-index notation|multi-index]] $(\alpha_1,\ldots,\alpha_n)$; $Du$ is a vector with components
 
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$$
the derivatives in equation (1) are of an order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076014.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076015.png" /> are the components of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076016.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076017.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076018.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076019.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076020.png" />. If strict inequality in equation (2) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076022.png" /> and for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076023.png" />, equation (1) is elliptic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076024.png" />. Equation (1) degenerates at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076025.png" /> at which inequality (2) becomes an equality for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076026.png" />. If equality holds only on the boundary of the domain under consideration, the equation is called degenerate on the boundary of the domain. The most thoroughly studied equations are second-order degenerate elliptic equations
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D^\alpha u = \frac{\partial^{\abs{\alpha}}u}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}};
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076027.png" /></td> </tr></table>
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the derivatives in equation \ref{eq1} are of an order not exceeding $m$; the $q_\alpha$ are the components of a vector $q$; $\xi$ is an $n$-dimensional vector $(\xi_1,\ldots,\xi_n)$; and $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n} $. If strict inequality in equation \ref{eq2} holds for all $x$ and $Du$ and for all real $\xi \neq 0$, equation \ref{eq1} is elliptic at $(x,Du)$. Equation \ref{eq1} degenerates at the points $(x,Du)$ at which inequality \ref{eq2} becomes an equality for any real $\xi \neq 0$. If equality holds only on the boundary of the domain under consideration, the equation is called degenerate on the boundary of the domain. The most thoroughly studied equations are second-order degenerate elliptic equations
 
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$$
where the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076028.png" /> is non-negative definite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030760/d03076029.png" />-values under consideration.
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\sum a^{ik}(x) u_{x_i x_k} + \sum b^i(x) u_{x_i} + c(x)u = f(x),
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$$
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where the matrix $\left[ a^{jk}(x) \right]$ is non-negative definite for all $x$-values under consideration.
  
 
See also [[Degenerate partial differential equation|Degenerate partial differential equation]] and the references given there.
 
See also [[Degenerate partial differential equation|Degenerate partial differential equation]] and the references given there.

Revision as of 00:10, 3 May 2012

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A partial differential equation \begin{equation} \label{eq1} F(x,Du) = 0 \end{equation} where the real-valued function $F(x,q)$ satisfies the condition \begin{equation} \label{eq2} \sum_{\abs{\alpha} = m} \frac{\partial F(x,Du)}{\partial q_\alpha} \xi^\alpha \geq 0 \end{equation} for all real $\xi$, and there exists a $\xi \neq 0$ for which \ref{eq2} becomes an equality. Here, $x$ is an $n$-dimensional vector $(x_1,\ldots,x_n)$; $u$ is the unknown function; $\alpha$ is a multi-index $(\alpha_1,\ldots,\alpha_n)$; $Du$ is a vector with components $$ D^\alpha u = \frac{\partial^{\abs{\alpha}}u}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}; $$ the derivatives in equation \ref{eq1} are of an order not exceeding $m$; the $q_\alpha$ are the components of a vector $q$; $\xi$ is an $n$-dimensional vector $(\xi_1,\ldots,\xi_n)$; and $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n} $. If strict inequality in equation \ref{eq2} holds for all $x$ and $Du$ and for all real $\xi \neq 0$, equation \ref{eq1} is elliptic at $(x,Du)$. Equation \ref{eq1} degenerates at the points $(x,Du)$ at which inequality \ref{eq2} becomes an equality for any real $\xi \neq 0$. If equality holds only on the boundary of the domain under consideration, the equation is called degenerate on the boundary of the domain. The most thoroughly studied equations are second-order degenerate elliptic equations $$ \sum a^{ik}(x) u_{x_i x_k} + \sum b^i(x) u_{x_i} + c(x)u = f(x), $$ where the matrix $\left[ a^{jk}(x) \right]$ is non-negative definite for all $x$-values under consideration.

See also Degenerate partial differential equation and the references given there.

How to Cite This Entry:
Degenerate elliptic equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Degenerate_elliptic_equation&oldid=25859
This article was adapted from an original article by A.M. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article