View source for Coerciveness inequality

An inequality providing a lower bound for a certain bilinear form, or providing an upper bound for the norm of a solution of a certain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let

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be a uniformly elliptic operator in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229403.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229404.png" />, with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229405.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229406.png" /> be a subregion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229407.png" />, and suppose that in some neighbourhood of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229408.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c0229409.png" /> one has differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294011.png" />, of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294012.png" />, the characteristics of which are not tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294013.png" /> at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294014.png" />. Then, in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294015.png" />, there exist differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294016.png" /> of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294018.png" />, such that

<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/c/c022/c022940/c02294019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294022.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294023.png" /> denotes the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294024.png" />.

The form

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is called a coercive form on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294027.png" />, if there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294029.png" /> such that

<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/c/c022/c022940/c02294030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294031.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294032.png" /> is the [[Sobolev space|Sobolev space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294033.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294034.png" /> consisting of all elements with compact support, i.e. elements vanishing in a neighbourhood of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294035.png" />. Inequality (2) is a coerciveness inequality for the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294036.png" />. If (2) remains valid with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294038.png" /> is said to be strongly coercive.

If a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294039.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294040.png" /> satisfies the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294042.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294043.png" />, then one has an inequality

<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/c/c022/c022940/c02294044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

for some constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294045.png" />. If a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294046.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294047.png" /> satisfies conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294050.png" />, then instead of (3) one has the inequality

<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/c/c022/c022940/c02294051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

This inequality provides an estimate for the norm of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294052.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294053.png" /> in the Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294054.png" /> in terms of its norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294055.png" /> and of the norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294058.png" /> in the appropriate spaces. Inequality (4) is a coerciveness inequality for the boundary value problem for an elliptic equation.

Using inequality (4) one obtains the more general inequality

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Coerciveness inequalities play an important role in the investigation of coercive boundary value problems and in proofs of the smoothness of solutions of elliptic equations; they are particularly important in analyticity proofs for solutions of analytic elliptic equations [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Agmon,   "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.B. Morrey,   L. Nirenberg,   "On the analyticity of the solutions of linear elliptic systems of partial differential equations"  ''Comm. Pure Appl. Math.'' , '''10''' :  2  (1957)  pp. 271–290</TD></TR></table>



====Comments====
Inequalities such as (3), (4), providing upper bounds for elliptic boundary value problems, are better known as boundary estimates for elliptic boundary value problems, instead of coerciveness inequalities. Lower bounds for bilinear forms are frequently encountered in the theory of variational inequalities (see also [[Variational equations|Variational equations]]). See also [[Coercive boundary value problem|Coercive boundary value problem]].

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294060.png" /> in (1) are obtained from the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294061.png" /> by integrating the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294062.png" /> by parts. Clearly, the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022940/c02294063.png" /> in (2) is inessential.