# 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

$$L = \sum_{\left| \alpha \right| \leqslant 2 m} a_{\alpha} \left( x \right) \partial^{\alpha},$$ $$\left( - 1 \right)^m Re \sum_{\left| \alpha \right| = 2 m} a_{\alpha} \left( x \right) \xi^{\alpha} \geqslant c \left| \xi \right|^{2 m},$$

be a uniformly elliptic operator in a region $\Omega _{1}$ in $\mathbf R ^{n}$, with coefficients $a _ \alpha (x) \in C ^ \infty ( \Omega _{1} )$; let $\Omega$ be a subregion of $\Omega _{1}$, and suppose that in some neighbourhood of the boundary $S$ of $\Omega$ one has differential operators $M _{j}$, $j = 0 \dots m - 1$, of orders $j$, the characteristics of which are not tangent to $S$ at any point of $S$. Then, in some neighbourhood of $S$, there exist differential operators $N _{j}$ of orders $j$, $j = m \dots 2m -1$, such that

$$\tag{1} \sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}} ( \partial ^ \alpha v ,\ a _{ {\alpha \beta}} \partial ^ \beta u ) - (v ,\ L (u))\ =$$

$$= \ \sum _ {j = 0} ^ {m - 1} \int\limits _{S} M _{j} (v) {N _{ {2m - 1 - j}} (u)} bar \ d \sigma$$

for all $v$ in $C ^ \infty ( \overline \Omega \; )$. Here $( \ ,\ )$ denotes the scalar product in $L _{2} ( \Omega )$.

The form

$$D (v,\ u ) \ = \ \sum _ { {| \alpha | \leq m, \atop | \beta | \leq m}} ( \partial ^ \alpha v,\ a _{ {\alpha \beta}} \partial ^ \beta u )$$

is called a coercive form on a space $X$, $W _{2c} ^ {m} ( \Omega ) \subset X \subset W _{2} ^{m} ( \Omega )$, if there exist constants $C > 0$ and $\lambda \geq 0$ such that

$$\tag{2} \mathop{\rm Re}\nolimits \ D (u ,\ u) \ \geq \ C \ \| u \| _{ {m,\ \Omega}} ^{2} - \lambda \ \| u \| _{ {0,\ \Omega}} ^{2}$$

for all $u \in X$. Here $W _{2} ^{m}$ is the Sobolev space and $W _{2c} ^ {m}$ is the subspace of $W _{2} ^{m}$ consisting of all elements with compact support, i.e. elements vanishing in a neighbourhood of the boundary of $\Omega$. Inequality (2) is a coerciveness inequality for the form $D (v,\ u)$. If (2) remains valid with $\lambda = 0$, then $D (v,\ u)$ is said to be strongly coercive.

If a solution $u$ of the equation $L (u) = f$ satisfies the conditions $M _{j} (u) = 0$, $j = 0 \dots m - 1$, on $S$, then one has an inequality

$$\tag{3} \| u \| _{ {2m,\ \Omega}} \ \leq \ C _{1} \ \| L (u) \| _{ {0,\ \Omega}} + \lambda _{1} \ \| u \| _{ {0,\ \Omega}}$$

for some constants $C _{1} > 0,\ \lambda _{1} \geq 0$. If a solution $u$ of the equation $L (u) = f$ satisfies conditions $M _{j} (u) = \phi _{j}$ on $S$, $j = 0 \dots m - 1$, then instead of (3) one has the inequality

$$\tag{4} \| u \| _{ {2m,\ \Omega}} \ \leq \ C \ \left \{ \| L (u) \| _{ {0,\ \Omega}} + \sum _ {j = 0} ^ {m - 1} \| \phi _{j} \| _{ {2m - j,\ s}} + \| u \| _{ {0,\ \Omega}} \right \} .$$

This inequality provides an estimate for the norm of the solution $u$ of the equation $L (u) = f$ in the Sobolev space $W _{2} ^{2m} ( \Omega )$ in terms of its norm in $L _{2} ( \Omega )$ and of the norms of $f$ and $\phi _{j}$, $j = 0 \dots m - 1$ 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

$$\| u \| _{ {2m + k,\ \Omega}} \ \leq \ C \ \left \{ \| L (u) \| _{ {k,\ \Omega}} + \sum _ {j = 0} ^ {m - 1} \| \phi _{j} \| _{ {2m - j + k,\ s}} + \| u \| _{ {0,\ \Omega}} \right \} .$$

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 .

How to Cite This Entry:
Coerciveness inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coerciveness_inequality&oldid=44363
This article was adapted from an original article by A.I. Yanushauskas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article