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 [2].

References

[1]	S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)
[2]	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

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). See also Coercive boundary value problem.

The functions $ a _{ {\alpha \beta}} $ in (1) are obtained from the functions $ a _ \alpha $ by integrating the expression $ (v,\ Lu) $ by parts. Clearly, the restriction $ \lambda \geq 0 $ in (2) is inessential.