View source for Coercive boundary value problem

A boundary value problem satisfying a [[Coerciveness inequality|coerciveness inequality]]. Coercive boundary value problems for elliptic equations are sometimes called elliptic boundary value problems [[#References|[4]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229301.png" />  $P(\xi_1,\dots,\xi_n)$ be a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229302.png" /> so 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/c022930/c0229303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

is an elliptic equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229304.png" />. Consider the boundary value problem for equation (1) in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229305.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229306.png" />, with boundary conditions

<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/c022930/c0229307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c0229309.png" />, are homogeneous polynomials of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293010.png" />.

The problem (1), (2) is coercive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293011.png" /> if the orders of all the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293012.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293013.png" /> are less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293014.png" /> and if the problem has no bounded solutions of the type

<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/c022930/c02293015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<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/c022930/c02293016.png" /></td> </tr></table>

As a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293018.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293019.png" /> roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293020.png" /> in the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293021.png" />. If these roots are distinct, the condition that problem (1), (2) has no bounded solutions of type (3) is equivalent to

<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/c022930/c02293022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

This condition is sometimes called the complementarity condition.

The boundary value problem

<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/c022930/c02293023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<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/c022930/c02293024.png" /></td> </tr></table>

for a linear elliptic equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293025.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293026.png" /> is coercive at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293027.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293029.png" /> if the problem

<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/c022930/c02293030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293032.png" /> are the homogeneous components of the highest degree of the corresponding polynomials, is coercive.

A good tool for the investigation of coercive boundary value problems is the method of boundary rectification. This method involves a homeomorphic mapping, defined in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293033.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293034.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293036.png" />, with the property that the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293037.png" /> is mapped onto some domain of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293038.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293039.png" /> is mapped onto the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293040.png" /> with points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293041.png" />. As a result, a coercive boundary value problem for an arbitrary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293042.png" /> with sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293043.png" /> is reduced to a problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293044.png" />.

In the case of problem (5), condition (4) means that the characteristic direction of the system of differential operators

<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/c022930/c02293045.png" /></td> </tr></table>

defined in a neighbourhood of the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293046.png" />, is not tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293047.png" /> at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293048.png" />. Under this condition, a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293049.png" /> of problem (5) satisfies the coerciveness estimate

<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/c022930/c02293050.png" /></td> </tr></table>

where the norm in the Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293051.png" /> is estimated in terms of the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293053.png" />, the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293055.png" /> and the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293059.png" /> being the order of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293060.png" />.

The concept of a coercive boundary value problem may also be generalized to elliptic systems. The methods of investigation and the results valid for coercive boundary value problems for a single elliptic equation may be generalized to coercive boundary value problems for elliptic systems [[#References|[3]]]. The investigation of coercive boundary value problems for elliptic equations is often reduced, e.g. by the [[Parametrix method|parametrix method]], to the investigation of a system of singular integral equations. This is a Noetherian system of integral equations (see [[#References|[2]]], [[#References|[6]]]), and condition (4) — or, in the case of a system, its analogue — guarantees normality of the system. A coercive boundary value problem is always Noetherian, i.e. the inhomogeneous problem is solvable subject to the observance of a finite number of orthogonality conditions imposed on the right-hand sides of the (system of) equations and on the functions that appear in the boundary conditions, while the homogeneous problem has finitely many linearly independent solutions.

One example of a problem which is coercive for any elliptic equation is the Dirichlet problem. In this case

<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/c022930/c02293061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022930/c02293062.png" /> denotes differentiation in the direction of the conormal for the elliptic operator in question. However, it is not true that the Dirichlet problem is a coercive boundary value problem for every elliptic system of equations. An example of systems for which the Dirichlet problem is not coercive is the system of two equations known in complex notation as the [[Bitsadze equation|Bitsadze equation]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ya.B. Lopatinskii,   "On a method of reducing problems for a system of differential equations of elliptic type to regular integral equations"  ''Amer. Math. Soc. Transl. (2)'' , '''89'''  (1970)  pp. 149–183  ''Ukrain. Mat. Zh.'' , '''5''' :  2  (1953)  pp. 123–151</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Hörmander,   "Linear partial differential operators" , Springer  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Hörmander,   ''Matematika'' , '''4''' :  4  (1960)  pp. 37–73</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Z.Ya. Shapiro,   "The first boundary value problem for an elliptic system of differential equations"  ''Mat. Sb.'' , '''28'''  (1951)  pp. 55–78  (In Russian)</TD></TR></table>



====Comments====
The complementarity condition is also called the complementing, complementary or admissibility condition. A Noetherian system of integral equations is, in Western literature, now more often known as a system associated with a Fredholm operator, or a Fredholm system.

Other basic references are [[#References|[a1]]], [[#References|[a2]]]. Additional important references are [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Agmon,   A. Douglis,   L. Nirenberg,   "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions"  ''Comm. Pure Appl. Math.'' , '''12'''  (1959)  pp. 623–727</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Agmon,   A. Douglis,   L. Nirenberg,   "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II"  ''Comm. Pure Appl. Math.'' , '''17'''  (1964)  pp. 35–92</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Lions,   E. Magenes,   "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations" , Holt, Rinehart &amp; Winston  (1969)</TD></TR></table>