Lax-Milgram's Lemma

The Lax-Milgram lemma , also the Lax-Milgram theorem , is a statement of functional analysis , a branch of mathematics named after Peter Lax and Arthur Milgram . These two mathematicians proved a first version of this lemma in 1954, which generalized the statement of the representation theorem of Fréchet-Riesz to continuous sesquilinear forms . A more general version of the lemma was proven by Ivo Babuška , which is why this statement is also known as the Babuška – Lax – Milgram theorem. These statements are used in the theory of partial differential equations. With their help, statements about existence and uniqueness can be made about solutions of partial differential equations.

formulation

requirements

Let it be a Hilbert space over and let it be a sesquilinear form . In addition, one of the following equivalent conditions applies: ${\displaystyle \left(H,\langle \cdot ,\cdot \rangle \right)}$ ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle B:H\times H\to \mathbb {K} }$

• ${\displaystyle B}$ is steady
• There is one with ${\displaystyle M\in \mathbb {R} }$

  ${\displaystyle |B(x,y)|\leq M\|x\|\|y\|,\quad \forall \,x,y\in H,}$

• ${\displaystyle y\mapsto B(x,y)}$is steady for everyone and is steady for everyone ${\displaystyle x\in H}$ ${\displaystyle x\mapsto B(x,y)}$ ${\displaystyle y\in H}$

statement

If the above conditions are met, then there is exactly one continuous , linear operator that gives the equation ${\displaystyle T\colon H\to H}$

${\displaystyle B(x,y)=\left\langle Tx,y\right\rangle }$

fulfilled for all . Furthermore, the following applies: The norm of is limited by. ${\displaystyle x,y\in H}$ ${\displaystyle T}$ ${\displaystyle M}$

Special case: coercive sesquilinear form

If the sesquilinear form is also coercive (often referred to as strongly positive or elliptical), i. H. is there so that ${\displaystyle B}$ ${\displaystyle m>0}$

${\displaystyle B(x,x)\geq m\|x\|^{2},\quad \forall x\in H,}$

holds, then is invertible with . ${\displaystyle T}$ ${\displaystyle \left\|T^{-1}\right\|\leq 1/m}$

Application to elliptic differential equations

The Lax-Milgram lemma is used in the theory of partial differential equations . In particular, for linear differential equations, the existence and uniqueness of a weak solution can be shown if the above conditions are met. This will now be illustrated using the example of a uniformly elliptical differential equation of the second order.

Be

${\displaystyle Pu:=-\sum _{i=1}^{n}\partial _{i}\left(\sum _{j=1}^{n}a_{ij}\partial _{j}u+h_{i}\right)+bu}$

a uniformly elliptic second order differential operator. That is, it applies to , with and there exists one , so that the main symbol for all and all the inequality ${\displaystyle a_{ij},h_{i}\in C^{1}(\Omega )}$ ${\displaystyle i,j=1,\ldots ,n}$ ${\displaystyle b\in L^{\infty }(\Omega )}$ ${\displaystyle b\geq 0}$ ${\displaystyle c_{0}>0}$ ${\displaystyle x\in \Omega }$ ${\displaystyle \xi \in \mathbb {R} ^{n}}$

${\displaystyle \sum _{i,j}^{n}a_{ij}(x)\xi _{i}\xi _{j}\geq c_{0}|\xi |^{2}}$

Fulfills. With the help of Lax-Milgram's lemma it can now be shown that the weak formulation of the Dirichlet boundary problem

${\displaystyle \left.{\begin{array}{cc}Pu=f&{\text{in}}\ \Omega \\u=g&{\text{auf}}\ \partial \Omega \end{array}}\right\}}$

has exactly one solution in the Sobolev area for and . This means that one considers the equation for all test functions ${\displaystyle u\in H_{0}^{1}(\Omega )}$ ${\displaystyle f\in L^{\infty }(\Omega )}$ ${\displaystyle g\in C(\partial \Omega )}$ ${\displaystyle \phi \in C_{c}^{\infty }(\Omega )}$

${\displaystyle \int _{\Omega }f(x)\phi (x)\mathrm {d} x=\int _{\Omega }-\sum _{i=1}^{n}\partial _{i}\left(\sum _{j=1}^{n}a_{ij}(x)\partial _{j}u(x)+h_{i}(x)\right)\phi (x)+b(x)u(x)\phi (x)\mathrm {d} x.}$

Partial integration of the right hand side of the equation yields

${\displaystyle \int _{\Omega }f(x)\phi (x)\mathrm {d} x=\int _{\Omega }\sum _{i=1}^{n}\partial _{i}\phi (x)\cdot \left(\sum _{j=1}^{n}a_{ij}(x)\partial _{j}u(x)+h_{i}(x)\right)\mathrm {d} x+\int _{\Omega }b(x)u(x)\phi (x)\mathrm {d} x.}$

If you set now

${\displaystyle a(u,v):=\sum _{i=1}^{n}\sum _{j=1}^{n}\int _{\Omega }\partial _{i}u(x)\cdot a_{ij}(x)\partial _{j}v(x)\mathrm {d} x+\int _{\Omega }u(x)b(x)v(x)\mathrm {d} x}$

so one obtains a real-valued bilinear form , the continuity of which can be shown with the help of the Hölder inequality . The form is also coercive, which follows from the condition . Hence the bilinear form fulfills the requirements of the Lax-Milgram lemma. So we are now looking for a solution to the equation ${\displaystyle a}$ ${\displaystyle \textstyle \sum _{i,j}^{n}a_{ij}(x)\xi _{i}\xi _{j}\geq c_{0}|\xi |^{2}}$ ${\displaystyle a}$

${\displaystyle a(u,v)=F(v),}$

in which

${\displaystyle F(v):=-\int _{\Omega }\sum _{i=1}^{n}\partial _{i}v(x)h_{i}(x)+v(x)f(x)\mathrm {d} x.}$

Since the expression is linear and continuous, i.e. it is an element of the dual space , one can apply the representation theorem of Fréchet-Riesz and get exactly one , so that applies to all . And based on Lax-Milgram's lemma, the equation has ${\displaystyle v\mapsto F(v)}$ ${\displaystyle (H_{0}^{1}(\Omega ))'}$ ${\displaystyle q\in H_{0}^{1}(\Omega )}$ ${\displaystyle \textstyle F(v)=\langle v,q\rangle _{H^{1}(\Omega )}}$ ${\displaystyle v\in H_{0}^{1}(\Omega )}$

${\displaystyle a(v,u)=\langle v,q\rangle _{H^{1}(\Omega )}}$

exactly one solution for everyone . ${\displaystyle v\in H_{0}^{1}(\Omega )}$ ${\displaystyle u\in H_{0}^{1}(\Omega )}$

In a similar way one can show the existence and uniqueness of Neumann boundary conditions .

Babuška – Lax – Milgram theorem

A generalization of the Lax-Milgram lemma is the Babuška – Lax – Milgram theorem. This was proven by Ivo Babuška in 1971 .

Let and be two Hilbert spaces and be a continuous bilinear form . Also, be weakly coercive, that is, there is one such that ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle B\colon U\times V\to \mathbb {R} }$ ${\displaystyle B}$ ${\displaystyle c>0}$

${\displaystyle \forall u\in U:\quad \sup _{\|v\|\leq 1}|B(u,v)|\geq c\|u\|}$

and

${\displaystyle \forall v\in V\setminus \{0\}:\quad \sup _{u\in U}|B(u,v)|>0}$

applies. Then there is exactly one continuous, linear operator that has the equation ${\displaystyle T\colon U\to V}$

${\displaystyle B(u,v)=\langle Tu,v\rangle }$

The inequality holds for all and and for the operator norm . In other words, there is exactly one solution for equations . ${\displaystyle u\in U}$ ${\displaystyle v\in V}$ ${\displaystyle \|T^{-1}\|\leq {\tfrac {\|f\|}{c}}}$ ${\displaystyle u}$ ${\displaystyle B(u,v)=\langle f,v\rangle ,\,v\in V}$

literature

• Hans Wilhelm Alt: Linear Functional Analysis . 5th edition. Springer, Berlin, Heidelberg, New York 2006, ISBN 978-3-540-34186-4 .
• I. Roşca: Lax – Milgram lemma . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• I. Roşca: Babuška – Lax – Milgram theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).