Meyers-Serrin's theorem

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The Meyers-Serrin Theorem or Meyers and Serrin Theorem , named after Norman George Meyers and James Serrin , is a theorem from the theory of partial differential equations . It says that the infinitely often differentiable functions are dense in Sobolev spaces .

formulation

Be an open , non-empty subset and be and numbers. Then the sub-vector space is close to the space where it denotes the Sobolev space .

Auxiliary sentences

It is open, coherent and limited . The (Friedrichssche) smoothing function (Mollifier) that goes back to Kurt Friedrichs is

,

where the constant should be chosen so that:

.

Also set

,

which is why also for and are met. The convolution integral, the smoothing of :

then exists and is arbitrarily often differentiable for .

Sentence 1

Be . We assign the regularized function (smoothing) to each function and each :

with ,

to. Then the mapping is linear from to and the following applies:

.

Sentence 2

The following statements apply:

  1. The following applies to:
    for .
  2. For with follows:
    for .

Sentence 3

Be continued through on . For designated:

With

the regularized function of the class . Then applies to all multi-indices with and all the identity:

with .

proof

We choose as open quantities with:

as well as with ,

so that:

.

Furthermore, let a decomposition of one be subordinate to the system of sets , i.e. i.e., there are:

and with .

For the given we now choose so that as well as:

according to the auxiliary clauses (in particular according to clause 1: the swapping of weak derivatives with the smoothing according to Kurt Friedrichs) is correct. Now apply:

such as

resp. along with the choice of :

.

There is also follows .

Remarks

The following inclusion applies :

.

The space is not closed with respect to the - norm . However, according to the Meyers-Serrin theorem, we can just consider it as the completion of under this Sobolev norm. The partial derivatives can be continued as continuous operators on these Sobolev spaces to the weak derivatives known to us .

meaning

  • In the older theory, the spaces were defined as the closings of in . The Meyers-Serrin theorem states that the H spaces coincide with the W spaces, which explains the short title of the original work given below.
  • The definition conditions for Sobolev spaces use the concept of the weak derivative, certain weak derivatives must lie in the L p space . Using the same conditions for the classical concept of derivative, one can construct the set of -functions that satisfy these conditions and then complete them. Meyers-Serrin's theorem says that in this way the same spaces are obtained; the term of the weak derivative can be avoided at this point.
  • It is noteworthy that, in contrast to other density theorems about Sobolev spaces, this theorem manages without additional regularity requirements at the edge .

literature

  • Norman George Meyers, James Serrin (Department of Mathematics, University of Minnesota): H = W . In: Proc. N. A. S . tape 51 , no. 6 . New York June 1, 1964, p. 1055-1056 ( pnas.org [PDF]).
  • Friedrich Sauvingy: Partial differential equations of geometry and physics . Functional analytical solution methods. tape 2 . Springer, Heidelberg 2005, ISBN 3-540-23107-2 , chap. X (weak solutions of elliptic differential equations), § 1 (Sobolev spaces), sentences 1–3 (Friedrichs) and sentence 4 (Meyers-Serrin), p. 182–185 ( limited preview in Google Book search).

Individual evidence

  1. a b Norman George Meyers, James Serrin: H = W . In: Proc. N. A. S . tape 51 , no. 6 . New York June 1, 1964, p. 1055-1056 ( PDF ).
  2. Giovanni Maria Troianiello: Elliptic differential equations and problem-obstacle . Plenum Press, New York 1987, ISBN 0-306-42448-7 , pp. 48 .
  3. Joseph Wloka : Partial differential equations . Teubner, Stuttgart 1982, ISBN 3-519-02225-7 , sentence 3.5 for rooms, p. 74/75 .
  4. Steffen Fröhlich: Weak derivatives and Sobolev spaces. (PDF; 93 kB) Lecture 15 (summer term 2009). (No longer available online.) In: Introduction to Functional Analysis. Department of Mathematics and Computer Science at Freie Universität Berlin, June 14, 2009, p. 1 , archived from the original on May 1, 2019 ; Retrieved December 27, 2012 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (on smooth functions: (PDF; 80 kB) ) @1@ 2Template: Webachiv / IABot / www.alt.mathematik.uni-mainz.de
  5. Friedrich Sauvingy: Partial Differential Equations and physics . Functional analytical solution methods. tape 2 . Springer, Heidelberg 2005, ISBN 978-3-540-23107-3 , chap. X, § 1, sentence 4, p. 184 f . ( limited preview in Google Book search).
  6. a b Steffen Fröhlich: The Meyers and Serrin theorem. (PDF; 104 kB) Lecture 16 (summer term 2009). (No longer available online.) In: Introduction to Functional Analysis. Department of Mathematics and Computer Science at the Free University of Berlin, June 9, 2009, p. 4 , formerly in the original ; Retrieved December 27, 2012 .  ( Page no longer available , search in web archives )@1@ 2Template: Toter Link / page.mi.fu-berlin.de