Trace operator and a nonlinear boundary value problem in a new space
© Pak and Park; licensee Springer 2014
Received: 1 February 2014
Accepted: 5 June 2014
Published: 11 September 2014
We develop a new function space and discuss trace operator on the same genealogical spaces. We also prove that the nonlinear boundary value problem with Dirichlet condition: in the given domain, on the boundary, possesses only a trivial solution if obeys the slope condition: , where is the anti-derivative of with .
There are two different attempts to generalize the classical Lebesgue spaces - Orlicz spaces and Lorentz spaces . The theory of Orlicz spaces has been well developed which is similar to our new spaces. The Orlicz space requires the convexity of the -function for the triangle inequality of the norm, whereas the norm for the space does not require the convexity of the Hölder function and it has indeed inherited the beautiful and convenient properties from the classical Lebesgue norm.
Throughout this paper, represents an open subset of and is an abstract measure space (Section 2). Also, denotes various real positive constants.
We introduce some terminologies to define the Lebesgue type function spaces which improve the original version introduced in . In this section, .
The following proposition and the proof may illustrate that the comparable condition (13) is not far-fetched.
A Hölder type inequality and a Minkowski inequality on the new space are presented as follows:
As an important application of a Hölder inequality, we have the Minkowski inequality on . We omit the proof.
(Generalized Minkowski inequality)
(Dual space of )
Let be the conjugate Hölder function of a Hölder function . Then the dual space is isomorphic to .
The proof is quite parallel to the classical Riesz representation theorem, so we omit the proof.
The two inequalities (31) and (32) explain the quasi-homogeneity of . That is, we have the following.
Trace operator on Sobolev type space
The completion of the space with respect to the norm is denoted by , where is the space of smooth functions on with compact support.
We introduce the trace operator on , which is important by itself and also useful in Section 4. We want to point out that the trace operator we present here is an improvement and a completion of the one briefly introduced in .
We prove that the boundary trace on can be extended to the space as follows.
(Trace map on )
Letbe a Hölder pair obeying the slope condition (40) andbe a bounded open subset ofwith smooth boundary. Then the trace operatoris continuous and uniquely determined byfor.
We first prove the theorem for the special case of flat boundary, and by using it, we take care of the general cases.
The general case - being bounded open in. In this section we restrict our attention to the case of being a bounded open subset. However, can be more general, such as unbounded domains satisfying the uniform -regularity condition (p.84 in ).
Ill-posedness of boundary value problems
The first author was supported by the research fund of Dankook University in 2012.
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