The trace theorem W p 2 , 1 ( Ω T ) ∋ f ↦ ∇ x f ∈ W p 1 - 1 / p , 1 / 2 - 1 / 2 p ( ∂ Ω T ) revisited

Peter Weidemaier

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 2, page 307-314
  • ISSN: 0010-2628

Abstract

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Filling a possible gap in the literature, we give a complete and readable proof of this trace theorem, which also shows that the imbedding constant is uniformly bounded for T ↓ 0 . The proof is based on a version of Hardy’s inequality (cp. Appendix).

How to cite

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Weidemaier, Peter. "The trace theorem $W^{2,1}_p(\Omega _T) \ni f \mapsto \nabla _{\!x} f \in W^{1-1/p,1/2-1/2p}_p(\partial \Omega _T)$ revisited." Commentationes Mathematicae Universitatis Carolinae 32.2 (1991): 307-314. <http://eudml.org/doc/247284>.

@article{Weidemaier1991,
abstract = {Filling a possible gap in the literature, we give a complete and readable proof of this trace theorem, which also shows that the imbedding constant is uniformly bounded for $T \downarrow 0$. The proof is based on a version of Hardy’s inequality (cp. Appendix).},
author = {Weidemaier, Peter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {trace theory; anisotropic Sobolev spaces; anisotropic Sobolev spaces; trace theorem; Hardy's inequality},
language = {eng},
number = {2},
pages = {307-314},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The trace theorem $W^\{2,1\}_p(\Omega _T) \ni f \mapsto \nabla _\{\!x\} f \in W^\{1-1/p,1/2-1/2p\}_p(\partial \Omega _T)$ revisited},
url = {http://eudml.org/doc/247284},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Weidemaier, Peter
TI - The trace theorem $W^{2,1}_p(\Omega _T) \ni f \mapsto \nabla _{\!x} f \in W^{1-1/p,1/2-1/2p}_p(\partial \Omega _T)$ revisited
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 2
SP - 307
EP - 314
AB - Filling a possible gap in the literature, we give a complete and readable proof of this trace theorem, which also shows that the imbedding constant is uniformly bounded for $T \downarrow 0$. The proof is based on a version of Hardy’s inequality (cp. Appendix).
LA - eng
KW - trace theory; anisotropic Sobolev spaces; anisotropic Sobolev spaces; trace theorem; Hardy's inequality
UR - http://eudml.org/doc/247284
ER -

References

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  1. Adams R.A., Sobolev Spaces, New York - San Francisco - London: Academic Press 1975. Zbl1098.46001MR0450957
  2. Besov O.V., Il'in V.P., Nikol'skii S.M., Integral Representations of Functions and Imbedding Theorems, Vol. I., Wiley, 1978. Zbl0392.46022
  3. Il'in V.P., The properties of some classes of differentiable functions of several variables defined in an n-dimensional region, Transl. AMS 81 (1969), 91-256 Trudy Mat. Inst. Steklov 66 (1962), 227-363. (1962) MR0153789
  4. Il'in V.P., Solonnikov V.A., On some properties of differentiable functions of several variables, Transl. AMS 81 (1969), 67-90 Trudy Mat. Inst. Steklov 66 (1962), 205-226. (1962) MR0152793
  5. Kufner A., John O., Fučik S., Function Spaces, Leyden, Noordhoff Int. Publ. 1977. MR0482102
  6. Ladyshenskaya O.A., Solonnikov V.A., Uralceva N.N, Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc., Providence, R.I. 1968. 
  7. Rákosník J., Some remarks to anisotropic Sobolev spaces, I. Beiträge zur Analysis 13 (1979), 55-68. (1979) MR0536217
  8. Weidemaier P., Local existence for parabolic problems with fully nonlinear boundary condition; an L p -approach, to appear in Ann. mat. pura appl. 
  9. Wheeden R. L., Zygmund A., Measure and Integral., New York - Basel: Dekker 1977. Zbl0362.26004MR0492146

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