# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Weakly monotone functions. (English) Zbl 0805.35013
The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spaces ${W}^{1,p}$, $p>n-1$. It is proven that such weakly monotone functions are continuous except in a singular set of $p$- capacity zero, that is empty in the case $p=n$. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.

##### MSC:
 35B50 Maximum principles (PDE) 74B20 Nonlinear elasticity 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35D10 Regularity of generalized solutions of PDE (MSC2000) 30C65 Quasiconformal mappings in ${ℝ}^{n}$ and other generalizations
##### References:
 [1] Ball, J. Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rational Mech. Anal. 63, 337–403 (1978). · Zbl 0368.73040 · doi:10.1007/BF00279992 [2] Bojarski, B., and Iwaniec, T. Analytical foundations of the theory of quasiconformal mappings in $ℝ$n.Ann. Acad. Sci. Fenn. Ser. A I Math. 8, 257–324 (1983). [3] Gehring, F. Rings and quasiconformal mappings in space.Trans. Amer. Math. Soc. 101, 499–519 (1961). · doi:10.1090/S0002-9947-1961-0132841-2 [4] Hayman, W., and Kennedy, P.Subharmonic Functions, Academic Press, 1976. [5] Heinonen, J., Kilpeläinen, T., and Martio, O.Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, 1993. [6] Lebesgue, H. Sur le problème de Dirichlet.Rend. Circ. Palermo 27, 371–402 (1907). · Zbl 02644079 · doi:10.1007/BF03015070 [7] Mostow, G. Quasiconformal mappings inn-space and the rigidity of hyperbolic space forms.Publ. Math. Inst. Hautes Études Sci. 34, 53–104 (1968). · Zbl 0189.09402 · doi:10.1007/BF02684590 [8] Manfredi, J., and Villamor, E. Traces of monotone Sobolev functions.Journal of Geometric Analysis, to appear. [9] Rešetnyak, J. Space mappings with bounded distortion.Sibirisk. Mat. Z. 8, 629–658 (1967). [10] Šverák, V. Regularity properties of deformations with finite energy.Arch. Rational Mech. Anal. 100, 105–127 (1988). · Zbl 0659.73038 · doi:10.1007/BF00282200 [11] Vodopyanov, S., and Goldstein, V. Quasiconformal mappings and spaces of functions with generalized first derivatives.Siberian Math. J. 17(3), 515–531 (1977).