A Baire's category method for the Dirichlet problem of quasiregular mappings

Author:
Baisheng Yan

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4755-4765

MSC (2000):
Primary 30C65, 35F30, 49J30

Published electronically:
July 24, 2003

MathSciNet review:
1997582

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Abstract | References | Similar Articles | Additional Information

Abstract: We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any and any piece-wise affine map with for almost every there exists a map such that

The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

**1.**Emilio Acerbi and Nicola Fusco,*Semicontinuity problems in the calculus of variations*, Arch. Rational Mech. Anal.**86**(1984), no. 2, 125–145. MR**751305**, 10.1007/BF00275731**2.**John M. Ball,*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rational Mech. Anal.**63**(1976/77), no. 4, 337–403. MR**0475169****3.**Bernard Dacorogna and Paolo Marcellini,*General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases*, Acta Math.**178**(1997), no. 1, 1–37. MR**1448710**, 10.1007/BF02392708**4.**Bernard Dacorogna and Paolo Marcellini,*Cauchy-Dirichlet problem for first order nonlinear systems*, J. Funct. Anal.**152**(1998), no. 2, 404–446. MR**1607932**, 10.1006/jfan.1997.3172**5.**F. W. Gehring,*The 𝐿^{𝑝}-integrability of the partial derivatives of a quasiconformal mapping*, Acta Math.**130**(1973), 265–277. MR**0402038****6.**Tadeusz Iwaniec,*𝑝-harmonic tensors and quasiregular mappings*, Ann. of Math. (2)**136**(1992), no. 3, 589–624. MR**1189867**, 10.2307/2946602**7.**Stefan Müller and Vladimir Šverák,*Attainment results for the two-well problem by convex integration*, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 239–251. MR**1449410****8.**S. Müller and M. A. Sychev,*Optimal existence theorems for nonhomogeneous differential inclusions*, J. Funct. Anal.**181**(2001), no. 2, 447–475. MR**1821703**, 10.1006/jfan.2000.3726**9.**Yu. G. Reshetnyak,*Space mappings with bounded distortion*, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR**994644****10.**Baisheng Yan,*Linear boundary values of weakly quasiregular mappings*, C. R. Acad. Sci. Paris Sér. I Math.**331**(2000), no. 5, 379–384 (English, with English and French summaries). MR**1784918**, 10.1016/S0764-4442(00)01654-2**11.**Baisheng Yan,*A linear boundary value problem for weakly quasiregular mappings in space*, Calc. Var. Partial Differential Equations**13**(2001), no. 3, 295–310. MR**1865000**, 10.1007/s005260000074**12.**B. Yan,*Relaxation and attainment results for an integral functional with unbounded energy-well*, Proc. Roy. Soc. Edinburgh Sect. A**132**(2002), no. 6, 1513-1523.

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Additional Information

**Baisheng Yan**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Email:
yan@math.msu.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03101-5

Keywords:
Baire's category method,
Dirichlet problem,
quasiregular mappings

Received by editor(s):
January 25, 2001

Published electronically:
July 24, 2003

Article copyright:
© Copyright 2003
American Mathematical Society