TY - JOUR

T1 - Mappings of generalized finite distortion and continuity

AU - Doležalová, Anna

AU - Kangasniemi, Ilmari

AU - Onninen, Jani

N1 - Publisher Copyright:
© 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

PY - 2024/1

Y1 - 2024/1

N2 - We study continuity properties of Sobolev mappings (Formula presented.), (Formula presented.), that satisfy the following generalized finite distortion inequality (Formula presented.) for almost every (Formula presented.). Here (Formula presented.) and (Formula presented.) are measurable functions. Note that when (Formula presented.), we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion (Formula presented.), where a sharp condition for continuity is that (Formula presented.) is in the Zygmund space (Formula presented.) for some (Formula presented.). We also show that one can slightly relax the boundedness assumption on (Formula presented.) to an exponential class (Formula presented.) with (Formula presented.), and still obtain continuous solutions when (Formula presented.) with (Formula presented.). On the other hand, for all (Formula presented.) with (Formula presented.), we construct a discontinuous solution with (Formula presented.) and (Formula presented.), including an example with (Formula presented.) and (Formula presented.).

AB - We study continuity properties of Sobolev mappings (Formula presented.), (Formula presented.), that satisfy the following generalized finite distortion inequality (Formula presented.) for almost every (Formula presented.). Here (Formula presented.) and (Formula presented.) are measurable functions. Note that when (Formula presented.), we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion (Formula presented.), where a sharp condition for continuity is that (Formula presented.) is in the Zygmund space (Formula presented.) for some (Formula presented.). We also show that one can slightly relax the boundedness assumption on (Formula presented.) to an exponential class (Formula presented.) with (Formula presented.), and still obtain continuous solutions when (Formula presented.) with (Formula presented.). On the other hand, for all (Formula presented.) with (Formula presented.), we construct a discontinuous solution with (Formula presented.) and (Formula presented.), including an example with (Formula presented.) and (Formula presented.).

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U2 - 10.1112/jlms.12835

DO - 10.1112/jlms.12835

M3 - Article

AN - SCOPUS:85176767860

SN - 0024-6107

VL - 109

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 1

M1 - e12835

ER -