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Removable Singularities and Quasilinear Parabolic Equations

出版	University of Sussex, 1985
URL	http://books.google.com.hk/books?id=gKSZnQEACAAJ&hl=&source=gbs_api

註釋The aim of this work is to obtain removable singularity results for quasilinear parabolic equations in divergence form satisfying a given number of structure inequalities. For this purpose we adapt a technique used by Gariepy and Ziemer in [11]. We characterize the singular set using definitions of s-capa city by Serrin [23] and Saraiva [21] Ziemer's definition in [33]), and of and Peletier [8]; in the degenerate extensions of these definitions. The rizations is that it allow us to use (which is very similar to (p, q)-null set by Edmunds case we use the natural essential of these charactesequences of C -functions converging pointwise to one almost everywhere, which are zero in a neighbourhood of part of the singular set and such that all first-order derivatives converge to zero in some Lebesgue space. In the degenerate case these sequences of derivatives also converge to zero in the corresponding weighted spaces. In obtaining the removable singularity theorems, the basic idea is to connect the capacity (or the nullity indices) of the singular set to the Lebesgue classes of the xi-derivatives of the weak solution, in the non-degenerate case, or to the Lebesgue classes of the weight p, of 1/u, and of the xi-derivatives of the weak solution in the degenerate case. Although the majority of the results provided is for locally bounded weak solutions, a non bounded case is also studied, using a theorem by Trudinger [31].. The techniques used in this work produce similar results for elliptic equations in divergence form satisfying equivalent structure inequalities.