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首页> 外文期刊>Siberian Mathematical Journal >On the W~l-regularity of solutions to systems of differential equations in the case when the equations are constructed from discontinuous functions
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On the W~l-regularity of solutions to systems of differential equations in the case when the equations are constructed from discontinuous functions

机译：由不连续函数构造微分方程组解的W〜正则性

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摘要

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C~l-smooth solution to a system of lth-order nonlinear partial differential equations constructed from C~1-smooth functions meet the local Holder condition with every exponent α, 0 < α < 1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W_(q,loc)~l-solution, q > 1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.
机译：从Sobolev空间理论的角度来看，从某种意义上来说，最终的一些解决方案是针对系统（通常为非线性）的偏微分方程组的解的正则性问题而提出的，该系统局部接近于椭圆形系统。常数系数的线性方程。该结果的主要结果是定理5和定理8。根据第一个定理，椭圆C〜l-光滑解对由C〜1-光滑函数构造的l阶非线性偏微分方程组的系统定理8满足每个指数α，0 <α<1的局部Holder条件。定理8声称，如果一个具有可测量系数和右手边的l阶线性偏微分方程组是均匀椭圆形，则在（充分地减小其前导系数的缓慢变化，每个W_（q，loc）〜l-解的q阶偏导数q> 1的局部可积程度与系统的较低系数的局部可积程度一致和右手边。

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