Existence of evolutionary variational solutions via the calculus of variations
Existence of evolutionary variational solutions via the calculus of variations
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DOI：
10.1016/j.jde.2014.03.005
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发表时间：
2014-06-15
影响因子：
2.4
通讯作者：
Marcellini, Paolo
中科院分区：
文献类型：
--
作者：
Boegelein, Verena;Duzaar, Frank;Marcellini, Paolo
本文引入了一种纯变分方法来求解时间依赖问题，当T > 0且φ是C-0（无穷大）（Ω x（0，T），R-N）的元素时，得到了全局抛物极小解的存在性，即[GRAPHICS].对于被积函数f：Omega x R-Nn -> [0，无穷大]，我们仅假设关于梯度变量的凸性和非凸性。这些进化变分解作为依赖于空间和时间的映射的极限而获得，从而最小化某些凸变分泛函。在最简单的情况下，在f满足某些增长条件的情况下，该方法给出了一类偏导数（t）u -divD（xi）f（x，Du）= 0在Omega x（0，无穷大）中的抛物型方程组的Cauchy-Dirichlet问题整体弱解的存在性. (C)2014 Elsevier Inc. All rights reserved.
In this paper we introduce a purely variational approach to time dependent problems, yielding the existence of global parabolic minimizers, that is[GRAPHICS],whenever T > 0 and phi is an element of C-0(infinity) (Omega x (0, T), R-N). For the integrand f:Omega x R-Nn -> [0, infinity] we merely assume convexity with respect to the gradient variable and coercivity. These evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals. In the simplest situation, with some growth conditions on f, the method provides the existence of global weak solutions to Cauchy-Dirichlet problems of parabolic systems of the typepartial derivative(t)u - div D(xi)f(x, Du) = 0 in Omega x (0, infinity).(C) 2014 Elsevier Inc. All rights reserved.