Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations
Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations
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DOI：
10.1002/cpa.21529
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发表时间：
2015-09
影响因子：
3
通讯作者：
D. Lupo;D. Monticelli;K. Payne
中科院分区：
文献类型：
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作者：
D. Lupo;D. Monticelli;K. Payne
对于混合椭圆-双曲型线性和非线性二阶偏微分方程，证明了Dirichlet问题的弱解具有变分原理。弱解为自然泛函的鞍点，由偏微分方程的散度形式表示。此外，泛函的自然域是解所属的加权Sobolev空间。此外，所有的临界水平将被表征为泛函的全局极值限制在适当的无限维线性子空间。这些子空间是根据鲁棒谱理论定义的，其权重与线性算子相关。这种谱理论是最近由作者发展起来的，它反过来利用了Morawetz和作者获得的弱适位性结果。©2015 Wiley期刊公司
For linear and nonlinear second‐order partial differential equations of mixed elliptic‐hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite‐dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well‐posedness results obtained by Morawetz and the authors. © 2015 Wiley Periodicals, Inc.