The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces
The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces
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DOI：
10.1007/978-3-0348-0516-2_21
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发表时间：
2013
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影响因子：
--
通讯作者：
L. Softova
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文献类型：
--
作者：
L. Softova
我们在有界光滑域中考虑具有 VMO 主系数的线性均匀椭圆方程的狄利克雷问题。其独特的强可解性在[5]和[6]中得到了证明。我们的目标是证明，对于属于广义莫雷空间的每个f $$ L^{p,\omega}(\Omega),p \in (1,\infty),\omega:\mathbb{R}^{n}\times\mathbb{R}_{+}\rightarrow \mathbb{R}_{+} \rm {the\; operator} \mathfrak{L}:W^{2,p,\omega}\cap W_{0}^{1,p}(\Omega)\rightarrow L^{p,\omega}(\Omega) $$ 是双射的，估计值 $$ \parallel D^{2}u \parallel _{L^{p,\omega}(\Omega)}\leq C(\parallel {f} \parallel_{L^{p,\omega}(\Omega)}+ \parallel{u}\parallel_ {L^{p,\omega}(\Omega)}) $$ 成立。
We consider the Dirichlet problem in a bounded smooth domainfor linear uniformly elliptic equationwithVMOprincipal coefficients. Its unique strong solvability is proved in [5] and [6]. Our aim is to show that for everyfbelonging to the generalized Morrey space $$ L^{p,\omega}(\Omega),p \in (1,\infty),\omega:\mathbb{R}^{n}\times\mathbb{R}_{+}\rightarrow \mathbb{R}_{+} \rm {the\; operator} \mathfrak{L}:W^{2,p,\omega}\cap W_{0}^{1,p}(\Omega)\rightarrow L^{p,\omega}(\Omega) $$ is bijective and the estimate $$ \parallel D^{2}u \parallel _{L^{p,\omega}(\Omega)}\leq C(\parallel {f} \parallel_{L^{p,\omega}(\Omega)}+ \parallel{u}\parallel_ {L^{p,\omega}(\Omega)}) $$ holds.