论文标题
关于椭圆形混合边界价值问题的批判性理论的发散形式
On criticality theory for elliptic mixed boundary value problems in divergence form
论文作者
论文摘要
该论文致力于研究以域中的二阶线性椭圆方程为二阶线性椭圆方程的阳性解决方案$ d \ subseteq \ subseteq \ mathbb {r}^n $，使在$ \ partial d $的一部分上满足斜边界条件。首先，我们研究退化的混合边界价值问题 $$ \ begin {case} pu = f＆\ text {in} d， \\ bu = 0＆\ text {on} \ partial d _ {\ mathrm {rob}}， \\ u = 0＆\ text {on} \ partial d _ {\ mathrm {dir}}， \ end {cases} $$ 其中$ d $是一个有界的lipschitz域，$ \ a部分d _ {\ mathrm {rob}} $是$ \ partial d $的相对开放部分D _ {\ Mathrm {Rob}} $。特别是，我们讨论了上述问题的独特解决性，主要特征值的存在以及阳性最小绿色功能的存在。然后，我们建立了一个批判性理论，用于在$ \ partial d _ {\ Mathrm {dir}} $上没有边界条件的一般域中运算符$（p，b）$的积极弱解决方案（p，b）$。该论文概括并扩展了Pinchover和Saadon（2002）获得此类问题的经典解决方案获得的结果，在这些解决方案中，对$（p，b）$和$ \ partial d _ {\ mathrm {rob {rob}} $的系数的规律性更强。
The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain $D\subseteq \mathbb{R}^n$ that satisfy an oblique boundary condition on a portion of $\partial D$. First, we study the degenerate mixed boundary value problem $$ \begin{cases} Pu=f & \text{in } D, \\ Bu = 0 & \text{on } \partial D_{\mathrm{Rob}}, \\ u=0& \text{on } \partial D_{\mathrm{Dir}}, \end{cases} $$ where $D$ is a bounded Lipschitz domain, $\partial D_{\mathrm{Rob}}$ is a relatively open portion of $\partial D$, $\partial D_{\mathrm{Dir}}$ is a closed set of $\partial D$, and $B$ is an oblique (Robin) boundary operator defined on $\partial D_{\mathrm{Rob}}$. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a positive minimal Green function. Then we establish a criticality theory for positive weak solutions of the operator $(P,B)$ in a general domain with no boundary condition on $\partial D_{\mathrm{Dir}}$ and no growth condition at infinity. The paper generalizes and extends results obtained by Pinchover and Saadon (2002) for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of $(P,B)$, and $\partial D_{\mathrm{Rob}}$.
