论文标题
具有非本地“梯度术语”的确定性KPZ型方程
Deterministic KPZ-type equations with nonlocal "gradient terms"
论文作者
论文摘要
本文的主要目的是证明存在涉及非本地“梯度术语”的确定性Kardar-Parisi-Zhang类型方程的存在和不存在结果。更准确地说，令$ω\ subset \ mathbb {r}^n $，$ n \ geq 2 $，为$ c^2 $的边界$ \ partial的边界域。对于$ s \ in（0,1）$，我们考虑表格\ [\ tag {kpz} \ left \ {\ oken {aligned}（-ugu）^s u＆=μ（x）| \ mathbb {d}（d}（d}）（d}（d}））（ \ mbox {in} \ mathbb {r}^n \ setMinusω，\ end {aligned} \ right。 \]其中$ q> 1 $和$λ> 0 $是真正的参数，$ f $属于合适的lebesgue空间，$μ$属于$ l^{\ infty}（ω）$和$ \ m m缩bbbbb {d} $代表非局限性的“渐变术语”。根据$λ> 0 $的大小，我们得出存在和不存在的结果。特别是，我们解决了[4，第6节]和[2，第7节]中提出的几个开放问题。
The main goal of this paper is to prove existence and non-existence results for deterministic Kardar-Parisi-Zhang type equations involving non-local "gradient terms". More precisely, let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial Ω$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form \[ \tag{KPZ} \left\{ \begin{aligned} (-Δ)^s u & = μ(x) |\mathbb{D}(u)|^q + λf(x), \quad && \mbox{ in } Ω,\\ u & = 0, && \mbox{ in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $q > 1$ and $λ> 0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $μ$ belongs to $L^{\infty}(Ω)$ and $\mathbb{D}$ represents a nonlocal "gradient term". Depending on the size of $λ> 0$, we derive existence and non-existence results. In particular, we solve several open problems posed in [4, Section 6] and [2, Section 7].
