arXiv 1807.061872018-07math.APmath.PR

具有有限程跳过程和衰减的非局部系统

On nonlocal systems with jump processes of finite range and with decays

Mostafa Fazly, Changfeng Gui

备注：: To appear in Journal of Differential Equations. 21 pages. The article arXiv:1506.03368 (never published) is included in this manuscript

摘要：

当$m\ge 1$，$u_i: \mathbb R^n \to \mathbb R$和$H=(H_i)_{i=1}^m$是一般非线性项序列时，我们研究了下列方程组$$ L_i(u_i) = H_i(u_1,\cdots,u_m) \quad \text{in} \ \ \mathbb R^n , $$.非局部算子$L_i$由$$L_i(f (x)):= \lim_{ε\to 0} \int_{\mathbb R^n \setminus B_ε(x) } [f(x) - f(z)] J_i(z-x) dz,$$给出，用于偶数，非负和可测量的跳跃核序列$J_i$。对于一般的跳跃核$J_i$，我们证明了上述系统稳定解的Poincaré不等式.特别地，对于标量方程的情况，即当$m=1$时，对于任何$η\in C_c^1(\mathbb R^{n})$和一些非负的$ \mathcal A_y(\nabla_x u)$和$ \mathcal B_y(\nabla_x u)$，它读取\begin{equation*}\label{} \iint_{ \mathbb R^{2n}} \mathcal A_y(\nabla_x u) [η^2(x)+η^2(x+y)] J(y) dx dy \le \iint_{ \mathbb R^{2n}} \mathcal B_y(\nabla_x u) [ η(x) - η(x+y) ] ^2 J(y) d x dy , \end{equation*}。这是一个对应的著名的不等式来自斯滕伯格和Zumbrun在\cite{sz}半线性椭圆方程，被广泛用于文献中建立德Giorgi型结果，研究相变和证明正则性。然后，我们将这个不等式应用到有限范围的跳跃过程和跳跃过程与衰减证明De Giorgi型结果在二维。此外，我们表明，无论何时$H_i(u)\ge 0$或$\sum_{i=1}^m u_i H_i(u)\le 0$，然后刘维定理持有每个$u_i$在一个和两个维度。最后，我们提供了一定的能量估计在各种假设下的跳跃核$J_i$和Liouville定理的$u$偏导数的商。

英文摘要：

We study the following system of equations $$ L_i(u_i) = H_i(u_1,\cdots,u_m) \quad \text{in} \ \ \mathbb R^n , $$ when $m\ge 1$, $u_i: \mathbb R^n \to \mathbb R$ and $H=(H_i)_{i=1}^m$ is a sequence of general nonlinearities. The nonlocal operator $L_i$ is given by $$L_i(f (x)):= \lim_{ε\to 0} \int_{\mathbb R^n \setminus B_ε(x) } [f(x) - f(z)] J_i(z-x) dz,$$ for a sequence of even, nonnegative and measurable jump kernels $J_i$. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel $J_i$. In particular, for the case of scalar equations, that is when $m=1$, it reads \begin{equation*}\label{} \iint_{ \mathbb R^{2n}} \mathcal A_y(\nabla_x u) [η^2(x)+η^2(x+y)] J(y) dx dy \le \iint_{ \mathbb R^{2n}} \mathcal B_y(\nabla_x u) [ η(x) - η(x+y) ] ^2 J(y) d x dy , \end{equation*} for any $η\in C_c^1(\mathbb R^{n})$ and for some nonnegative $ \mathcal A_y(\nabla_x u)$ and $ \mathcal B_y(\nabla_x u)$. This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in \cite{sz} for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever $H_i(u)\ge 0$ or $\sum_{i=1}^m u_i H_i(u)\le 0$ then Liouville theorems hold for each $u_i$ in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel $J_i$ and a Liouville theorem for the quotient of partial derivatives of $u$.

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摘要：

英文摘要：

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