Regularity of extremal solutions of nonlocal elliptic systems
Regularity of extremal solutions of nonlocal elliptic systems
复制标题
非局部椭圆系统极值解的正则性
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Mostafa Fazly
中科院分区：
文献类型：
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作者：
Mostafa Fazly
We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray}
\left\{ \begin{array}{lcl} \hfill \mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v &=& \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminus\Omega , \end{array}\right.
\end{eqnarray}
with an integro-differential operator, including the fractional Laplacian, of the form
\begin{equation*}\label{} \mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*}
when $J$ is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of $J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$ where $s\in (0,1)$ and $a$ is any nonnegative even measurable function in $L^1(\mathbb {S}^{n-1})$ that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions $n 1$ (with positive and negative exponents), respectively. When $s\to 1$, these dimensions are optimal. However, for the case of $s\in(0,1)$ getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions $n<4s$. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.
We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray}
\left\{ \begin{array}{lcl} \hfill \mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v &=& \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminus\Omega , \end{array}\right.
\end{eqnarray}
with an integro-differential operator, including the fractional Laplacian, of the form
\begin{equation*}\label{} \mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*}
when $J$ is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of $J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$ where $s\in (0,1)$ and $a$ is any nonnegative even measurable function in $L^1(\mathbb {S}^{n-1})$ that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions $n 1$ (with positive and negative exponents), respectively. When $s\to 1$, these dimensions are optimal. However, for the case of $s\in(0,1)$ getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions $n<4s$. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.