论文标题
关于分数扩散的准线性方程的注释
A note on quasilinear equations with fractional diffusion
论文作者
论文摘要
在本文中，我们研究了以下非本地椭圆问题的分布解决方案的存在\ begin {eqnarray*} \左\ lbrace \ begin {array} {l} （-Δ）^{s} u + | \ nabla u |^{p} = f \ quad \ text {in}ω \ qquad \ qquad \ qquad \，\，\，\，u = 0 \，\，\，\，\，\，\，\，\，\，\ text {in} \ mathbb {r}^{n}^{n} \ setMinusω \ end {array} \正确的。 \ end {eqnarray*} 我们对源任期$ f $的规律性与相应解决方案的规律性之间的关系感兴趣。如果$ p <2s $，那就是自然增长，我们可以在l^1（Ø）$中显示所有$ f \的存在。在亚临界情况下，也就是说，对于$ p <p _ {*}：= n/（n-2s+1）$，我们表明解决方案为$ \ nathcal {c}^{1，α} $ for $ f \ for $ f \ in L^{m} $，$ m $ gum fame $ m $。在一般情况下，我们在源尺寸的条件下达到相同的结果。作为应用程序，我们可能会表明，对于常规来源，分销解决方案是粘度解决方案，而是相反。
In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*} \left\lbrace \begin{array}{l} (-Δ)^{s}u + |\nabla u|^{p} =f \quad\text{ in } Ω \qquad \qquad \qquad \,\,\, u=0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus Ω, \quad s \in (1/2, 1). \end{array} \right. \end{eqnarray*} We are interested in the relation between the regularity of the source term $f$, and the regularity of the corresponding solution. If $p<2s$, that is the natural growth, we are able to show the existence for all $f\in L^1(Ø)$. In the subcritical case, that is, for $p < p_{*}:=N/(N-2s+1)$, we show that solutions are $\mathcal{C}^{1, α}$ for $f \in L^{m}$, with $m$ large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.
