非局部2-Hessian方程的几何方法
A geometric approach to nonlocal 2-Hessian equations
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中文总结 AI 辅助
本文通过几何方法研究非局部2-Hessian方程，刻画系数矩阵类，证明在弱假设下的一致椭圆性及正则性，结果对s∈(0,1)成立且当s→1时恢复局部情形。
中文摘要 AI 辅助
我们研究一个非局部2-Hessian方程，该方程由分数阶拉普拉斯算子的线性变形的下确界给出，即 $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$。我们刻画了系数矩阵类 ${A}_2$，该矩阵类决定了算子的行为，并提供了可能退化情形的详细几何描述。我们的主要定理表明，对于严格正的右端项，非局部2-Hessian方程保持一致椭圆性，这导致正则性估计。所得结果在比文献中先前考虑的假设更弱的条件下成立，并且适用于完整范围 $s\in(0,1)$。特别地，不需要解具有凸性。我们的假设可以被解释为局部半凹性和2-凸性概念的非局部对应物。此外，所有结果在 $s\to1$ 时是稳定的，恢复局部情形。这里发展的几何方法是新的，即使在局部情形下也是如此，并且可能适用于更广泛的非局部完全非线性方程和曲率型问题。
英文摘要
We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}Δ^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.
发表机构
- Wayne State University（韦恩州立大学）
- Universidad Autónoma de Madrid（马德里自治大学）
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