一般边界条件下混合阶偏微分方程的时间周期解的$L^p$理论
From Whole-Space Theory to Boundary Value Problems for Mixed-Order PDEs
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中文总结 AI 辅助
该研究建立了一般边界条件下混合阶偏微分方程时间周期解的$L^p$理论，引入抽象框架与补条件，应用于Cahn--Hilliard--Gurtin系统等问题，得到适定性显式准则。
中文摘要 AI 辅助
我们针对混合阶偏微分方程及方程组的时间周期边值问题建立了$L^p$理论。该方法基于由阶函数及其关联牛顿多边形描述的各向异性函数空间。我们建立了半空间上的迹理论，包括通过实插值对迹空间的刻画。对于一般边界条件，我们引入了一个抽象框架，其中适定性由基于解的迹表述的补条件刻画。特别地，包含边界算子诱导的相容性条件的容许数据空间自然地从该抽象框架中产生。对于混合阶微分算子，补条件简化为补边界矩阵的可逆性，从而得到基于$L^p$空间的适定性的显式准则。所得理论适用于一般牛顿多边形结构，且允许包含时间导数的边界算子。作为应用，我们建立了Cahn--Hilliard--Gurtin系统及带动态边界条件的抛物型问题的时间周期$L^p$适定性。
英文摘要
We develop a general principle for passing from the analysis of partial differential operators on the whole space to boundary value problems on the half-space. The reduction separates the boundary problem into restricted invertibility of the interior operator and a complementing problem formulated entirely on the trace space. For a broad class of operators admitting suitable factorizations, the restricted invertibility is governed by the whole-space theory of the factors, while the complementing problem determines both well-posedness and the admissible combinations of interior and boundary data. In this way, boundary value theory is reduced to whole-space operator theory together with an explicit analysis on the trace space. We realize this principle for time-periodic partial differential equations and systems of mixed order, developing the required Newton-polygon function spaces, interpolation theory, and $L^p$-trace theory. For mixed-order differential operators under general boundary conditions, well-posedness and the admissible data space are determined by an explicit complemented boundary matrix. This yields new $L^p$-well-posedness results for Cahn--Hilliard equations with dynamic boundary conditions and for Cahn--Hilliard--Gurtin systems. Most notably, we obtain the first maximal $L^p$-regularity result for a genuinely mixed-order interior system coupled to dynamic boundary conditions carrying a distinct mixed-order structure.
发表机构
- Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学）
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