Schauder estimates for parabolic equations with degenerate or singular weights

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-20 DOI:10.1007/s00526-024-02809-2

Alessandro Audrito, Gabriele Fioravanti, Stefano Vita

引用次数: 0

Abstract

We establish some $C^{0,\alpha }$ and $C^{1,\alpha }$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $\Sigma $ as a power $a > -1$ of the distance to $\Sigma $. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar $C^{1,\alpha }$ estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

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具有退化或奇异权重的抛物方程的 Schauder 估计数

我们为一类发散形式的加权抛物线问题建立了一些（C^{0,\alpha }\ ）和（C^{1,\alpha }\ ）正则性估计。主要的新颖之处在于权重可能会消失或在特征超平面 $\Sigma $上爆炸，作为到 $\Sigma $的距离的幂 $a > -1$ 。对于系数和数据的假设，我们得到的估计值非常精确。我们的方法依赖于方程的正则化和一些均匀正则性估计，并结合了Liouville定理和近似论证。作为我们主要结果的一个推论，当权重的退化/奇异性发生在一个规则的圆柱型超曲面上时，我们会得到类似的（C^{1,\alpha }\ ）估计值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.

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