We establish a local Lipschitz regularity result of solutions to the Cauchy–Dirichlet problem associated with evolutionary partial differential equations

We do not impose any growth assumptions from above on the function \(f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) and only require it to be convex and coercive. The domain \(\Omega \) is required to be bounded and convex, and the time-independent boundary datum \(u_0\) is supposed to be convex and Lipschitz continuous on \({\overline{\Omega }}\). It can be seen as an evolutionary analogue to the one-sided bounded slope condition. Additionally, assuming \(\Omega \) to be uniformly convex, we establish global continuity on \(\overline{\Omega _T}\) of the solution.

我们建立了与演化偏微分方程有关的柯西-狄利克雷问题解的局部Lipschitz正则性结果

我们没有在函数\（f：{\ mathbb {R}} ^ n \ rightarrow {\ mathbb {R}} \）上强加任何增长假设，仅要求它是凸的和强制的。要求域\（\ Omega \）是有界的和凸的，并且与时间无关的边界数据\（u_0 \）应该是凸的，并且Lipschitz在\（{\ overline {\ Omega}} \\）上是连续的。可以将其视为单边有界边坡条件的进化模拟。另外，假设\（\ Omega \）是一致凸的，我们在解的\（\ overline {\ Omega _T} \）上建立全局连续性。