论文标题
非均匀双相方程的弱和粘度解的等效性
Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation
论文作者
论文摘要
我们建立了弱和粘度解决方案与非均匀双相方程之间的等价性，低阶项$$ - {\ rm div}（| du |^{p-2} du+a（x）| du+a（x）| du |^{q-2} du） $$我们在系数$ a（x）$，指数$ p，q $和非线性项$ f $上找到一些适当的假设，以表明具有{\ em a a Ligi} lipChitz连续性的粘度解决方案是由于$ \ inf $ \ \ sup $（$ \ sup $）的薄弱解决方案。可以通过比较原则结论反向含义。此外，我们验证有界的粘度解决方案完全是Lipschitz的连续，这也是独立的。
We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$ -{\rm div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0. $$ We find some appropriate hypotheses on the coefficient $a(x)$, the exponents $p, q$ and the nonlinear term $f$ to show that the viscosity solutions with {\em a priori} Lipschitz continuity are weak solutions of such equation by virtue of the $\inf$($\sup$)-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest.
