论文标题
粘度解决方案理论在高阶微分方程中的应用
An Application of the Theory of Viscosity Solutions to Higher Order Differential Equations
论文作者
论文摘要
我们将粘度解决方案的理论直接应用于大于两个以上的阶差分方程。我们证明，对于$ c^{2，α}（b_r）\ cap c（\ overline {b_r}）$的解决方案，用于ball $ b_r \ b_r \ subset \ subet \ subbb {r}^n $：$ b_r \ subset {r}^n $： | d（ΔU）|^2 = f（x）$$带有navier边界条件（$ u = g \ in c（\ partial b_r），ΔU= 0 \ textrm {on} \ partial b_r $）。我们还证明，在$ c^{1，α}中存在一个解决方案（\ mathbb {r}^n）$ for $ \ \ m athbb {r}^n $：$ \ mathbb {r}^n $：$ \ m}连续，边界和支撑在环上。
We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,α}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian equation on a ball $B_R\subset \mathbb{R}^n$: $$Δ_\infty^2 u:=(Δu)^3 |D(Δu)|^2 =f(x)$$ with Navier Boundary conditions ($u=g\in C(\partial B_R), Δu =0 \textrm{ on } \partial B_R$). We also prove that there exists a solution in $C^{1,α}(\mathbb{R}^n)$ for all $α>0$ to the eigenvalue problem on $\mathbb{R}^n$: $$Δ_\infty^2 u =-λu+f(x)$$ whenever $n\geq 3, λ<0,$ and $f(x)$ is continuous, bounded, and supported on an annulus.
