论文标题
具有Neumann边界条件的规定平均曲率方程的多界变化溶液
Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions
论文作者
论文摘要
我们证明了Neumann问题的多个正面BV分解$$ \ begin {case} \ displayStyle- \ left（\ frac {U'} {\ sqrt {\ sqrt {1+u'^2}}}}} \ right） u'（0）= u'（1）= 0，＆{cases} $$其中$ a（x）> 0 $和$ f $属于一类非线性函数，其原型示例由$ f（u）=-λu + u^p $，$λ> 0 $ 0 $和$ p> 1 $给出。特别是，$ f（0）= 0 $和$ f $具有唯一的正零，以$ u_0 $表示。解决方案的区别在于与恒定解决方案$ u = u_0 $的相交数量（从广义上讲）。我们进一步证明，所发现的解决方案具有连续的能量，我们还为非线性提供了足够的条件以获得经典的解决方案。使用平均曲率算子和射击方法的近似进行分析。
We prove the existence of multiple positive BV-solutions of the Neumann problem $$ \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} $$ where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -λu + u^p$, for $λ> 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method.
