论文标题
关于自由边界问题解决方案的独特性和单调性
On the uniqueness and monotonicity of solutions of free boundary problems
论文作者
论文摘要
对于任何$ω\ subset \ mathbb {r}^n $平滑且有限的域，我们证明了在整齐间隔的$ω$上出现的自由边界问题的肯定解决方案的独特性，这仅取决于Sobolev $ h^{1} _0} _0} _0（$ hookirt $ l^whook whook r^l^2p） [1，\ frac {n} {n-2}）$，并表明边界密度和适当定义的能量具有通用的单调行为。至少据我们所知，对于$ p> 1 $，这是关于不是二维球的独特性的第一个结果，尤其是解决方案的单调性的第一个结果，即使对于$ p = 1 $，这似乎是新的。对于$ p = 1 $来说，阈值是尖锐的，产生了一种新条件，可以保证$ω$内部没有自由边界。作为推论，在同一范围内，我们解决了一个长期的开放问题（可以追溯到1980年的Berestycki-Brezis的工作），涉及变异解决方案的独特性。此外，在二维球上，我们描述了阳性溶液的完整分支，也就是说，我们证明了沿正溶液的曲线曲线的单调性，直到边界密度消失。
For any $Ω\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $Ω$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(Ω)\hookrightarrow L^{2p}(Ω)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for $p=1$. The threshold, which is sharp for $p=1$, yields a new condition which guarantees that there is no free boundary inside $Ω$. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.
