本文研究在边界条件$\partial$ u $\partial$ $下，在$Ω$中寻找一个验证$Δ$ u = 0的函数u的问题> n + g（u）= $\partial$ $Ω$上的$μ$其中$Ω$ $\subset$ RN是光滑域，n是$Ω$的法向单位向外向量，$μ$是对$\partial$ $Ω$的度量，g是连续非减函数。我们给出了这个问题对任意测度可解的关于g的充分条件。当g（r）= |r| p--1 r，p > 1时，我们给出了$\partial$ $Ω$上孤立奇异性可消除的条件。我们还给出了测度$μ$上的容性条件，使得g（r）= |r| p-1 r的问题对于某些$μ$是可解的.本文还研究了满足- $Δ$ u = 0的函数在$Ω$上的孤立奇点和满足$\partial$ u $\partial$ n + g（u）= 0的函数在$\partial$ $Ω$ \ {0}上的孤立奇点.
英文摘要：
We study the problem of finding a function u verifying --$Δ$u = 0 in $Ω$ under the boundary condition $\partial$u $\partial$n + g(u) = $μ$ on $\partial$$Ω$ where $Ω$ $\subset$ R N is a smooth domain, n the normal unit outward vector to $Ω$, $μ$ is a measure on $\partial$$Ω$ and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r) = |r| p--1 r, p > 1, we give conditions in order an isolated singularity on $\partial$$Ω$ be removable. We also give capacitary conditions on a measure $μ$ in order the problem with g(r) = |r| p--1 r to be solvable for some $μ$. We also study the isolated singularities of functions satisfying --$Δ$u = 0 in $Ω$ and $\partial$u $\partial$n + g(u) = 0 on $\partial$$Ω$ \ {0}.
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We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory. At the interface of theoretical physics, Riemannian geometry and nonlinear analysis
