Regularity of boundary quasi-potentials for planar systems
The author considers a first order differential system in \(R^ 2: x'= b(x)\), where the vectorfield \(b\) is smooth and bounded. Let \(D\) be a bounded region invariant for the differential system, which is attracted to an asymptotically stable critical point \(0\in D\). The main purpose of the paper is to study the value function \[ W(x)= \inf\{S_{0T}(\phi)\mid 0< T<\infty,\;\phi\in C([0,T];\;\bar D),\;\phi(0)= x,\;\phi(T)\in \partial D\}, \] where \(S_{0T}(\phi)= \int^ T_ 0 L(\phi,\phi')dt\) if \(\phi\) is absolutely continuous, and \(S_{0T}(\phi)= \infty\) otherwise, \(L(x,v)= {1\over 2}(b(x)- v)^ T a(x)^{-1}(b(x)- v)\), \(a(x)\) is symmetric and positive definite. It is proved that if \(\partial D\) is a single \(T\)-periodic solution and if \(\text{div}(b)> 0\) everywhere on \(\partial D\), then \(W\) is a smooth function in some strip around \(\partial D\) and \(W\) satisfies the Hamilton-Jacobi equation \(H(x,-dW(x))= 0\), where \(H\) is the Legendre transform of \(L\). This problem is connected with the study of small random perturbation of planar dynamical systems. Other results are also given when \(\partial D\) satisfies other conditions than the previous one.
scientific article; zbMATH DE number 617361