On semicontinuous solutions for general Hamilton-Jacobi equations
In this short note the authors announce several recent results concerning a new type of semicontinuous viscosity solutions for problems of the form: \[ u_t+H(x,u_x)=0 \quad \text{on } (0,T)\times \mathbb{R}^n, \qquad u(0,x)\equiv u_0(x) \] in the case \(u_0\) is upper semicontinuous and the Hamiltonian \(H\) satisfies the usual Lipschitz and growth conditions and, for which, in addition, the existence of the ``recession function'': \[ H_\infty(x,p):=\lim_{\lambda \to 0_+}\lambda _0H\biggl(x,{{p}\over {\lambda}}\biggr) \] is assumed. The new concept of \(L\)-solution is defined as follows: If \(K_0:= \{(x,y)\); \(x\in \mathbb{R}^n\), \(y\leq u_0(x)\}\) is the subgraph of \(u_0\), \(\psi_0(x,y):=-\min\{d((x,y),K_0),1\}\) and if \(\psi\) is a continuous viscosity solution of the associated ''level-set equation'': \[ \psi_t-\psi_yH(x,-\psi_x/\psi_y)=0 \quad \text{in } (0,T)\times \mathbb{R}^{n+1}, \qquad \psi (0,x,y)\equiv \psi_0(x,y) \] then the function \(u\) defined by: \(u(t,x):= \sup\{y\in \mathbb{R}\); \(\psi(t,x,y)\geq 0\}\) is said to be an L-solution of the problem. The authors give the statements of several theorems concerning existence, uniqueness, comparison, consistency and ``right-accessibility'' of this type of generalized solutions, the details being published in preprint form.
scientific article; zbMATH DE number 1443389