Dynamic boundary conditions for Hamilton-Jacobi equations

A non standard dynamic boundary condition for a Hamilton-Jacobi equa­tion in one space dimension is studied in the context of viscosity solutions. A comparison principle and, hence, uniqueness is prm·ed by consideration of an equivalent notion of vis­cosity solution for an alternative formulation of the boundary condition. The relationship with a l\eumann condition is established. Global existence is obtained by consideration of a related parabolic approximation with a dynamic boundary condition. The problem is motivated by applications in superconductivity and interface evolution

Hamilton-Jacobi equations with discontinuous source terms

We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is dis- continuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinu- ous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function

Dynamic boundary conditions for Hamilton-Jacobi equations

A non standard dynamic boundary condition for a Hamilton-Jacobi equa­tion in one space dimension is studied in the context of viscosity solutions. A comparison principle and, hence, uniqueness is prm·ed by consideration of an equivalent notion of vis­cosity solution for an alternative formulation of the boundary condition. The relationship with a l\eumann condition is established. Global existence is obtained by consideration of a related parabolic approximation with a dynamic boundary condition. The problem is motivated by applications in superconductivity and interface evolution

Dynamic boundary conditions for Hamilton-Jacobi equations

A nonstandard dynamic boundary condition for a Hamilton--Jacobi equation in one space dimension is studied in the context of viscosity solutions. A comparison principle, and hence uniqueness, is proved by consideration of an equivalent notion of viscosity solution for an alternative formulation of the boundary condition. The relationship with a Neumann condition is established. Global existence is obtained by consideration of a related parabolic approximation with a dynamic boundary condition. The problem is motivated by applications in superconductivity and interface evolution