A rigorous setting for the reinitialization of first order level set equations

In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function. We also propose another simpler equation whose solution has a gradient bound away from zero

On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general

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

Eikonal equations in metric spaces

A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to in- nite dimensional setting as well as topological networks, surfaces with singularities

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学位の種別:課程博士University of Tokyo(東京大学

Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations

We present a new Harnack inequality for non-negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices. This estimate applies to all supersolutions; instead the Harnack constant depends on the graph distance on lattices. For the proof we modify the proof of the weak Harnack inequality. Applying the same idea to elliptic equations in a Euclidean space, we also derive a Harnack type inequality for non-negative viscosity supersolutions

Uniqueness and existence of viscosity solutions under a degenerate dynamic boundary condition

We consider the initial boundary value problem for a fully-nonlinear parabolic equation in a half space. The boundary condition we study is a degenerate one in the sense that it does not depend on the normal derivative on the boundary. A typical example is a stationary boundary condition prescribing the value of the time derivative of the unknown function. Our setting also covers the classical Dirichlet boundary condition. We establish a comparison principle for a viscosity sub- and supersolution under a weak continuity assumption on the solutions on the boundary. We also prove existence of solutions and give some examples of solutions under several boundary conditions. We show among other things that, in the sense of viscosity solutions, the stationary boundary condition can be different from the Dirichlet boundary condition which is obtained by integrating the stationary condition

An improvement of level set equations via approximation of a distance function

In the classical level set method, the slope of solutions can be very small or large, and it can make it difficult to get the precise level set numerically. In this paper, we introduce an improved level set equation whose solutions are close to the signed distance function to evolving interfaces. The improved equation is derived via approximation of the evolution equation for the distance function. Applying the comparison principle, we give an upper- and lower bound near the zero level set for the viscosity solution to the initial value problem.Mathematics Subject Classification 2010: 35D40; 35F21; 35F2