In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,{\xi }^0)\in {ℂ}_x^n\times {ℂ}_u\times {ℂ}_{\xi }^n({\xi }^0=D_{x}u\left(0\right))$ and $f(0,0,{\xi }^0)=0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $\left(\xi \in {ℂ}^n\right)$. The criterion of convergence of a formal solution $u\left(x\right)={\sum }_{\left|\alpha \right|\ge 1}{u}_{\alpha }{x}^{\alpha }$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal...

Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z):=(y{\partial }_x-z{\partial }_y)u(x,y,z)=f(x,y,z){\in }_{x,y,z}$, where $P=y{\partial }_x-z{\partial }_y{:}_{x,y,z}{\to }_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.