Global continuity of variational solutions weakening the one-sided bounded slope condition

We study regularity properties of variational solutions to a class of Cauchy–Dirichlet problems of the form { ∂ t ⁡ u - div x ⁡ ( D ξ ⁢ f ⁢ ( D ⁢ u ) ) = 0 in ⁢ Ω T , u = u 0 on ⁢ ∂ 𝒫 ⁡ Ω T . \left\{\begin{aligned} \displaystyle\partial_{t}u-\operatorname{div}_{x}(D_{% \xi}f(Du))&\displaystyle=0&&\displaystyle\phantom{}\text{in }\Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on }\partial% _{\mathcal{P}}\Omega_{T}.\end{aligned}\right. We do not impose any growth conditions from above on f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\to\mathbb{R}} , but only require it to be convex and coercive. The domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is mainly supposed to be bounded and convex, and for the time-independent boundary datum u 0 : Ω ¯ → ℝ {u_{0}\colon\overline{\Omega}\to\mathbb{R}} we only require continuity. These requirements are weaker than a one-sided bounded slope condition. We prove global continuity of the unique variational solution u : Ω T → ℝ {u\colon\Omega_{T}\to\mathbb{R}} . If the boundary datum is Lipschitz continuous, we obtain global Hölder continuity of the solution.

On the existence and uniqueness of solution to a stochastic 2D Allen–Cahn–Navier–Stokes model

We study, in this paper, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional (2D) bounded domain. The model consists of the Navier–Stokes equations (NSEs) for the velocity, coupled with a Allen–Cahn model for the order (phase) parameter. We prove the existence and the uniqueness of a variational solution.

A Variational Solution of the Schrödinger Equation in an Inhomogeneous Central Field

The present work is devoted to the computer modeling of emission processes from the graphene surface which is a promising material for modern applications. We examine the effect of the ion field inhomogeneity on the variational solution of the Schrödinger equation for the ground state of a loosely bound electron of a hydrogen-like carbon atom, which simulates the graphene model. This shows a significant impact of the allowance for the field inhomogeneity to the ground state of the electron.