10 problems

Let $\Omega\subset\mathbb{R}^n$ be the domain, let $u_\varepsilon$ and $f_\varepsilon$ satisfy the hypotheses of the main convergence theorem, and let $u$ be the limiting phase fun…

Let $\Omega$ be the spatial domain, let $T>0$, and consider the two-phase model with weak solutions defined by quadruples … where $L_0^2(\Omega)$ is the space of square-integrable…

Sharpness conjecture. The sharpest condition should instead be

Let $\theta(t)$ and $\eta(t)$ denote the angle and phase-field variables of the one-dimensional KWC system on $\Omega$, and let $I_*\in\Omega$ be a closed interval. Revised discont…

Let $h$ denote the spatial mesh size and let $\tau$ denote the time-step size. For the displacement field $u$, measure the numerical error in the $l^\infty(L^2)$ norm and the…

Consider the first time step of the numerical method applied to an elliptic initial configuration, with a hyperbolic-tangent profile used as initial data for the order parameter. T…

De Giorgi's conjecture. The functionals $\mathcal{A}_\epsilon$ $\Gamma$-converge in $L^1(\Omega)$ to $\sigma\mathbf{E}_A$:

Consider a one-dimensional phase-field configuration with one or two interfaces and pinned obstacles located at points of $d_\varepsilon\mathbb Z$, where the interfaces are…

Let $V(t)$ solve the interface velocity law … where $F(t)$ is the forcing, and let $\mathcal{S}$ be the set of stable velocities. Theorem 1Dinterface describes the continuous evolu…

Let $c_0$ be the constant and let $\beta$ be the parameter in the velocity law. For a velocity $V$, define the stable-velocity set … where $\sigma(\mathcal{A}_V)$ is the spectrum o…