Related papers: An asymptotic analysis for a generalized Cahn-Hill…
We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to…
We prove the existence of unique weak solutions to an extension of a Cahn--Hilliard model proposed recently by C.~Liu and H.~Wu (2019), in which the new dynamic boundary condition is further generalised with an affine linear relation…
A variational analysis is presented for the generalized spiked harmonic oscillator Hamiltonian operator H, where H = -(d/dx)^2 + Bx^2+ A/x^2 + lambda/x^alpha, and alpha and lambda are real positive parameters. The formalism makes use of a…
We consider fractional operators of the form $$\mathcal{H}^s=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s,\ (x,t)\in\mathbb R^n\times\mathbb R,$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$ is an accretive,…
In this paper, we study the prototypical model of liquid-liquid phase separation, the Cahn-Hilliard functional, in a highly irregular setting. Specifically, we analyze potentials with low regularity vanishing on space-dependent wells. Under…
A model of strongly inhomogeneous medium with simultaneous perturbation of rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a…
We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the…
Current performance bounds for randomized iterative methods are often considered tight under per-iteration analyses, yet they are notoriously loose in practice. We derive asymptotic performance bounds that narrow this theory-practice gap,…
We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of…
We prove an integral representation result for variational functionals in the space $BV^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous…
This paper investigates asymptotic fixed point results for nonlinear contractions, with emphasis on Kirk-type theorems and their generalizations. A central difficulty in the literature has been the requirement that the mapping possesses a…
We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $\Omega\times…
Consider non-linear time-fractional stochastic heat type equations of the following type, $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where…
The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation…
We investigate the nonlocal version of the Abels-Garcke-Gr\"{u}n (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system…
In this survey we prove H\"older regularity results for viscosity solutions of fully nonlinear nonlocal uniformly elliptic second order differential equations with local gradient terms. This extends the nonlocal counterpart of the work of…
This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that…
The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is asymptotically free for $N>2$: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in…
The asymptotic properties of integral operators with the generalized sine kernel acting on the real axis are studied. The formulas for the resolvent and the Fredholm determinant are obtained in the large x limit. Some applications of the…
In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 & on\quad \partial\Omega\\ \end{array} \right.…