In this article we study the scaling limit of interface model on \({{\,\mathrm{{\mathbb {Z}}}\,}}^d\) where Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension Gaussian free field. discuss appropriate spaces which convergence takes place. While infinite volume proof based Fourier analytic methods, finite rely some discrete PDE techniques involving f...

In this paper, we study the following prototypical two-species chemotaxis system with Lotka-Volterra competition and signal production: $$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot (u\nabla w)+u(1-u^{\theta -1}-v), &x\in \Omega , t>0, \\ v_{t}=\Delta v-\nabla (v\nabla w)+v(1-v-u), & x\in w_{t}=\Delta w-w+ u+ v, t>0. \end{array}\displaystyle \right .(\ast ) We sho...

Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues fourth order Lamm–Rivière system $$\begin{aligned} \Delta ^{2}u=\Delta (V\cdot \nabla u)+\mathrm{div}(w\nabla u)+(\nabla \omega +F)\cdot u+f \end{aligned}$$ in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher sharp Hölder co...

We consider singular problems associated with the weighted anisotropic $p$-Laplace operator $$ H_{p,w}u=\text{div}(w(x)(H(\nabla u))^{p-1}\nabla H(\nabla u)), where $H$ is a Finsler-Minkowski norm and weight $w$ belongs to class of $p$-admissible weights, which may vanish or blow up near origin. discuss existence regularity properties weak solutions for mixed exponential nonlinearities. In part...