Related papers: Characterization of stress concentration in two-di…
It is an interesting and important topic to study the motion of small particles in a viscous liquid in current applied research. In this paper we assume the particles are convex with arbitrary shapes and mainly investigate the interaction…
This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation $$\partial_t u-\Delta_p u=|\nabla u|^q,$$ with Dirichlet boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$, where $p>2$ and…
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We…
In composite material, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lam\'{e} system of linear elasticity. The aim of…
In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of…
Under some conditions we give a blow-up analysis for solutions of an equation with Dirichlet boundary condition.
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between…
On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} -…
We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient…
When inclusions with extreme conductivity (insulator or perfect conductor) are closely located, the gradient of the solution to the conductivity equation can be arbitrarily large. And computation of the gradient is extremely challenging due…
We study the insulated conductivity problem which involves two adjacent convex insulators embedded in a bounded domain. It is known that the gradient of solutions may blow up as the distance between the two inclusions tends to zero.…
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is…
In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and…
We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta_p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become…
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on…
This paper is devoted to establishing results for semilinear elliptic boundary value problems where the solvability of problems subject to {\it No Flux} boundary conditions follows from the solvability of related {\it Dirichlet} boundary…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
We consider the diffusive Hamilton-Jacobi equation $u_t - \Delta u = |\nabla u|^p$ in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for $p>2$, the solutions to this problem can exhibit gradient blow-up…
Using a conjecture that allows to approach separable-variables conductivity functions, the elements of the Modern Pseudoanalytic Function Theory are used, for the first time, to numerically solve the Dirichlet boundary value problem of the…
We investigate the 2D combustion model with Dirichlet boundary conditions and slip boundary conditions in bounded domains. The global existence of weak and strong solutions and the uniqueness of strong solutions are obtained provided the…