Related papers: Effective numerical computation of $p(x)-$Laplace …
In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation
$L^1$ based optimization is widely used in image denoising, machine learning and related applications. One of the main features of such approach is that it naturally provide a sparse structure in the numerical solutions. In this paper, we…
We introduce an algebraic multiscale method for two--dimensional problems. The method uses the generalized multiscale finite element method based on the quadrilateral nonconforming finite element spaces. Differently from the…
We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a $p$-structure, including degenerate diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain decomposition is…
The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14,…
This paper mainly investigates the analytic solutions for the approximation of $p$-Laplacian problem. Through an approximation mechanism, we convert the nonlinear partial differential equation with Dirichlet boundary into a sequence of…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle…
We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained…
The explicit solution to the Dirichlet problem for a class of mean value equations on the real line is derived. It shed some light on the behavior of solutions to general nonlocal elliptic equations.
We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise…
In this paper, we present a problem involving fully nonlinear elliptic operators with Hamiltonian, which can present a singularity or degenerate as the gradient approaches the origin. The model studied here, allows the appearance of plateau…
In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calder\'on-Zygmund type estimates in the variable exponent setting. In a recent work of…
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and…
In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only.…
This paper studies a maximal $L^q$-regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having superlinear and sub-quadratic growth, complemented with Dirichlet boundary…
In this paper we study the asymptotic behavior of the solutions of a class of nonlinear elliptic problems posed in a 2-dimensional domain that degenerates into a line segment (a thin domain) when a positive parameter $\varepsilon$ goes to…
A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the…