Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

In this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

Comparison Principle for Elliptic Equations with Mixed Singular Nonlinearities

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

Multiplicity results for sublinear elliptic equations with sign-changing potential and general nonlinearity

In this paper, we study the following elliptic boundary value problem: $$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x, u),\quad x\in \Omega , \\ u=0, \quad x \in \partial \Omega , \end{cases} $$ { − Δ u + V ( x ) u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset {\mathbb {R}}^{N}$ Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that $V\in L^{N/2}(\Omega )$ V ∈ L N / 2 ( Ω ) with $N\geq 3$ N ≥ 3 and that $V\in C(\Omega , \mathbb {R})$ V ∈ C ( Ω , R ) with $\inf_{\Omega }V(x)>-\infty $ inf Ω V ( x ) > − ∞ , we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem.

An adaptively enriched coarse space for Schwarz preconditioners for P1 discontinuous Galerkin multiscale finite element problems

In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.