On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Michele Matzeu ◽  
Raffaella Servadei
Keyword(s):  
Variational Inequalities ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Divergence Form ◽  
Elliptic Operators

AbstractIn this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled bywhere Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given byHere a

On a family of torsional creep problems involving rapidly growing operators in divergence form

2018 ◽  
Vol 149 (2) ◽  
pp. 495-510 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu
Keyword(s):  
Boundary Condition ◽  
Uniform Convergence ◽  
Bounded Domain ◽  
Distance Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Divergence Form ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Real Numbers

Let Ω⊂ℝN (N≥2) be a bounded domain with smooth boundary and {pn} be a sequence of real numbers converging to+∞ as n→∞. For each integer n>1, we define the function $\varphi_{n}(t)=p_{n} \vert t \vert^{p_{n}-2}te^{ \vert t \vert^{p_{n}}}$, for all t∈ℝ, and we prove the existence of a unique nonnegative variational solution for the problem−div(((φn(|∇ u(x)|))/(|∇ u(x)|))∇ u(x))=φn(1), when x∈Ω, subject to the homogeneous Dirichlet boundary condition. Next, we establish the uniform convergence in Ω of the sequence of solutions for the above family of equations to the distance function to the boundary of Ω. Our result complements the earlier developments on the topic obtained by Payne and Philippin [26], Kawohl [21], Bhattacharya, DiBenedetto and Manfredi [2], Perez-Llanos and Rossi [27] and Bocea and Mihăilescu [4].

scholarly journals Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xiongtao Wu ◽  
Wenyu Tao ◽  
Yanping Chen ◽  
Kai Zhu
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Square Function ◽  
Fractional Differentiation ◽  
Square Functions ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

An Example of Fractal Singular Homogenization

2007 ◽  
Vol 14 (1) ◽  
pp. 169-193
Author(s):  
Umberto Mosco ◽  
Maria Agostina Vivaldi
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Open Domain ◽  
Weighted Energy ◽  
Spectral Convergence

Abstract We construct a sequence of quadratic weighted energy forms in an open domain of the plane, that 𝑀-converges to an energy form with a singular fractal term. The weights belong to the Muckenoupt class 𝐴2 and have pointwise singularities. The result implies the spectral convergence of a sequence of second-order weighted elliptic operators in divergence form in the plane to a singular elliptic operator with a second order fractal term.

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

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