Variational framework and Lewy-Stampacchia type estimates for nonlocal operators on Heisenberg group

The Lewy-Stampacchia Inequality for the Fractional Laplacian and Its Application to Anomalous Unidirectional Diffusion Equations

In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.

A Lewy-Stampacchia Estimate for quasilinear variational inequalities in the Heisenberg group

We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $ε\ge0$ and $\bar u_ε$ minimizes the functional $$ \int_Ω(ε+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $ψ$, then 0 \le ÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}\bar u_ε|^2)^{(p/2)-1} \nabla_{\H^n}\bar u_ε\Big) \qquad \le (÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}ψ|^2)^{(p/2)-1} \nabla_{\H^n}ψ\Big))^+.

随机伪单调抛物障碍问题：适定性 $\u0026 Lewy-Stampacchia不等式

Stochastic pseudomonotone parabolic obstacle problem: well-posedness $\&$ Lewy-Stampacchia's inequalities

研究了Leray-Lions型非线性伪单调算子主导的随机抛物障碍问题，通过奇异摄动、鞅解和Gyöngy-Krylov方法证明了强解的适定性，并以此为基础建立了变分不等式和Lewy-Stampacchia不等式。

The obstacle problem for nonlinear elliptic equations with variable growth and L^1-data

The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy--Stampacchia inequalities to the general framework of L^1.

On the A-Obstacle Problem and the Hausdorff Measure of its Free Boundary

In this paper we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L^1 data. We also extend the Lewy-Stampacchia inequalities to the general framework of L^1 data, and show convergence and stability results. We then prove that the free boundary has finite N-1 Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p>2.

On the Two Obstacles Problem in Orlicz-Sobolev Spaces and Applications

We prove the Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya-Uraltseva type, including p(x)-Laplacian type operators, we derive new results of $C^{1,α}$ regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz-Sobolev spaces.