Mouna Kraiem
In this paper, we consider the analogous of the obtacle problem in $H_0^1(Ω)$, on the space $W^{1,p}_0(Ω)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated. The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard
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In this paper,we will study the homogenization of $p$-Laplacian with obstacles in perforated domain, where the holes are periodically distributed and have random size. And we also assume that the $p$-capacity of each hole is stationary ergodic.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N \geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.
We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle
该文研究带内部固定球形障碍物的第一Steklov-拉普拉斯特征值问题，证明内球和体积固定时球壳在近球形集合中局部最大化第一特征值。
Convergence results for the solutions of $(p,q)$-Laplacian double obstacle problems on irregular domains
本文研究 $(p,q)$-Laplace 型算子双障碍问题，并分析分形及预分形边界域上解的渐近收敛性。
We study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution. Moreover, we prove that if the obstacle is continuous, so is the solution.
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take int…
Gradient continuity for $p$-Laplacian obstacle problems under mean oscillation conditions
研究p-拉普拉斯型方程障碍问题解的C^1正则性，核心方法是在数据的迪尼平均振荡条件下证明解的梯度连续，主要贡献是放宽了经典迪尼连续性假设至更一般的平均振荡条件。
Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
本文研究了分数阶拉普拉斯算子障碍问题，通过推广自由边界精细结构分析并证明新的频率间隙，建立了几乎对所有障碍在维度3以内自由边界仅含正则点的正则性结果。
Obstacle problems for the fractional $p$-Laplacian on fractal domains: well-posedness and asymptotics
本文研究了在分形边界上分数p-拉普拉斯方程的障碍问题，证明了解的well-posedness及渐进行为。
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.
An implicit two-phase obstacle-type problem for the $p$-Laplacian with the fractional gradient
本文研究了与 $s$-分数阶 $p$-Laplacian 相关的非齐次两相障碍型问题，证明了解的存在性与唯一性，探讨了解在 $s\ o 1$ 时向经典问题的收敛性，并基于对水平集函数 $v$ 的连续依赖性，得出了隐式水平集两相障碍型问题的存在性结果。
Limit of p-Laplacian Obstacle problems
本文研究$p$-拉普拉斯障碍问题当$p\to\infty$时解的渐近行为，给出了一维和径向情形的极限显式表达式，并为$n$维情形提供了保证整个解族一致收敛的充分条件。
We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.
A two phase boundary obstacle-type problem for the bi-Laplacian
本文研究上半单位球中双调和算子的两相边界障碍型问题，建立了解的适定性与最优正则性，并利用 Almgren 型和 Monneau 型单调性公式分析了自由边界结构。
On a class of thin obstacle-type problems for the bi-Laplacian operator
本文研究了生物拉普拉斯算子的一类薄障碍型问题，通过单调性公式和放大分析，证明了自由边界的分层结果。
We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $φ$ satisfies $Δφ\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,α}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical …
We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{(-Δ)^s u + b \cdot \nabla u,\,u -φ\right\} = 0$ in $\mathbb{R}^n$, in the critical regime $s = \frac{1}{2}$. Our main result establishes the $C^{1,α}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \[ u(x)-φ(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tildeγ(x_0)} + o\left(|x-x_0|^{1+\tildeγ(x_0)+σ}\right), \] \[ \tildeγ(x_0) = \frac{1}{2}+\frac{1}π \arctan (b\cdot e), \] where $e \in \mathbb{S}^{n-1}$ is the normal vector to the free boundary, $σ>0$, and $c_0> 0$. We al…
In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the $p$-obstacle problem: Find the smallest $u$ such that $$ \hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad u\geq ϕ, \qquad \hbox{in } B_1, $$ with $ϕ\in C^{1,1}(B_1)$ and given boundary datum on $\partial B_1$. We prove that the solution is uniformly $C^{1,1}$ at free boundary points. Similar results are obtained in the c…
We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-Δ_p u,\,u-φ\bigr\}=0$ in $Ω\subset\mathbb R^n$. Here, $Δ_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\nabla φ\neq0$, the operator $Δ_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla φ=0$ then $Δ_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $\…
Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle
本文研究固定盒内移除小体积障碍以最小化 Dirichlet 拉普拉斯第一特征值的问题，证明最优障碍聚集于边界上首特征函数梯度最小处，并刻画特征值、特征函数与自由边界的收敛。
We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ Δ_{H}u=fχ_{\{u\ne0\}}, \] where $Δ_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*Γ$ is $C_{H}^{1,1}$, where $Γ$ is the fundamental solution of $Δ_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is …
A game-theoretic approach to the parabolic normalized p-Laplacian obstacle problem
该研究采用博弈论方法，通过带噪声的零和随机拔河博弈为抛物型正规化p-Laplacian障碍问题的解建立概率表示，证明值函数的存在性、动态规划原理及向唯一粘性解的一致收敛性。
We study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving Dirichlet-Laplacian of arbitrary real order $m>0$.
We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.
Epiperimetric inequalities in the obstacle problem for the fractional Laplacian
本文利用周长度量不等式方法研究了分数阶拉普拉斯算子的障碍问题，证明了Weiss能量的正、负周长度量不等式及对数不等式，推导出频率间隙与爆破刻画，并给出了自由边界点的正则性另证与结构描述。
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.
In this paper we obtain symmetry and monotonicity results for positive solutions to some $p$-Laplacian cooperative systems in bounded domains involving first order terms and under zero Dirichlet boundary condition.
In this paper we investigate a two dimensional free boundary problem involving the A-Laplacian. We show that the free boundary is represented locally by graphs of a family of continuous functions.
We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.
We prove the uniqueness for viscosity solutions of a differential equation involving the infinity-Laplacian with a variable exponent. A version of the Harnack's inequality is derived for this minimax problem.
We present a probabilistic approach to the obstacle problem for for the $p$-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the step size go to zero, not unlike the case when Brownian motion is approximated by random walks. Rather than stopping the process when the boundary is reached, the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain.
The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.
In this paper we prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the $A$-Laplacian. We also show that the free boundary is represented locally by graphs of a family of lower semi-continuous functions.
We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.
Quasilinear Choquard equations involving N-Laplacian and critical exponential nonlinearity
该研究聚焦于含N-Laplacian与临界指数非线性项的拟线性Choquard方程，主要探讨了这类方程正解的存在性问题。
In the present paper we study the existence of solutions for some nonlocal problems involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution method
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-Δ_Ω\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $Ω\subset\mathbb R^n$.
In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
This paper is concerned with the study of a nonlinear problems involving the fractional p(x)-Laplacian operator. By means of the Berkovits degree theory, we prove the existence of nontrivial weak solutions for this problem. The appropriate functional framework for this problems is the fractional Sobolev spaces with variable exponent.
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
In the present paper we study the existence of solutions for a class of nonlocal system involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution result.
In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.
In this paper we study strongly indefinite systems involving the fractional Laplacian on bounded domains. We obtain existence and non-existence results, $a priori$ estimates of Gidas-Spruck type, and the symmetric property.
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
In this article, we study the boudary blow-up solutions for semilinear fractional equations with power absorption. Our main purpose is to obtain the existence, nonuniqueness and behavior asymptotic near the boundary.
On the solutions of a double-phase Dirichlet problem involving the 1-Laplacian
针对含1-Laplacian的非齐次Dirichlet双相问题，论文证明了弱解的存在唯一性，并通过相应极小化问题给出其变分刻画。
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal …
We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.
An inverse problem for semilinear equations involving the fractional Laplacian
本文研究带零阶非线性扰动的分数阶拉普拉斯热方程与波动方程反问题，利用分数阶 Dirichlet-to-Neumann 映射、Runge 逼近和唯一延拓性质恢复非线性项。
Regularity and Pohozaev identity for the Choquard equation involving the $p$-Laplacian operator
该研究针对含$p$-Laplacian算子的非线性Choquard方程，开展了弱解正则性分析并建立了Pohozaev型恒等式，丰富了相关方程的理论结果。
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