Variational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations, arising in numerous applications. In the linear case many refined results are known, while in the nonlinear setting our understanding is still at a preliminary stage.
In this paper we prove $C^1$ regularity for the solutions to a general class of quasi-linear variational inequalities with thin obstacles and $C^{1, α}$ regularity for variational inequalities under Signorini-type conditions on the boundary of a domain.
We show that signed weak solutions to parabolic obstacle problems with porous medium type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous.
In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{σ,σ/2}$-regularity for every $0<σ<1$, as well as $H^{β,β/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<β<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.
This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a ingenious Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions' regularity that we intended to study. We assume the same Sobolev regularity both for the gra…
In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary.
In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$ regularity of the free boundary. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle $ψ$, $F(D^2 u,x) =fχ_{Ω(u) \cap\{ u< ψ\} } + F(D^2ψ,x) χ_{Ω(u)\cap \{u=ψ\}}, u\le ψ\text { in } B_1$, where $Ω(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$.
In this paper we study the regularity properties of $Λ$-minimizers of the capillarity energy in a half space with the wet part constrained to be confined inside a given planar region. Applications to a model for nanowire growth are also provided.
We revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem, we show that each stratum can be further decomposed into a `good' part and an `anomalous' part, where the former is covered by $C^{1,1-}$ manifolds, and the latter is of lower dimension.
We analyse the obstacle problem for the nonlocal parabolic operator \[\partial_t u + (-Δ)^{s} u - b \cdot \nabla u - \mathcal{I}u - ru,\] where $b\in\mathbb{R}^n$, $r\in\mathbb{R}$, and $\mathcal{I}$ is a nonlocal lower order diffusion operator with respect to the fractional Laplace operator $(-Δ)^{s}$. This model appears in the study of American options pricing when the stochastic process governing the stock price is assumed to be a purely jump process. We study the existence and the uniqueness of solutions to the obstacle problem, and we prove optimal regularity of solutions in space, and al…
The interior free boundary theory for linear elliptic operators in higher dimensions was developed by Caffarelli in the low regularity context. In these notes, the up-to-the boundary free boundary regularity is discussed for nonlinear elliptic operators based on a different approach.
The Variable Coefficient Thin Obstacle Problem: Higher Regularity
在这篇文章中，我们继续我们的调查薄障碍问题的变系数，这是在\cite{KRS14}，\cite{KRSI}。利用部分Hodograph-Legendre变换和隐函数定理，证明了正则自由边界的高阶Hölder正则性，如果相关系数具有相应的正则性.对于零障碍物，与度量的规则性相比，这产生了针对自由边界规则性的\emph{full derivative}的改进。在存在非零障碍或不均匀性的情况下，我们得到了自由边界正则性相对于不均匀性的正则性的\emph{three halves of a derivative}。进一步证明了解析度量的正则自由边界的解析性。我们还讨论了低正则性设置的$W^{1,p}$度量与$p>n+1$有和没有（$L^p$）的不均匀性。在我们的分析中，关键的新成分是广义Hölder空间的引入，它允许将变换的完全非线性，退化（子）椭圆型方程解释为Baouendi-Grushin算子的扰动，与适当的算子相关的内在几何的各种用途，隐函数定理的应用推导出（更高）的正则性和分裂技术从\cite{KRSI}。
In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^2 v= f(y)χ_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^2 u = 1/f (Du)+δ_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic eq…
We prove $C^{1, α}$ regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory (\cite{Kim11,GKTT1}). In this paper, we prove an Hölder regularity in $C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary…
We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian $(-Δ)^s$ (and more general integro-differential operators) in the regime $s>\frac{1}{2}$. We prove that once the free boundary is $C^1$ it is actually $C^{2,α}$. To do so, we establish a boundary Harnack inequality in $C^1$ and $C^{1,α}$ (moving) domains, providing that the quotient of two solutions of the linear equation, that vanish on the boundary, is as smooth as the boundary. As a consequence of our results we also establish for the first time optimal regularity of such solutions to…
We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension $n-1$ which is also $\varepsilon$-flat in space, for any $\varepsilon>0$.
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $\mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $n\leq4$, the free boundary is generically a $C^\infty$ manifold. Thi…
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 …
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-Δ)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free boundary is $C^1$ then it is $C^\infty$, whenever $s\not\in\mathbb{Q}$. In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that $s$ is irrational plays …
