Damião J. Araújo, Rafayel Teymurazyan, Vardan Voskanyan
We obtain sharp local $C^{1,α}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ Δ_p u=γ(u-φ)^{γ-1}\,\text{ in }\,\{u>φ\}, $$ for $0<γ<1$ and $p\ge2$. At the free boundary $\partial\{u>φ\}$, we prove optimal $C^{1,τ}$ regularity of solutions, with $τ$ given explicitly in terms of $p$, $γ$ and smoothness of $φ$, which is new even in the linear setting.
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We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.
We construct a monotone quantity for the classical obstacle problem with non-smooth obstacle, and show that the blow-ups are homogeneous functions of degree $α<2$.
In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.
We study some regularity issues for solutions of non-autonomous obstacle problems with $(p,q)$-growth. Under suitable assumptions, our analysis covers the main models available in the literature.
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.
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.
Partial regularity for Double Phase obstacle problems
本文研究了在满足适当拓扑性质的向量障碍型约束下，双相泛函的向量值局部极小子的部分正则性。
This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp $C^{1,1/3}$--regularity at the free boundary.
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 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 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.
These notes record and expand the lectures for the `Journées Équations aux Dérivées Partielles 2018' held by the author during the week of June 11-15, 2018. The aim is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary.
In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.
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 article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson \cite{An16} and the epiperimetric inequality from \cite{FS16}, \cite{GPSVG15}, we prove the optimal $C^{1,\min\{α,1/2\}}$ regularity of solutions in the presence of $C^{0,α}$ coefficients $a^{ij}$ and $C^{1,α}$ obstacles $ϕ$. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $C^{1,γ}$ manifold for some $γ\in (0,1)$.
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 $Ω(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$.
Regularity for obstacle problems to anisotropic parabolic equations
本文沿用Dibenedetto的内在标度法，在障碍仅满足Hölder连续性的条件下，证明了部分各向异性抛物方程障碍问题弱解的局部Hölder连续性。
Optimal Hölder regularity for solutions to Signorini-type obstacle problems
本文研究一类障碍问题弱解的存在性、唯一性和正则性，建立了Signorini型问题（障碍条件施加在余维1子集上）的最优Hölder正则性，并应用于不规则障碍的经典障碍问题。
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.
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.
Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
本文研究Monge-Ampère障碍问题自由边界直到边界的正则性，发现边界值增长阶影响正则性，且二次增长下高维正则性优于二维。
In this paper, we prove local $C^{1}$ regularity of free boundaries for the double obstacle problem with an upper obstacle $ψ$, \begin{align*} Δu &=fχ_{Ω(u) \cap\{ u< ψ\} }+ Δψχ_{Ω(u)\cap \{u=ψ\}}, \qquad u\le ψ\quad \text { in } B_1, \end{align*} where $Ω(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for $u$ and $ψ$.
The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
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 $s\in(0,1)$.
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 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…
Existence, uniqueness, and regularity of solutions to nonlinear and non-smooth parabolic obstacle problems
研究非线性抛物障碍问题解的存在性、唯一性及$W^{1,2,p}$正则性，适用于凸障碍，为停止问题提供决策者不选择凸角的条件。
In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible (NTA) domain, we derive the $C^{1,α}$-regularity of the free boundary in a small ball centered at the origin. From the $C^{1,α}$-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $ C^{3,α}$ up to the free boundary, and therefore $C^{2,1} $. In the end we study an exa…
In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is Hölder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$.
In this paper non-transversal intersection of the free and fixed boundary is shown to hold in any dimension for obstacle problems generated by fully nonlinear uniformly elliptic operators. Moreover, $C^1$ regularity results of the free boundary are obtained and a classification of blow-up solutions is given.
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.
The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results…
The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.
In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and optimal regularity of solutions, and establish structural properties of the free boundary. The proofs are based on tailor-made monotonicity formulas of Almgren, Weiss, and Monneau-type, combined with the classical theory of oblique derivative problems.
In this paper we address the following question regarding the regularity of geodesics in the space of Kähler potentials. Given a geodesic which is highly regular, and has smooth boundary value, can we expect that it is actually smooth? We construct an example to show that the answer to the question is ``no".
This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,α}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1…
Regularity theory for nonlocal obstacle problems with critical and subcritical scaling
本文提出了一种新颖且统一的方法，解决了具有临界与次临界缩放的非局部障碍问题中的正则性理论开放难题，并恢复了大多数已知结果。
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…
Regularity to Thin Obstacle Problem in Orlicz spaces
本文研究了Orlicz空间中薄障碍问题的极小解正则性，证明了梯度的Lipschitz连续性和Hölder连续性，并揭示了极小解零点集的结构特征。
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.
In this article we investigate the higher regularity properties of the regular free boundary in the fractional thin obstacle problem. Relying on a Hodograph-Legendre transform, we show that for smooth or analytic obstacles the regular free boundary is smooth or analytic, respectively. This leads to the analysis of a fully nonlinear, degenerate (sub)elliptic operator which we identify as a (fully nonlinear) perturbation of the fractional Baouendi-Grushin Laplacian. Using its intrinsic geometry and adapted function spaces, we invoke the analytic implicit function theorem to deduce analyticity of…
For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a $C^{1,α}$-manifold, and the lower strata are covered by $C^{1,\log^\varepsilon}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.
In this article we continue our investigation of the thin obstacle problem with variable coefficients which was initiated in \cite{KRS14}, \cite{KRSI}. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove higher order Hölder regularity for the regular free boundary, if the associated coefficients are of the corresponding regularity. For the zero obstacle this yields an improvement of a \emph{full derivative} for the free boundary regularity compared to the regularity of the metric. In the presence of non-zero obstacles or inhomogeneities, we gain \emph{three…
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 study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in $C^{k,α}$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\infty$) and by the second author and Serra (for $k=1$).
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。
We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= fχ_{u\ne 0}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution $u$. Both the regularity and the assumptions are optimal. Using this result and assuming that $f$ is Dini continuous, we prove that the free boundary is, near so called low energy points, a $C^1$ graph. Our result completes the theory for this type of problems for the heat operator.
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.
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.
In this paper we prove the optimal $C^{1,1}(B_\frac12)$-regularity for a general obstacle type problem $$ \lap u = fχ_{\{u\neq 0\}}\textup{in $B_1$}, $$ under the assumption that $f*N$ is $C^{1,1}(B_1)$, where $N$ is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$-regularity. As a by-product of the $C^{1,1}$-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$, the free boundary is locally a $C^1$-graph close to $x^0$, provided $f$ is Dini. This completely settles the que…
Gradient regularity for a class of elliptic obstacle problems
该研究针对带障碍约束的非自治积分泛函局部极小值问题，在偏映射系数满足合适Sobolev正则性的条件下，证明了局部极小值的更高可微性与Lipschitz连续性，推进了椭圆障碍问题的梯度正则性研究。
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 $\varepsilon>0$.
Free boundary partial regularity in the thin obstacle problem
本文研究 $\mathbb{R}^n$ 中薄障碍问题的自由边界正则性，证明除一个 $(n-3)$ 维集合外，解与其 blow-up 之差具有高阶修正，并由此得到余维 3 集合之外的 $C^{1,1}$ 型自由边界正则性。
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 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.
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 …
Lipschitz regularity for the parabolic $(s,p)-$obstacle problem
研究退化范围2 < p < 2/(1 - s)内抛物型分数阶p - 拉普拉斯方程的障碍问题，证明粘性解在空间上局部Lipschitz连续、时间上Hölder连续，p > 1/(1 - s)时时间正则性提升为Lipschitz连续。
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