Thin Obstacle Problem Research Articles

The parabolic thin obstacle problem for the weighted biLaplacian

The parabolic thin obstacle problem for the weighted biLaplacian

Free boundary regularity in the fully nonlinear parabolic thin obstacle problem

Abstract We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 C^{1} graph in x near any regular free boundary point.

Optimal regularity for the fully nonlinear thin obstacle problem

In this work, we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent \alpha_{F} such that u is C^{1,\alpha_F} on either side of the obstacle. In order to do that, we prove the uniqueness of blow-ups at regular points, as well as an expansion for the solution there. Finally, we also prove that if the operator is rotationally invariant, then \alpha_{F} \ge \frac{1}{2} and the solution is always C^{1,1/2} .

Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle, and establish their C^{1,\beta} regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.

Optimal regularity of the thin obstacle problem by an epiperimetric inequality

Optimal regularity of the thin obstacle problem by an epiperimetric inequality

Generic regularity of free boundaries for the thin obstacle problem

The free boundary for the Signorini problem in Rn+1 is smooth outside of a degenerate set, which can have the same dimension (n−1) as the free boundary itself.In [15] it was shown that generically, the set where the free boundary is not smooth is at most (n−2)-dimensional. Our main result establishes that, in fact, the degenerate set has zero Hn−3−α0 measure for a generic solution. As a by-product, we obtain that, for n+1≤4, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in R3 and R4 for the thin obstacle problem.

Regularity of almost minimizers for the parabolic thin obstacle problem

In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their Hσ,σ/2-regularity for every 0<σ<1, as well as Hβ,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.

Free boundary partial regularity in the thin obstacle problem

AbstractFor the thin obstacle problem in , , we prove that at all free boundary points, with the exception of a ‐dimensional set, the solution differs from its blow‐up by higher order corrections. This expansion entails a C1, 1‐type free boundary regularity result, up to a codimension 3 set.

Contact points with integer frequencies in the thin obstacle problem

AbstractFor the thin obstacle problem, we develop a unified approach that leads to rates of convergence to blow‐up profiles at contact points with integer frequencies. For these points, we also obtain a stratification result.

Half-Space Solutions with 7/2 Frequency in the Thin Obstacle Problem

For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.

The thin obstacle problem for some variable coefficient degenerate elliptic operators

In this paper, we establish the optimal interior regularity and the C1,γ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.

H1+α estimates for the fully nonlinear parabolic thin obstacle problem

We study the regularity of the viscosity solution to the fully nonlinear parabolic thin obstacle problem. In particular, we prove that the solution is local H1+α on each side of the smooth obstacle for some small α>0. Following the method which was first introduced for the harmonic case by Caffarelli in 1979, we extend the results of Fernández-Real (2016) who treated the fully nonlinear elliptic case. Our results also extend those of Chatzigeorgiou (2019) in two ways. First, we do not assume solutions nor operators to be symmetric. Second, our estimates are local, in the sense that do not rely on the boundary data.

On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem

In this work, we consider the singular set in the thin obstacle problem with weight $|x_{n+1}|^a$ for $a\in (-1, 1)$, which arises as the local extension of the obstacle problem for the fractional Laplacian (a non-local problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single $C^2$ manifold, up to a lower dimensional subset, and the top stratum is locally contained in a $C^{1,\alpha}$ manifold for some $\alpha > 0$ if $a < 0$. In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension two lower dimensional obstacle problem (or fractional thin obstacle problem) when $a < 0$, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and $a$-harmonic polynomials when $a \geq 0$. To do so, we establish regularity results for this codimension two problem, what we call the very thin obstacle problem. Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many $C^2$ manifolds, up to a lower dimensional subset.

Almost minimizers for the thin obstacle problem

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their \(C^{1,\beta }\) regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the \(C^{1,\gamma }\) regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetric inequalities for the solutions of the thin obstacle problem.

Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian

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≥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.

The regular free boundary in the thin obstacle problem for degenerate parabolic equations

This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the H 1 + γ , 1 + γ 2 H^{1+\gamma ,\frac {1+\gamma }{2}} regularity of the free boundary near such regular points.

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight |y|^a for a in (-1,1). Such problem arises as the local extension of the obstacle problem for the fractional heat operator (partial _t - Delta _x)^s for s in (0,1). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation (a=0).

New boundary Harnack inequalities with right hand side

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in Lq with q>n. Our approach is based on the scaling and comparison arguments of [13], and we show that all our assumptions are sharp.As a consequence of our results, we deduce the C1,α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

Regularity for the fully nonlinear parabolic thin obstacle problem

We prove [Formula: see text] 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.

How a minimal surface leaves a thin obstacle

We prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.