Hector Chang-Lara, Ovidiu Savin
We consider the Bernoulli one-phase free boundary problem in a domain $Ω$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial Ω$. We achieve this by relating the behavior of $F$ near $\partial Ω$ to a Signorini-type obstacle problem.
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In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.
In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
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。
We prove Wolf's regularity condition up to the boundary for solutions to the Navier-Stokes equations satisfying non-slip boundary condition.
We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.
We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to 1/2 + N, as it involves nonlocal compatibility conditions.
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.
Boundary regularity for non-local operators with symmetric kernels and vanishing horizon
本文研究了具有奇异对称核且依赖于到域边界距离的非局部算子，证明了解的最优 Hölder 边界正则性及更高边界正则性。
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.
We prove the partial regularity of the boundary suitable weak solutions to the MHD system near the plane part of the boundary.
Boundary regularity for the Mumford-Shah functional with Dirichlet boundary conditions
本文研究带 Dirichlet 边界条件的 Mumford-Shah 泛函极小元，通过分析边界爆破证明了 epsilon-正则性定理。
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.
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.
Boundary characteristic point regularity is studied for a class of semilinear heat equations and an ODE criterion of regularity is obtained. Extensions to higher-order semilinear parabolic problems are discussed.
We prove local regularity up to flat part of boundary, for certain classes of distributional solutions that are $L_{\infty}L^{3,q}$ with $q$ finite.
We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.
Boundary partial regularity for a class of Dirac-harmonic maps
本文研究一类满足特定能量单调不等式的耦合 Dirac-harmonic 映射，证明其在边界附近具有部分正则性。
We obtain up to a flat boundary regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.
We study geometric and regularity properties of the largest subsolution of a one-phase free boundary problem under a very general free boundary condition in R2. Moreover, we provide density bounds for the positivity set and its complement near the free boundary.
F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd dimension case and study how the higher regularity is obstructed. We introduce the logarithm of the distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.
In a bounded domain, we consider a variable range nonlocal operator, which is maximally isotropic in the sense that its radius of interaction equals the distance to the boundary. We establish $C^{1,α}$ boundary regularity and existence results for the Dirichlet problem.
We study the boundary regularity for the normalised $\infty$-heat equation $u_t = Δ_{\infty}^Nu$ in arbitrary domains. Perron's Method is used for constructing solutions. We characterize regular boundary points with barrier functions, and prove an Exterior Ball result. A Petrovsky-like criterion is established.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
We consider the Hölder continuity for the Dirichlet problem at the boundary. Almgren introduced the multivalued; Q-valued functions for studying regularity of minimal surfaces in higher codimension. The Hölder continuity in the interior for Dirichlet minimizers is an outcome of Almgren's original theory, to which C. De Lellis and E.N. Spadaro's work have given a simpler alternative approach. We extend the Hölder regularity for Dirichlet minimizing Q-valued functions up to the boundary assuming enough regularity of the domain and Hölder regularity of the boundary data with exponent >1/2.
The embeddings theorem of space-boundary-type DeGiorgi-Ladyzhenskaya parabolic classes into Holder spaces is presented, which is useful for regularity considerations for parabolic boundary value problems. Additionaly, the application of this theory to Navier-Stokes-s swirl is presented.
We show that the solutions of SG elliptic boundary value problems defined on the complement of compact sets or on the half-space have some regularity in Gelfand-Shilov spaces. The results are obtained using classical results about Gevrey regularity of elliptic boundary value problems and Calderón projectors techniques adapted to the SG case. Recent developments about Gelfand-Shilov regularity of SG pseudo-differential operators on $\mathbb{R}^{n}$ appear in an essential way.
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.
We prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully nonlinear operators.
We show that the quotient of two positive harmonic functions vanishing on the boundary of a $C^{k,α}$ domain is of class $C^{k,α}$ up to the boundary.
Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations
本文综述了有界区域中一致退化椭圆方程解的最优正则性，建立了解及其导数直至边界的Hölder连续性。
The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies a non-degeneracy condition. This is the optimal regularity for this problem.
In this paper we show boundary monotonicity formulae for rectifiable varifolds having a $C^{1,α}$ "boundary". In particular, we show that the area ratios of balls centered at this "boundary'' satisfy a nice monotonicity formula, similar to that for interior balls (proved in Allard's paper "On the first variation of a varifold''). This extends the boundary monotonicity formulae of Allard (see "On the first variation of a varifold- boundary behavior''), which require that the boundary is $C^{1,1}$. As a corollary, Allard's boundary regularity results extend to this case and provide a regularity …
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,α}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.
We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any $L^p$ with $1<p$ locally near boundary. On the other hand, we present criteria of solutions of the Stokes equations near boundary to imply that the gradients of solutions are bounded (in fact, even further Hölder continuous). Finally, we provide examples of solutions whose local regularity near boundary is optimal.
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.
We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or $H$-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Riviere, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisym…
We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are $C^{1,γ}$. In particular, viscosity solutions are indeed classical.
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.
We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.
We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.
In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
In this note we consider the initial boundary value problem for the heat equation on cylinders based on Lipschitz domains with Besov data. We obtain a regularity exponent for the solution that improves the rate of convergence of nonlinear approximation methods.
We investigate the regularity in $L^p$ ($p>2$) of the gradient of any weak solution of a Cauchy problem with mixed Neumann-power type boundary conditions. Under suitable assumptions we prove the existence of weak solutions that satisfy explicit estimates. Some considerations on the steady-state regularity are discussed.
We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show the $C^{1,α}$ regularity of the free boundary, when the solution is directionally monotone in space variable in a local neighborhood. The main challenge lies in establishing a local non-degeneracy estimate (Theorem 1.3 and Proposition 1.5), which appears new even for the zero drift case.
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.
We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius.
Existence and regularity in the fully nonlinear one-phase free boundary problem
本文研究一般完全非线性算子及法向量依赖自由边界条件下单相问题的粘性解，用Perron方法证明存在性，通过平坦度二次改进和高度图变换分别得到$C^{2,α}$及更高正则性。
This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ($C^{3, a_0}$ for a positive $a_0$ suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general we…
We establish several boundary $\varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the Dirichlet boundary condition on the flat boundary. Our proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur [18], Choi-Vasseur [2], and Phuc-Guevara [6].
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian;…
We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1<p<2$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points.
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…
In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.
Maximal $L^p$-$L^q$ regularity is proved for the strong, weak and very weak solutions of the inhomogeneous Stokes problem with Navier-type boundary conditions in a bounded domain $Ω$, not necessarily simply connected. This extends previous results of the authors (2017).
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