Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, Tung To
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.
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We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its Hölder continuity at regular free boundary points.
We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the free boundary can be reduced to the study of the free boundary in the parabolic Signorini problem. We show that locally there is always a sign on the boundary; therefore, the optimal regularity of the solutions as well as the study of the free boundary is reduced to the one phase parabolic Signorini problem.
Optimal regularity for the variable coefficients parabolic Signorini problem
本文针对变系数抛物型Signorini问题，利用Carleman估计和epiperimetric不等式，证明了在特定系数和非齐次项条件下解及正则自由边界的最优正则性。
In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the "vanishing order" of a solution at a free boundary point is close to $3/2$ or an even integer, then the solution is asymptotically homogeneous. Furthermore, one can derive a convergence rate estimate towards the asymptotic homogeneous solution. As a consequence, we obtain the regularity of the regular free boundary as well as the frequency gap.
In this note we discuss the (higher) regularity properties of the Signorini problem for the homogeneous, isotropic Lamé system. Relying on an observation by Schumann \cite{Schumann1}, we reduce the question of the solution's and the free boundary regularity for the homogeneous, isotropic Lamé system to the corresponding regularity properties of the obstacle problem for the half-Laplacian.
Free boundary regularity for almost minimizers of the parabolic Signorini problem
本文针对抛物型Signorini问题中的几乎极小化子，通过建立Weiss型单调性公式和Almgren型频率公式，结合epiperimetric不等式，证明了正则自由边界点的优化增长及正则集的正则性。
We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity $C^{1,1/2}$. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary $C^{1,β}$ hypersurface, $β>1/2$, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren's monotonicity formula and the optimal regularity of global solutions.
In this paper we establish the $C^{1,β}$ regularity of the regular part of the free boundary in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is a continuation of the recent paper [GSVG14], where two of us established the interior optimal regularity of the solution. Two of the central results of the present work are a new monotonicity formula and a new epiperimetric inequality.
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。
We show that the quotient of two caloric functions which vanish on a portion of an $H^{k+ α}$ regular slit is $H^{k+ α}$ at the slit, for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+α}$ if the slit is assumed to be space-time $C^{1, α}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DSS14a], whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in [DGPT] with zero obstacle is $C^{\infty}$ regular in space and time.
In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph-Legendre transformation we show that the regular part of the free boundary is real analytic. The first complication in the study is the invertibility of the hodograph transform (which is only $C^{0,1/2}$) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The second and main complication in the study is that the equation satisfied by the Legendre transform is degenerate. However, the equation has a …
We investigate the regularity of the free boundary for the Signorini problem in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^\infty$. However, even for $C^\infty$ obstacles $φ$, the set of non-regular (or degenerate) points could be very large, e.g. with infinite $\mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when $φ$ is analytic, and when $Δφ< 0$. However, even in these cases, the set of degenerate points is in general $(n-1)$-dimensional (as large as the set of reg…
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.
We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3)$ minimizes $$ J(\u)=\int_{B_1^+}|\nabla \u+\nabla^\bot \u|^2+λ÷(\u)^2 $$ in the convex set $$ K=\big\{\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3);\; u^3\ge 0 \textrm{on}Π, $$ $$ \u=f\in C^\infty(\partial B_1) \textrm{on}(\partial B_1)^+ \big\}, $$ where $λ\ge 0$ say. Then $\u\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by $$ Γ_\u=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $$ will be a $C^{1,α}$ graph close to points where $\u$ is …
We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,α}$ estimates on each side of the smooth obstacle, for some small $α> 0$. Our results extend those of Milakis-Silvestre in two ways: first, we do not assume solutions nor operators to be symmetric, and second, our estimates are local, in the sense that do not rely on the boundary data. As a consequence, we prove $C^{1,α}$ regularity even when the problem is posed in general Lipschitz domains.
The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general $r_i^{π/(2α_i)}$ at transition points of Signorini to Dirichlet or Neumann conditions but $r_i^{π/α_i}$ at kinks of the Signorini boundary, with $α_i$ being the internal angle of the domain at these critical points.
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 consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.
The $C^{1,α}$ boundary Harnack principle in a slit domain and its application to the Signorini problem
本文研究了具有变系数的Signorini问题自由边界$C^{2,α}$正则性，利用裂隙域中的$C^{1,α}$边界Harnack不等式，通过非标准退化椭圆方程的研究获得$C^{1,α}$Schauder估计。
We study the Signorini problem near a fixed boundary, where the solution is "clamped down" or "glued." We show that in general the solutions are at least $C^{1/2}$ regular and that this regularity is sharp. We prove that near the actual points of contact of the free boundary with the fixed one the blowup solutions must have homogeneity $κ\geq 3/2$, while at the non-contact points the homogeneity must take one of the values: $1/2, 3/2, \ldots, m-1/2, \ldots$.
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.
We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs $E=\{(x,t): x_{n-1}\leq f(x'',t),x_n=0\}\subset \mathbb{R}^{n-1}\times\mathbb{R} $ for parabolically Lipschitz functions $f$ on $\mathbb{R}^{n-2}\times\mathbb{R}$. We are motivated by applications to parabolic free boundary problems with thin (i.e co-dimension two) free boundaries. In particular, at the end of the paper we show how to prove the spatial $C^{1,α}$ regularity of the free boundary in the parabolic …
We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the $C^\infty$ regularity of the free boundary in the Signorini problem near non-degenerate points.
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory with some proofs: from the optimal regularity of solutions and classification of free boundary points, to more recent results on the non-regular part of the free boundary and generic regularity.
Sensitivity analysis of a Signorini-type history-dependent variational inequality
本文研究描述粘弹性体与覆盖软材料层的刚性障碍物无摩擦接触的Signorini型历史依赖变分不等式，通过将其等价转化为不动点问题，证明解算子的方向可微性并给出方向导数的变分不等式刻画，同时建立相关最优控制问题存在性与两类适定性结果。
Parabolic boundary Harnack inequalities with right-hand side
本文用爆破技术建立允许非零右端项的抛物型边界 Harnack 不等式，并据此推出抛物障碍与 Signorini 问题中平坦 Lipschitz 自由边界的 $C^{1,\alpha}$ 正则性。
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.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
This paper studies Hölder regularity property of bounded weak solutions to a class of strongly coupled degenerate parabolic systems.
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.
In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<σ<2. And we show a Harnack inequality, Hölder regularity, and C^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets.
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.
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 give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent.
This note presents a regularity result with proof for an initial-boundary value problem of a linear parabolic system involving curl of the unknown vector field, subjected to the boundary condition of prescribing the tangential component of the solution.
We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
We establish Hölder estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,α}$ estimates.
We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be $C^{2,1}$.
Higher Hölder regularity for nonlocal parabolic equations with irregular kernels
本文研究了具有不规则核系数的非局部抛物型方程，在对非齐次项施加适当高阶可积性且对核系数作最小正则性假设的条件下，建立了该方程的高阶Hölder正则性。
We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a (strictly positive) finite time.
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,γ}$ 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 epiperim…
Regularity theory of a gradient degenerate Neumann problem
本文研究由均匀化中奇异各向异性自由边界问题引出的临界梯度退化 Neumann 问题，证明二维最优 $C^{1,1/2}$ 正则性、高维在退化值不聚集条件下的同类结果，并建立刻画最小上解的比较原理。
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
Here we describe a simple and fundamental approach to the maximal L^p regularity of parabolic equations, which only uses the concept of singular integrals of Volterra type. Knowledge of analytic semigroups, R-boundedness or H^\infty-functional calculus are not required.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form $u_t-Iu=f$, where the operator $I$ is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain $C^{1,α}$ interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.
We establish Hölder estimates for the time derivative of solutions of non-local parabolic equations under mild assumptions for the boundary data. As a consequence we are able to extend the Evans-Krylov estimate for rough kernels to parabolic equations.
Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems
本文针对非光滑域上带混合边界条件的非自治拟线性抛物系统，建立了$L^p$-极大正则性，通过外推技术获得一致估计，仅需温和边界正则性及有界可测复系数，并允许时间非局部非线性，涵盖稳态不满足椭圆性的情形。
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.
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.
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.
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.
Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations
该文研究一致退化抛物方程解的全局Hölder正则性，并在极限一致退化椭圆方程特征指数的附加假设下，分析时间趋于无穷时解的收敛性。
In this paper, we study the second order Sobolev regularity of solutions to the parabolic $p$-Laplace equation. For any $p$-parabolic function $u$, we show that $D(|Du|^{\frac{p-2+s}{2}}Du)$ exists as a function and belongs to $L^{2}_{\text{loc}}$ with $s>-1$ and $1<p<\infty$. The range of $s$ is sharp.
This paper deals with a class of nonlinear anisotropic parabolic equations with degenerate coercivity. Using the anisotropic Gagliardo-Nirenberg-type inequality, we prove some existence and regularity results for the solutions under the framework of anisotropic Sobolev spaces, which generalize the previous results of [10,18,23].
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
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