Optimal regularity for the Signorini problem

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.     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.     

Analytic regularity of a free boundary problem

In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most n − 8.     

Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.Supported by Schweizerischer Nationalfonds     

Maximal space regularity in nonhoomogeneous initial boundary value parabolic problem

Maximal regularity results for second order linear parabolic nonhoomogeneous initial-boundary value problems are established. They are used to show existence, uniqueness and C¹ dependence on the initial value of the solution of general fully nonlinear problems.     

Asymptotic solution to the Signorini problem with small parts of the free boundary

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 2, pp. 258–277, March–April, 1994.     

A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

One-phase parabolic free boundary problem in a convex ring

This paper studies a free boundary problem for the heat equation in a convex ring. It is proved that the considered problem has unique solution under some conditions on the initial data.     

The regularity of the free boundary in a multidimensional filtration problem

We prove the regularity of the free boundary for a filtration problem with capillarity in more than one space dimension. The free boundary is the interface between the saturated region (in which the governing equation is elliptic) and the unsaturated region (where a degenerate parabolic equation is to be solved).This work was partially supported by National Project Equazioni di Evoluzione e Applicazioni Fisico Matematiche (M.U.R.S.T.).     

Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary , arising in the following complementary problem:

Here denotes the parabolic boundary, is a parabolic operator with certain properties, is the upper half of the unit cylinder in , and the equation is satisfied in the viscosity sense. The obstacle is assumed to be continuous (with a certain smoothness at , ), and coincides with the boundary data at time zero. We also discuss applications in financial markets.

     

Partial regularity for a minimum problem with free boundary

Regularity of the free boundary ?{u > 0} of a non-negative minimum u of the functional $\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)} $ , where Ω is an open set in ?ⁿ and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3. By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ?{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets. This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ?{u > 0} is locally in Ω a C^1,α-surface, for n = k* the singular set Σ:= ?{u > 0} ? ?_red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.     

A qualitative study on a parabolic free boundary problem

In memoriam Lothar Collatz     

A two phase stefan problem: regularity of the free boundary

We proved the infinite differentability of the function x=s(t) giving, for all t, the abscissa of the interface plane for a two phase Stefan problem in a plane infinite slab. The proof applies in both cases of temperature or thermal fluxes prescribed on the two limiting planes.

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Riassunto Con riferimento ad un problema di Stefan a due fasi in uno strato piano indefinito, viene dimostrata la infinita differenziabilità della funzione x=s(t) che rappresenta ad ogni istante la ascissa del piano di separazione tra le due fasi. La trattazione è valida sia per il caso in cui si assegni la temperatura sulle facce dello strato, sia per quello in cui venga assegnato il fiusso.

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The research was supported in part by the National Science Foundation contract GP 15724 and the NATO Senior Fellowship program.

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Entrata in Redazione il 14 settembre 1970.     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

On the viscoelastodynamic problem with Signorini boundary conditions

This note focuses on a viscoelastodynamic problem being subject to unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, the problem can be reduced to the pseudo-differential linear complementarity problem through Fourier analysis. We prove that this problem possesses a solution, which is obtained as the limit of a sequence of solutions of penalized problems and we establish that the energy losses are purely viscous. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The regularity of minimizers of a radially symmetric free boundary problem

We treat a variational problem for a functional with a characteristic function term which causes the free boundary, and investigate the regularity of minimizers in the radially symmetric case. The regularity results depend upon the quantity of the coefficient of the term.     

The boundary regularity of non-linear parabolic systems II

This is the second part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we establish higher fractional differentiability of solutions up to the boundary. Based on the necessary and sufficient condition for regular boundary points from the first part of Bögelein et al. (in this issue)[7] we achieve dimension estimates for the boundary singular set and eventually the almost everywhere regularity of solutions at the boundary.