Radially Symmetric Functions as Fixed Points of some Logarithmic Operators

In this note we show that for f C((0,); R₊) C¹ ((0,)) with support in [0,), if a function u C¹(R²) is such that support (u⁺) is compact and u(x) = R² f(u(y)) log 1/(|x-y|)dy x, then u is radial. This result is important for some free boundary problems in R² or some axisymmetric ones in Rⁿ.     

A free boundary problem for quasilinear degenerate parabolic equations with general degeneracy

In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)–(1.4). The existence of generalized solutions inBV ₁, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C¹⁺ smoothness for the free boundary is obtained in the parabolic case.     

A singular perturbation free boundary problem for elliptic equations in divergence form

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem , where A = A(x) is Holder continuous, β _ɛ converges to the Dirac delta δ₀. By studying some suitable level sets of u _ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C ^1,γ surface around a.e. point on the free boundary.     

A priori estimates for minimal surfaces with free boundary,which are not minima of the area

We estimate the length of the free boundary, the Dirichlet integral and theC ^k, -norms of a stationary minimal surface whose free boundary lies on a given supporting surface.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. II

In this paper we extend the results of the first one to solutions of some obstacle problem in the semilinear elliptic case that are used as a model for a gas problem. More precisely, we prove that the points of the free boundary, where the zero set has no density, lie in a Lipschitz surface. Furthermore, we get the C ¹ regularity for singular points with some (n ? 1)-density.

We also investigate the free boundary at points with density. We show that the set of these points is locally a C ¹ surface. This result is an extension of those achieved by Alt and Phillips [3] Alt, H. W. and Phillips, D. 1986. A free boundary problem for semilinearelliptic equations. J. Reine Angew. Math., 368: 63–107.  [Google Scholar], where it is used a concept stronger than the “density” applied here.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C¹(∂Ω) boundary data û. We show here that the same result holds for ûC^0,μ(∂Ω) if and it cannot be extended to cover the case . The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.     

A One-Dimensional Parabolic Problem Arising in Studies of Some Free Boundary Problems

The paper is concerned with a one-dimensional parabolic problem in a domain bounded by two lines x = 0 and x = kt, k > 0, (x, t) ², with the Neumann boundary condition on the line x = 0 and with dynamic boundary condition on the line x = kt. For the solution of this problem, a coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems. Bibliography: 7 titles.     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free Boundary Regularity in the Parabolic Fractional Obstacle Problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when , the free boundary is a C^1,α graph in x and t near any regular free boundary point . Furthermore, we also prove that solutions u are C^{1 + s} in x and t near such points, with a precise expansion of the form (1) with , and . © 2018 Wiley Periodicals, Inc.     

Regularity for Shape Optimizers: The Nondegenerate Case

We consider minimizers of (1) where F is a function strictly increasing in each parameter, and is the k^th Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C^1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter ?², where ? takes arbitrary values in the half-open interval (0, 1]. When ? → 0, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in the radial and time variables, respectively.     

The spectrum of the Leray transform for convex Reinhardt domains in

The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in . Our class is self-dual; it contains some domains with less than C²-smooth boundary and also some domains with smooth boundary and degenerate Levi form. L²-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.     

A two phase elliptic singular perturbation problem with a forcing term

We study the following two phase elliptic singular perturbation problem:

Δu^ε=β_ε(u^ε)+f^ε,