相关论文: An anisotropic Alt-Caffarelli problem of higher or…
In this paper we prove a $C^{1,\alpha}$ regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As…
For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of…
We examine a variational free boundary problem of Alt-Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior C2-regularity of minimizers and that the free boundary consists of…
We study a natural biharmonic analogue of the classical Alt-Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and $C^{1,\alpha}$-regularity of minimisers. For the Navier…
We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity…
The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the…
In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth…
We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric…
We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N$,…
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second…
This paper aims at solving an optimal control problem governed by an anisotropic Allen-Cahn equation numerically. Therefore we first prove the Fr\'echet differentiability of an in time discretized parabolic control problem under certain…
In this paper we introduce a new type of symmetrization, which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function, in order to prove sharp comparison results for solutions of a class of homogeneous…
In this paper we prove a classification result for axially symmetric one phase minimizers of the Alt-Phillips free boundary problem in dimensions 3, 4, and 5. To accomplish this, we establish a stability inequality that extends the one for…
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet…
In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the…
We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we…
We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…
We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by…
A free boundary problem arising from materials science is studied in one-dimensional case. The problem studied here is an obstacle problem for the non-convex energy consisting of a bending energy, tension and an adhesion energy. If the…
This paper considers the Alt-Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the…