Related papers: Free boundaries subject to topological constraints
The statistical mechanics of flexible two-dimensional surfaces (membranes) appears in a wide variety of physical settings. In this talk we discuss the simplest case of fixed-connectivity surfaces. We first review the current theoretical…
In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity…
We study a class of two-phase inhomogeneous free boundary problems governed by elliptic equations in divergence form. In particular we prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. Our results apply to the classical…
We investigate the regularity of the free boundaries in the 3 elastic membranes problem. We show that the two free boundaries corresponding to the coincidence regions between consecutive membranes are $C^{1,\log}$-hypersurfaces near a…
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…
This treats the base-point-freeness of the adjoint bundles on normal surfaces with a boundary. This is an extension of the non-relative version of the theorem of Ein-Lazarsfeld-Masek and the theorem of Kawachi-Masek.
We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their…
In this paper we investigate a two dimensional free boundary problem involving the A-Laplacian. We show that the free boundary is represented locally by graphs of a family of continuous functions.
We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and…
We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,\alpha}$…
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational…
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…
Let $u$ be a weak solution of the free boundary problem $$\mathcal L u=\lambda_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal L u={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(\xi)$ is a given…
Caffarelli-Hardt-Simon used the minimal surface equation on the Simons cone $C(S^3\times S^3)$ to generate newer examples of minimal hypersurfaces with isolated singularities. Hardt-Simon proved that every area-minimizing quadratic cone…
If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the…
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the…
In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds…
In this paper, we have defined the free boundary formulation for two extended Blasius problems. These problems are of interest in boundary layer theory and are deduced from the governing partial differential equations by using appropriate…
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always…
In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $\Delta u = \lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}, \lambda_\pm>0$. We prove that the free boundary touches the fixed one in…