We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the rate at which the fixed boundary converges to its limit cone at each boundary point.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are assumed on that part of the lateral boundary.
Consider the stationary Boltzmann equation in 2D convex domains with diffusive boundary condition. In this paper, we establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier-Stokes-Fourier system with non-slip boundary conditions. Our contribution focuses on novel weighted $W^{1,\infty}$ estimates for the Milne problem with geometric correction. Also, we develop stronger remainder estimates based on an $L^{2m}-L^{\infty}$ framework.
In a recent paper [7], we were led to consider a distance over a bounded open convex domain. It turns out to be the so-called Thompson metric, which is equivalent to the Hilbert metric. It plays a key role in the analysis of existence and uniqueness of solutions to a class of elliptic boundary-value problems that are singular at the boundary.
The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
Given a mean-convex domain $Ω\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial Ω} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean curvature of $\partial Ω$ and $f$ is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We also show that the …
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke's mean curvature flow with a generalized right angle condition on the boundary of the domain.
We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.
In this paper we prove a strong maximum principle for certain parabolic systems of equations. In particular, our methods place no restriction on the regularity of the boundary of the convex set in which the system takes its values, and therefore our results hold for any convex set. We achieve this through the use of viscosity solutions and their corresponding strong maximum principle.
We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.
We establish the global-wellposedness and stability of the Boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the Maxwellian with or without a small external potential. In particular, we have completely solved the long standing open problem after an announcement of [20] in 1977.
Let $Ω\subset\mathbb{R}^{d}$ be an open set. Given a boundary datum $g$ on $\partialΩ$ and a function $K:\bar Ω \to\mathcal{K}$, the family of all compact convex subsets of $\mathbb{R}^{d}$, we prove the existence of functions $u:Ω\to\mathbb{R}$ such that $u=g$ on $\partialΩ$ and $\nabla u(x)\in K(x)$ a.e. and we investigate the regularity of such solutions on the set $\mathcal{U} \subset \barΩ$ of points at which they all coincide.
In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blowup rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of matrix, we establish the gradient estimates for Lamé systems with partially infinite coefficients and find that they are bounded for some boundary data $φ$ while some $φ$ wi…
The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the regularity of both dynamical and steady Boltzmann equation in strictly convex domain with the Cercignani-Lampis (C-L) boundary condition. The C-L boundary condition describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We construct local weighted $C^1$ dynamical solution using repeated interaction through the characteristic. When we assume small fluctuation to the wall temperature and accommodat…
This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain is convex, or on each of the two components of the boundary when the domain is a convex ring. A function is called quasiconcave if its superlevel sets, defined in a suitable way when the domain is a convex ring, are all convex. In this paper, we prove that the superlevel sets of the solutions do not always inherit the convexity or ring-convexity of the doma…
In this work, we consider the following elliptic partial differential equations: \begin{equation*} \left\{ \begin{aligned}{} - b_{ij} \; \frac{\partial^{2} w}{\partial x_{i} \partial x_{j}} &= g \;\;\; \text{in} \;\; Ω, w &= 0 \;\;\;\text{on} \;\partial Ω, \end{aligned} \right. \end{equation*} \noindent where the domain $Ω\subset \mathbb{R}^{n}$ is convex, the matrix $\big(b_{ij}\big)_{n \times n}$ satisfies the uniform ellipticity conditions. For $g$ in the scaling critical Lorentz space $ L(n,\; 1)(Ω)$, we establish boundary differentiability of solutions to the above problem. We also prove …
We present in this paper a result about existence and convexity of solutions to a free boundary problem of Bernoulli type, with non constant gradient boundary constraint depending on the outer unit normal. In particular we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.
We introduce a notion of convexity with respect to a one-dimensional operator and with this notion find a one-parameter family of different convexities that interpolates between classical convexity and quasiconvexity. We show that, for this interpolation family, the convex envelope of a continuous boundary datum in a strictly convex domain is continuous up to the boundary and is characterized as being the unique viscosity solution to the Dirichlet problem in the domain for a certain fully nonlinear partial differential equation that involves the associated operator. In addition we prove that t…
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} Δu = λϕ(x)\Sum_{i=1}^n H(u-μ_i )& \quad \mbox{ in }\ Ω=Ω_2\setminus \overlineΩ_1, \\[0.3cm]u =0 &\quad \mbox{ on } \partial Ω_2, \\[0.3cm]u =M &\quad \mbox{ on } \partial Ω_1. \end{array} \right. $$ The domain $Ω$ is a convex ring where $Ω_1$ and $Ω_2$ are bounded convex domain in $\mathbb{R}^N,$ $N\geq 2,$ $H$ is the Heaviside step function, $λ, M,μ_i$ $(i=1,..,n)$ are a given positive real parameters and $ϕ$ is a given function . We show that under suitable conditions, there exists a solution to pro…
A basic question about the existence and stability of the Boltzmann equation in general non-convex domain with the specular reflection boundary condition has been widely open. In this paper, we consider cylindrical domains whose cross sections are general non-convex analytic planar domain. We establish the global-wellposedness and asymptotic stability of the Boltzmann equation with the specular reflection boundary condition in such domains. Our method consists of sharp classification of billiard trajectories which bounce infinitely many times or hit the boundary tangentially at some moment, an…
In the perfect conductivity problem of composites, the electric field may become arbitrarily large as $\varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[φ]$ introduced in [29] by picking the boundary data $φ$ of $k$-order growth. In particular, the smoothness of inclusions requi…
We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.
We give a lower bound for the Gaussian curvature of convex level sets of minimal graphs and the solutions to semilinear elliptic equations with the norm of boundary gradient and the Gaussian curvature of the boundary.
In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of this body, a single pair of the temperature and heat flux is given as an observation datum for the inverse problem. It is found the minimum length of broken paths connecting arbitrary fixed point in the outside, a point on the boundary of the cavities and a point on the outside boundary in this order, if the minimum path is not line segment.
A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical $C^{1}$ solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct $W^{1,p}$ solutions for $1< p<2$ and wei…
In the perfect conductivity problem, it is interesting to study whether the electric field can become arbitrarily large or not, in a narrow region between two adjacent perfectly conducting inclusions. In this paper, we show that the relative convexity of two adjacent inclusions plays a key role in the blowup analysis of the electric field and find some new phenomena. By energy method, we prove the boundedness of the gradient of the solution if two adjacent inclusions fail to be locally relatively strictly convex, namely, if the top and bottom boundaries of the narrow region are partially "flat…
We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.
This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexi…
We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry-Mather set associated with the Neumann type boundary problem and establish some properties of the Aubry-Mather set including the existence results for the ``calibrated'' extremals for the corresponding action functional (or variational problem).
We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.
The existence, multiplicity and nonexistence of nontrivial radial convex solutions of a system of two weakly coupled Monge-Ampere equations are established with asymptotic assumptions for an appropriately chosen parameter. The proof of the results is based on Krasnoselskii's fixed point theorem in a cone.
We study the distribution of resonances for smooth strictly convex obstacles under general boundary conditions. We show that under a pinched curvature condition for the boundary of the obstacle, the resonances are separated into cubic bands and the distribution in each bands satisfies Weyl's law.
In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.
We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set contain all minimal disks with boundary on it? Does it contain the solutions of Plateau's problem? We answer this question negatively and characterize the least barrier enclosing all the minimal hypersurfaces with boundary on a given set.
We study the Dirichlet problem for a graph $Σ$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $Γ=\partial Ω$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $Γ$ satisfies $h\geq H$, also suffices when $Ω$ is strictly convex, to prove the strict convexity of $Σ$.
